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Application of Single Party and Multiple Party Decision Making under Risk and Uncertainty to Water Resources Allocation...

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APPLICATION OF SINGLE PARTY AN D MULTIPLE PARTY DECISION MAKING UNDER RISK AND UNCERTAINTY TO WATER RESOURCES ALLOCATION PROBLEMS By GHINA M. YAMOUT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Ghina Yamout

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This document is dedicated to my family; my mother, father, sister, brothers, and husband, I thank you.

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iv ACKNOWLEDGMENTS At last! The “there” is “here” I started school at the age of three, finished at the age of ( ), let’s just say by the year 2005! Twen ty-four years of schooling (do your math)! As people set high standards for me, I set even higher ones for myself. You start something, you finish it, completely and perfectly; you star t school, you finish it: kindergarten / PhD! It might not make sense to many; for me, ther e was no other way! Goal s, trivial ones and less trivial ones, all but small steps towards an ultimate aspiration, that, once seeming so far away, now is unbelievably close. I look back at this journey of self-fulfillment, and I realize that everythi ng happens for a reason. I thank God for what seemed to be “a dverse” circumstances at the time and for what still does. God grant me the bl essing of being a good servant of His. This long longed for “end” is but a begi nning; it is called commencement for a reason! At the onset of my “new” life, I w ould like to express my utmost gratitude to many, whose guidance and comfort had a hand in getting me where I am today. Firstly, I dedicate this work to my mother and father; their self-sacrifice was my drive. My achievement is theirs. Throughout my life, they made me recognize that when there is a will, there is always a way. It is with this will that I stand here today. I would like to express my d eepest gratitude to my advisor, Professor Kirk Hatfield, whose character and energy added considerably to my graduate experience. I thank him for his unyielding understanding, patience, and fait h in me when I was at my best and my worst. It has been and will always be a pr ivilege. I doubt that I will ever be able to

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v convey my appreciation fully, but I owe hi m my life long gratitude, for providing me with the opportunity to be part of an excepti onal university, the Univer sity of Florida, in this exceptional country. An advice for new doctoral candidates: listen to the advisor! When he says stop taking courses, stop! Ther e is always a “one more” course that needs to be taken! I would like to thank the other member s of my committee: Professor Edwin Romeijn, for his unyielding patience and indispensable tutoring which added considerably to the value of this work ; and Dr. Clayton Clark, Professor Warren Viessman, and Professor Scot Smith for their valuable comments. I owe my most loving th anks to the man whose support, encouragement, and persistence were behind the completion of this work, my husband, Husam Jumaa. A woman is so lucky to meet her better half as young as I was, as his touch can bring back the starlight and glow of years ago, for me, the first day I met this amazing man, when I was still twenty-two; with him, I have and will grow older and build memories, but I will always be 22. His high expectations from me and total devotion to me only better me, to become, one day, as amazing as he is. When people think that their prayers ar e not being answered, they should look around them, they might not be seeing clea rly. Thank God everyday for his blessings, because she is a blessing; no lesser word can describe her. Zeina Najjar’s unconditional love and guidance, so innate to her, at my worst and my best, are my sunshine in the happiest and the darkest moments. I will not le t words define what she means to me and will use Martin Luther King’s words as he said “Occasionally in life there are those moments of unutterable fulfillment which cannot be completely explained by those

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vi symbols called words. Their meanings can onl y be articulated by th e inaudible language of the heart.” I would also like to lovingly recognize Amal and Wafic Dabbous, my youngest sister and brother, whose genuineness and affection had the deepest effect on my heart and mind. I would also like to express my most affectionate gratit ude to my sister, Aya, and brothers, Abdel-Ghani and Mohammad. I hope I have and will always be there for them; I hope they forgive me if at any time I have n’t. The responsibility of being their older sister and the care I have for them were my biggest motivations. I thank my dearest sister, Aya, for being the older sister at every step of the way. I thank her for opening to me a world of possibilities, for loving me at my best and my worst, for knowing when I am smiling even in the dark, for being my teacher my attorney, my stylist, even my shrink. I would also like to extend my loving a ppreciation to my mother, father, and brother in laws, Abla, Wafi c, and Mohammad Luay Jumaa, for their watching eyes, indispensable prayers, and much needed a dvice which helped alleviate the heaviest obstacles. It is amazing how much comfort th e mere realization of the presence of a caring and supporting hand brings to you. I would like to extend my warmest gratitude to three very dear people to my heart: my uncle, Mohammad El-Wali, who answered my calls in the latest, or shall I say earliest, hours; my gentle aunt, Hoda Yam out Kandil, whose rare warm and gracious nature is a model very hard to follow; a nd my dear departed uncle, Talal Yamout, who saluted me with the word “Dr.” befo re I even thought of becoming one. I would like on this oc casion to extend my profound gr atitude to another very dear person to my heart, Mrs. Leila Knio, whose admirable graciousness will be a model for

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vii me throughout my life. I thank her for giving me the privilege of being part of my life; rare are the people who mark one’s life with unconditional guardianship and love. I would also like to express my warmest th anks to all my friends from the Water Resources and Hydrology group, Nebiyu Tirune h, Sudheer Satti, Anirudha Guha, Ali Sedighi, Mark Neuman, Nanette Conrey, Qi ng Suny, and Sherish Bhat, with whom I shared many laughs, debates, exchange of knowledge, and venting of frustration, making my stay at UF unforgettable. I pray we all stay in touch. I would also like to extend my gratitude to Ms. Sonja Lee, Doretha Ray, and Carol Hipsley, who answered, over and over, ev ery question and concern I had; their welcoming faces and ready assistance had an immense part in easing my transition as I came to the United States and my experien ce through my time at the University of Florida. Thanks also go out to Mr. Anthony Mu rphy for all of his computer and technical assistance throughout my gradua te program. Finally, this work would not have been submitted on time without the tremendous assi stance from the editorial office at UF. I also recognize that this research w ould not have been possible without the financial assistance of the Ci vil Engineering Alumni Fellows hip at the University of Florida. I must also acknowledge the Saint Johns River Water Management District for providing me with the data used in part of this study. In particular, I would like to take the opportunity to extend my gratitude to Mr. Ron Wycoff, private consultant for the district, for generously giving me of his time and answering all my concerns. For all your guidance, I wish to express my sincerest appreciation. If I forgot somebody please forgive me, I thank you as well.

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viii TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................xii LIST OF FIGURES.........................................................................................................xiv ABSTRACT....................................................................................................................xvi i CHAPTER 1 INTRODUCTION........................................................................................................1 2 DECISION MAKING UNDER UNCERTAI NTY: A COMPARATIVE REVIEW OF METHODS AND APPLICATIO NS TO WATER RESOURCES MANAGEMENT.........................................................................................................5 On the Origin of Risk...................................................................................................6 Definition of Risk.........................................................................................................8 Definition of Risk Management.................................................................................13 Our Definition.............................................................................................................16 Risk Management Techniques....................................................................................17 Mathematical Notations.......................................................................................19 Non-Probability-Based RM Techniques.............................................................20 Sensitivity analysis.......................................................................................20 Decision making criteria..............................................................................20 Analytic hierarchy proce ss or decision matrix.............................................21 Utility and game theory................................................................................21 Multiobjective optimization.........................................................................22 Probability-Based RM Techniques......................................................................22 Scenario analysis..........................................................................................22 Moments and quantiles.................................................................................23 Decision trees...............................................................................................23 Stochastic optimization................................................................................24 Bayesian analysis.........................................................................................25 Fuzzy sets.....................................................................................................25 Information gap............................................................................................26 Downside risk metrics..................................................................................26 The Different RM Methods: A Discussion.................................................................27

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ix Value-at-Risk and Conditional Value-at-Risk............................................................30 Scenario Tree.......................................................................................................34 Discretization.......................................................................................................37 Risk in the Water Resources Management Literature................................................41 Conclusion..................................................................................................................42 3 COMPARISON OF RISK MANAGE MENT TECHNIQUESFOR A WATER ALLOCATION PROBLEM WITH UNCERTA IN SUPPLIES A CASE STUDY: THE SAINT JOHNS RIVER WA TER MANAGEMENT DISTRICT.....................46 Model Formulation.....................................................................................................47 Objective Function..............................................................................................48 Decision Variables...............................................................................................49 Problem Data.......................................................................................................49 Constraints...........................................................................................................49 Deterministic Expected Value Model.................................................................50 Scenario Model....................................................................................................51 Two-Stage Stochastic Model with Recourse.......................................................52 CVaR Objective Function Model......................................................................53 CVaR Constraint Model....................................................................................54 Scenario Generation....................................................................................................55 Case Study Area.........................................................................................................59 Water Demand.....................................................................................................59 Water Supply.......................................................................................................65 Water Cost...........................................................................................................67 Scenario Generation....................................................................................................68 Results and Discussion...............................................................................................72 5% Standard Deviation........................................................................................73 10% Standard Deviation......................................................................................85 Analysis...............................................................................................................96 Conclusions and recommendations............................................................................98 4 UTILITY, GAME, AND WATER: A REVIEW.....................................................101 Theory of Preference................................................................................................103 Preference Comparison Relationship................................................................104 Expected Utility Theory....................................................................................105 Bernoulli’s utility theory............................................................................105 Linear expected utility theory.....................................................................107 Subjective linear expected utility theory....................................................111 Multiattribute expected utility....................................................................112 Descriptive Limitations of LEUT and SEUT....................................................113 Violation of independence.........................................................................114 Violation of transitivity..............................................................................116 Probability judgment..................................................................................120 Non-Archimedean preferences...................................................................121

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x Alternatives to Expected Utility Theory............................................................121 Linear generalizations................................................................................122 Non-linear generalizations.........................................................................122 Strategic Decision Making.......................................................................................131 Game Theory.....................................................................................................131 Mathematical formulations........................................................................132 Classification of games..............................................................................135 Solutions concepts......................................................................................137 Extensions to Standard Game Theory...............................................................145 Metagame analysis.....................................................................................146 Hypergame theory......................................................................................147 Analysis of options.....................................................................................147 Conflict analysis.........................................................................................148 Drama theory..............................................................................................148 Graph model for conflict resolution...........................................................148 Theory of moves.........................................................................................149 Alternatives to Standard Game Theory.............................................................149 Limited thinking models............................................................................150 Learning models.........................................................................................150 Social preferences models..........................................................................151 Game Theory in Water Resources Management......................................................152 Conclusion................................................................................................................155 5 A MULTIAGENT MULTIATTRIBU TE WATER ALLOCATION GAME MODEL....................................................................................................................157 Power and Preferences..............................................................................................158 Salience Theories......................................................................................................160 Issue Linkage Theories.............................................................................................162 Equity Theories.........................................................................................................162 Coalition Formation Theories...................................................................................163 Minimum Resource Theory...............................................................................163 Balance Theory..................................................................................................164 Minimum Power Theory...................................................................................164 Bargaining Theory.............................................................................................165 Equal Surplus Theory........................................................................................165 Policy-Distance Minimization Theory..............................................................165 Outcome Grouping Theory................................................................................166 Option Preferences Theory................................................................................167 Ordinal Deduction Selection System Theory....................................................167 Graph Model Theory.........................................................................................168 Triads Theory....................................................................................................168 Probability of Coalition Formation...........................................................................168 Size-Probability Model Theory.........................................................................169 Johansen-C Probability Model Theory..............................................................169 Central Union Theory........................................................................................170 Willingness and Oppor tunity Theory................................................................170

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xi Cohesion Theory...............................................................................................171 Stochastic Communication Structures...............................................................171 Risk Attitudes...........................................................................................................171 Pratt-Arrow Model............................................................................................172 Risk Aversion Matrix........................................................................................173 Risk-Value Theory............................................................................................174 Moments Risk-Value Model.............................................................................175 De Mesquita’s Risk Model................................................................................175 Model Development.................................................................................................176 Utility Function.................................................................................................181 Relative Gain.....................................................................................................184 Coalition Formation...........................................................................................184 Political Uncertainty..........................................................................................186 Modified Utility Function..................................................................................187 Hypothetical Application..........................................................................................187 Conclusion................................................................................................................200 6 CONCLUSION.........................................................................................................202 APPENDIX SJRWMD COSTS DEPRECIATION.......................................................207 LIST OF REFERENCES.................................................................................................217 BIOGRAPHICAL SKETCH...........................................................................................266

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xii LIST OF TABLES Table page 1-1 Possible decision making environment combinations...............................................2 2-1 The use of DMuU techniques in water resources management...............................44 3-1 Standard normal discrete distribu tion approximation for N=10 and up to the 2nd, 4th, 6th, and 8th moments constraints.........................................................................60 3-2 Moments constraints................................................................................................60 3-3 Least square regres sion analysis results...................................................................61 3-4 SJWRMD caution area water demand projections..................................................65 3-5 Supply sources, capacities, and costs.......................................................................69 3-6 Scenarios of supply capacities at 5% and 10% standard deviation..........................71 4-1 Choice model classification...................................................................................102 4-2 Applications of GT in water res ources management grouped into area of application..............................................................................................................156 5-1 Pattern of risk attitudes...........................................................................................172 5-2 Definition of model components............................................................................179 5-3 Player description...................................................................................................188 5-4 Input data................................................................................................................1 88 5-5 Players ranking matrix...........................................................................................194 5-6 Coalitions...............................................................................................................19 4 5-7 Coalition combination scenarios............................................................................194 5-8 Measuring the probability of fo rmation of different coalitions..............................196 5-9 Input data................................................................................................................1 96

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xiii 5-10 Optimum resources alloca tion under the different scen arios, with the ratio of gain constraint........................................................................................................199 5-11 Optimum resources allocati on under the different scenar ios, without the ratio of gain constraint........................................................................................................199 A-1 CCI ENR (1908-2005)...........................................................................................209 A-2 CCI ENR projection and relative change(2000-2030)...........................................210 A-3 Discounted costs of the SJRWMD pr oposed projects for the caution area............211

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xiv LIST OF FIGURES Figure page 1-1 Dissertation organizational diagram...........................................................................3 2-1 Chapter 2 organizational diagram..............................................................................6 2-2 Generic decision tree................................................................................................24 2-3 A visualization of VaR and CVaR concepts............................................................32 2-4 Measure of CVaR scenario generation...................................................................36 2-5 Scenario tree for a two-stage stochastic problem with recourse..............................36 3-1 Chapter 3 organizational diagram............................................................................47 3-2 Discretized standard no rmal distribution (moments in parentheses, i.e., (qth), are the unmatched moments).........................................................................................62 3-3 Discretized standard no rmal distribution (moments in parentheses, i.e., (qth), are the unmatched moments).........................................................................................63 3-4 Moments least-square re gression analysis plots.......................................................64 3-5 Priority water resource caution ar eas in the SJRWMD, Florida, USA....................64 3-6 Approximate locations of potential alternative water supply projects.....................66 3-7 Efficient frontier for 50, 75, 80, 85, 90, 95, and 99 percent, 5% STD.............75 3-8 Change of CVaR with (A) (B) and (C) cost for 50%, 5% STD.........78 3-9 Change of CVaR with (A) (B) and (C) cost for 75%, 5% STD.........79 3-10 Change of CVaR with (A) (B) and (C) cost for 80%, 5% STD.........80 3-11 Change of CVaR with (A) (B) and (C) cost for 85%, 5% STD.........81 3-12 Change of CVaR with (A) (B) and (C) cost for 90%, 5% STD.........82

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xv 3-13 Change of CVaR with (A) (B) and (C) cost for 95%, 5% STD.........83 3-14 Change of CVaR with (A) (B) and (C) cost for 99%, 5% STD.........84 3-15 Efficient frontier for 50, 75, 80, 85, 90, 95, and 99 percent, 10% STD...........86 3-16 Change of CVaR with (A) (B) and (C) cost for 50%, 10% STD.......89 3-17 Change of CVaR with (A) (B) and (C) cost for 75%, 10% STD.......90 3-18 Change of CVaR with (A) (B) and (C) cost for 80%, 10% STD.......91 3-19 Change of CVaR with (A) (B) and (C) cost for 85%, 10% STD.......92 3-20 Change of CVaR with (A) (B) and (C) cost for 90%, 10% STD.......93 3-21 Change of CVaR with (A) (B) and (C) cost for 95%, 10% STD.......94 3-22 Change of CVaR with (A) (B) and (C) cost for 99%, 10% STD.......95 3-23 Efficient frontiers for 50, 75, 80, 85, 90, 95, and 99 percent, (A) 5% STD and (B) 10% STD.....................................................................................................97 3-24 Comparison of CVaR CVaR, CVaR, and VaR values calculated using model 3, (A) 5% STD and (B) 10% STD................................................................99 3-25 Comparison of CVaR values calculated using model 5% and 10% STD............99 4-1 Chapter 4 organizational diagram..........................................................................103 4-2 Expected utility indifference curves.......................................................................109 4-3 Fanning-out effect..................................................................................................115 5-1 Chapter 5 organizational diagram..........................................................................158 5-2 Change of the utility function for coalition 1 w. r. t. issue 1 (horizontal axis) and issue 2 (vertical axis) for increasing values of x3 (0 – 1, top to bottom, left to right).......................................................................................................................19 1 5-3 Change of the utility function for coalition 2 w. r. t. issue 1 (horizontal axis) and issue 2 (vertical axis) for increasing values of x3 (0 – 1, top to bottom, left to right).......................................................................................................................19 2

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xvi 5-4 Change of the utility function for coalition 3 w. r. t. issue 1 (horizontal axis) and issue 2 (vertical axis) for increasing values of x3 (0 – 1, top to bottom, left to right).......................................................................................................................19 3 5-5 Change of the utility function with the relative gain constraint.............................200 5-6 Change of the utility function wi thout the relative gain constraint........................200 A-1 CCI ENR historical data.........................................................................................208

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xvii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy APPLICATIONS OF SINGLE PART Y AND MULTIPLE PARTY DECISION MAKING UNDER RISK AND UNCERTA INTY TO WATER RESOURCES ALLOCATION PROBLEMS By Ghina Yamout December 2005 Chair: Kirk Hatfield Cochair: Edwin Romeijn Major Department: Civil and Coastal Engineering Decision theory refers to the analysis formalization, and prediction, through mathematical models, of optimal and real decision-making; it i nvolves the process of selection of perceived soluti ons, actions, and outcomes to a given problem from a set of possible alternatives. Depending on the extent of possible quantification, presence of uncertainty, number of decisi on makers, and number of obj ectives, decision theory is classified as quantitative or qualitative, dete rministic or stochastic, single or multiple party, and single or multiobjective. The manage ment of water resources systems involves unavoidable natural and social conditions of risk and uncertainty and multiple competing or conflicting parties and obj ectives, which introduce the risks of high economic and social costs due to wrong decisions, nece ssitating the formulation of models that adequately represent a give n situation by incorporating all factors affecting it.

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xviii Hence, the adequate modeling of such systems should inco rporate risk and uncertainty; decision theory under risk and uncer tainty is called decision analysis or risk management. Risk management approach es may be broadly categorized as nonprobability and probability based techniques. Non-probability based techniques include sensitivity analysis, decision criteria, analytical hierarc hy, and game theory; probability based techniques include scenario analysis, moments, decision trees, expected value, stochastic optimization, Bayesian and fuzzy anal ysis, and downside risk measures such as Value-at-Risk and Conditional Value-at-Ris k metrics. Many of these methods, though very useful, suffer from critical shortcomi ngs: sensitivity and scenario analysis only provide some intuition of risk, expected value fails to highlight extreme-event consequences, decision trees and hierarchical approaches fail to generate robust and efficient solutions in highly uncertain e nvironments, central mo ments do not account for fat-tailed distributions and penalize positive and negative deviations from the mean equally, recourse does not provide means to control risk, and Value-at-Risk does not provide information about the extent and dist ribution of the losses that exceed it and is not coherent. Game theory, used in situati ons of multiple party decision making, suffers from several systematic violations, such as the common consequence, preference reversal, and framing effects. To account for some of these shortcomings, several extensions and alternatives have been sugge sted such as the conditional-Value-at-Risk and behavioral game theory.

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1 CHAPTER 1 INTRODUCTION Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in many fields such as ma thematics, statistics, economics, philosophy, management, sociology, political science, and psychology. Its main concern is the analysis, formalization, and prediction, thr ough conceptual, physical, and mathematical models, of optimal and real decision-making, defined as the process of selection of a perceived solution, action, and co rresponding outcome, to a give n problem, from a set of possible alternatives, in a give n situation. A situation is us ually described by the extent of possible quantification, the presence of uncertainty, the number of decision makers, and the number of issues or objectives (Hipel et al., 1993; Radford et al., 1994); these four factors result in sixteen combinations depicted in Table 1-1. Water resources systems management involves unavoidable na tural and social condi tions of risk and uncertainty and multiple competing or conflic ting parties and objectives, which introduce the risks of high economic and social cost s due to wrong decisi ons, necessitating the formulation of models that adequately repres ent a given situation by incorporating all the factors affecting it (Haimes, 2004). Hence, this research is focused on the formal representation of multiple objective situations involving risk and uncertainty, for single party and multiple parties decision-making (the bolded sections of Table 1-1). The diagram in Figure 1-1 previews the plan of this dissertation. Ch apter 2 reviews the concepts of uncertainty, risk, and probability.

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2Table 1-1. Possible decision ma king environment combinations Factors Uncertainty Quantification Objectives Decision makers Combination Absent Present Qualitative Quantitative Single Multiple Single Multiple 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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3 Figure 1-1. Dissertation organizational diagram The chapter also compares different probability and non-probability based analytical techniques used in risk manage ment, focusing on the conditional value-at-risk method, CVaR. The chapter also presents the di fferent methods used for scenario generation representation of uncertainty. The chapter concludes with an extensive review of the application of risk management techniques to water resources problems. Chapter 3 applies and compares different single party risk management techniques, presented in Chapter 2, to a water resour ces management problem, where risk is quantified as cost. These methods are the e xpected value, scenario model, two-stage stochastic programming with recourse, and CVaR. They were built into a mixed integer fixed cost linear programming framework. Uncertainty was introduced via water Chapter 1 Chapter 2 Chapter 3: Plan of dissertation Decision making under uncertainty: methods comparative review and applications to water resources management Comparison of single decision-maker risk management techniques using a wate r allocation case study Chapter 4: Utility theory, game theory, and water resources management applications Chapter 5: Multiagents multattributes water allocation game model: development and application Chapter 6 Conclusions and future research recommendations

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4 supplies and results were compared for two di screte distributions with equal means, and different standard deviations, developed us ing one of the scenar io generation methods presented in Chapter 2. The models were applied to a case study, using the Saint Johns River Water Management District (SJRWMD) Priority Water Resource Caution Areas (PWRCA), in East-Central Florida (ECF), as a study area. Chapter 4 presents a review of the development of utility theory and game theory, as theories of individual a nd strategic decision making under risk and uncertainty. The chapter starts with a summary of the form al conception of exp ected utility theory, followed by its critique as a human choice pr edictive tool in deci sion making situations, and an overview of its alternatives, and conti nues with the examination of standard game theory, its main taxonomy, and solution concepts setting the stage fo r behavioral game theory. The chapter conclude s with an extensive review of the application of game theory to water resources decision making problems. Chapter 5 presents the main concepts of research in the allocation of multiple resources between multiple competing or conflicting parties, such as preferences, risk, equity, salience, and power. Subsequentl y, the chapter proposes an applied modelling tool for common pool resources conflict resolution that combines essential concepts in an n parties game theoretic framework. Specifi cally, these concepts are utility, ideal position, issue linkage, equity, salience, risk propensity, conflict le vel, and political uncertainty. The chapter concludes with the illustration of this model using a hypothetical conflict situati on over water, land, and fina ncial resources among three different parties.

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5 CHAPTER 2 DECISION MAKING UNDER UNCERTAINTY: A COMPARATIVE REVIEW OF METHODS AND APPLICATIONS TO WATER RESOURCES MANAGEMENT Decision Theory, DT, is a subset of Operations Research, OR, and Systems Analysis (SA);1 it is a body of knowledge and related analytical techniques, of different degrees of formality, designed to help a decision-maker choose among a set of alternatives in light of thei r possible consequences. Decisi on Analysis (DA) is a subset of DT that refers to th e discipline of Decision-Maki ng under Uncertainty (DMuU) (Haimes, 2004; Winston, 1994). The proce ss of Decision-Making under Uncertainty is also equivalent to another process, name ly, Risk Management, RM (Haimes, 2004). Decision Analysis, Decision-Ma king under Uncertainty, and Ri sk Management are used interchangeably in the rest of the text. Although DMuU and RM refer to the same processes, their designations employ terms, namely uncertainty and risk, which, hi storically, have genera ted a great deal of argument. In the next sections we clar ify the concepts of uncertainty, risk, and probability in the framework of RM. Subseque ntly, we present the general structure of the latter. These overviews ar e not aimed at presenting the field specific definitions and applications of risk and its management; rath er they are meant to provide a summary of the general definitions. Following, we descri be and compare different probability and 1 Originally, OR and SA referred to the use of quantita tive techniques as a scientific approach to decisionmaking and the analysis of complex interdependent el ements of a system in a holistic interdisciplinary manner, respectively, to aid in deci sion-making. Currently, the analyti cal tools and evaluative techniques that both sciences utilize overlap; as a result, these terms are sometimes used interchangeably (Winston, 1994).

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6 non-probability based analytical techniques used in RM, focusing on the conditional value-at-risk method. As a major issue in DM uU is the representation of uncertainty, we also present the different methods used for scen ario generation. We conclude this chapter with an extensive review of the applica tion of RM techniques to water resources problems. The diagram in Figure 2-1 e xhibits the plan of this chapter. Figure 2-1. Chapter 2 organizational diagram On the Origin of Risk The first recorded practice of risk analysis qualitative, dates to as back as early Mesopotamia, about 3200 B.C. in the Tigris -Euphrates valley, wh ere the Ashipu served as consultants for difficult decisions in anci ent Babylonia; the Ashipu created ledgers of alternatives, their corresponding outcome s, and favorability (Oppenheim, 1977). Risk Risk Management Risk Management Techniques Origin and Definition Definition and Structure Comparison of probability and non-probability based Tools Scenario Generation Methods Conditional Valueat-Risk Formal definition Application of RM to Water Resources Review

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7 Evidence of games of chance have been found in archeological ancient Egyptian, Sumerian, and Assyrian sites, where talis, the predecessor of dice, was used. Marcus Aurelius was regularly accompanied by his master of games (Covello and Mumpower, 1986). Quantitative risk analysis can be traced to early religious ideas concerning the probability of an afterlife. Plato addresse d it in Phaedo in the fourth century B.C. Arnobius the Elder, a major church figure in the fourth century A.D. in North Africa, proposed a two-by-two matrix to analyze the a lternatives of God’s existence/inexistence versus accepting Christianity /being a pagan (Covello and Mu mpower, 1986). In 1518, in response to a cash flow problem, the Catholic Church sanctioned us ury as long as there was risk on the part of the lender (this defi nition was rescinded in 1586 to be resanctioned later in 1830) (Grier, 1981). The formal mathematical theories of pr obability, however, did not appear until the time of Pascal, who introduced probability th eory in 1657. Following the steps of Pascal, the late seventeenth century witnessed a surg e of related intellectual activity by authors such as Arbuthnot, who argued that probabilitie s of an event’s causes can be calculated; Graunt, in 1662, and Halley, in 1693, who published life expect ancy tables; and Hutchinson, who studied the trade-offs between pr obability and utility in risky situations. In the early eighteenth century, 1738, Cramer and Bernoulli proposed solutions for the Saint Petersburg paradox.2 In 1792, LaPlace analyzed the probability of death as a 2 In probability theory and decision theory the St. Pete rsburg paradox is a paradox that exhibits a random variable whose value is probably very small, and yet has an infinite expected value. This poses a situation where decision theory may superficia lly appear to recommend a course of action that no rational person would be willing to take. That appearance evaporates when utilities are taken into account. The paradox is named from Daniel Bernoulli's original solution, published in 1738 in the “Commentaries of the Imperial Academy of Science of Saint Petersburg.”

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8 function of smallpox vaccination (Covello and Mumpower, 1986). In the nineteenth century, Von Bortkiewicz examined ten year records of Prussian soldiers’ mortality to study the event of death by kicks from horse s (Campbell, 1980). Before the century was out, with the need to quantify risk, authors, mainly in the fields of economics, finance, and accounting, had begun to explore the rela tionship between uncer tainty, probability, and risk. The next section shows how that ta sk was revealed to be not as simple and straightforward a matter as it appears (McGoun, 1995). Definition of Risk Historically, the concept of risk has been far from easy to define. Its comprehension and quantification challenged and confused professionals such as philosophers, psychologists, economists, social scientists, physical scientists, natural scientists, mathematicians, and engineers (Haimes, 2004). The association between uncertainty, probability, and risk was a matter of great debate in the late nineteenth and early twentieth cen turies (McGoun, 1995). Haynes (1895) argued that for many risks, hi storical relative frequency statistics are unreliable or inexistent and th at risk exists even when st atistics are not. Ross (1896) distinguished between variation, as the unqua ntifiable descriptive of possible outcomes with no statistics, and uncertainty as the conse quence of this variation; he argued that the latter is equivalent to risk. Willett (1901) differentiated between probability or chance, uncertainty, and risk: uncertainty is the degree of rational ambivalence between two alternatives and also is the deviation of a probability from its normal value; risk, related to both uncertainty and probability, is the quant ification of uncertainty in the form of mean absolute deviation. Willett also defined risk and uncertainty as the objective and subjective aspects of apparent variability, where the former is the effect of the

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9 psychological effect of the latter. Fisher (1906) described probability as a measure of ignorance; as such it is subjective and, in many cases, undefined; it is necessary information to assess risk, but it is not a measure of risk (McGoun, 1995). Lavington (1912) and Pigou (1920) were the fi rst to explicitly define risk, as measurable, as the dispersion of relative frequency distribution. Knight (1921) underlined risk as “measurable uncertain ty” where objective probability exists and uncertainty, or “unmeasurable uncertainty” as the cases where no quantitative distribution or only subjective probability exists. Lavingt on (1925) defined risk as the probability of a loss and uncertainty as the confidence in that probability. Florence (1 929) elicited three values associated with uncertainty and risk: th at of probability itself, the meaning of that value, and its quality or preci sion as objective statistics or subjective confidence. Fisher (Fisher, 1930) rejected the probabilistic measure of ris k, which is the synonym of uncertainty or lack of knowledge. By 1930, probabilistic measurement of risk had been rejected (McGoun, 1995). It was with Fisher (1930) and Hicks (1931), in economic research, th at the probabilistic measure of risk returned. Makower a nd Marschak (Makower and Marschak, 1938; Marschak, 1938) continued the movement to ward an objective probabilistic measure of risk. In 1944, Domar and Musgrave (1944) acknowledged the skepticism toward the probabilistic measurement of ri sk; however, they recognized that, with the need of the quantification of values and th eir associated risks and with the absence of a satisfactory alternative approach to the subject of risk, this method should be adopted. Lawrence (1976) defined risk as a measur e of the probability and severity of adverse effects. He also di stinguished between risk and sa fety; whereby measuring risk

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10 is an empirical, quantitative, and scientific activity, j udging safety is judging the acceptability of risk, a normative, qualitativ e, political activity. His definition was adopted by Haimes (2004) who stated that risk is a complex composition of two components: (1) “real” potential adverse eff ects and consequences, and (2) “imagined” mathematical intangible construct of probability. The Principles, Standards, and Procedur es (PSP) published in 1980 by the U.S. Water Resources Council (USWRC, 1980) made a clear distinction between risk, uncertainty, imprecision, and variability as foll ows. In a situation of risk, the potential outcomes can be described using a reasonabl e well-known probability distribution. In situations of uncertainty, the potential out comes cannot be described in terms of objectively known probability dist ributions or subjective probabi lities. In s ituations of imprecision, the potential outcomes cannot be described in terms of objectively known probability distributions, but can be estimat ed by subjective probabilities. Finally, variability is the result of inherent fluctuati ons or differences in the quantity concerned. The PSP identified two major sources of risk and uncertainty: (1) measurement errors of the variable complex natural, social, and ec onomic situations and (2 ) unpredictability of future events that are subject to random influences. Kaplan and Garrick (1981) defined risk as a set i i iX L S R, , where iS, iL, and iX denote the risk scenario i, its likelihood, and its dama ge vector, respectively. Subsequently, Kaplan (1991; 1993) added a subscript c to indicate that the set of scenarios should be complete. He also a dded the idea of “success” or “as planned” scenario 0S; the risk of a scenario iS is then visualized as the deviation from 0S. This idea matured into the theory of scenario structuring (TSS) (Kaplan et al., 2001; Kaplan et

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11 al., 1999). Haimes (2004) refine d this risk definition as P i i i PX L S R, , where PR is an approximation to R based on the partition P of the underlying risk space, stating that the scenarios iS are finite in number, disjoint, and complete. Lave (1986) defined hazard as referri ng to some undesirable event that might occur; the probability of occu rrence of a hazard is how fre quently this hazard would be expected to occur. He defined risk as the expected loss or the sum of all products of each possible hazard and its probability of occurrence. Morgan and Henrion (1990) defined proba bility as a formal quantification of uncertainty. They distinguished between the classical, objectiv e, or frequentistic and the Bayesian, subjective, or pers onalistic views of probabil ity; the former defines the probability of an event occurr ing in a particular trial as the frequency with which it occurs in a long sequence of similar trials, wh ile the latter defines a probability of an event as the degree of belief that a person ha s that it will occur gi ven all the relevant information known by that person at that time, so it is a function not only of the event, but also of the state of information. Note that regardless of the view, probability assignments must be consistent with the axioms of probability. In his book, Technical Risk Management, Micheals (1996) distinguished between the terms hazard, peril, and risk. A hazard is a condition or action, following a decision, which may result in perilous conditions. Pe ril is the undesirable event resulting from a hazard. The peril probability is the probability of occurrence of that peril. The number of possible hazard – peril combinations is an indication of the complexity of the system. In a system with four hazard factors such as product, proc ess, intrinsic, and extrinsic hazard factors, and four impacts such as on quality, functionality, affordability, and

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12 profitability, there exists 4 1! ! 4rr n r n in this case sixty, hazard – peril combinations. Peril recovery is the correctiv e action cost and time to recover from that peril. In that framework, Michael s (1996) defined risk as th e uncertainty surrounding the loss from a given peril and risk cost and tim e as the product of corrective action costs and times, respectively, for recovery from pe ril and probability of the peril; they are probabilistic measures of risk to the cost and time, respectiv ely, of corrective actions. Risk exposure is the sum of risk costs and ri sk times for a given system. Risk avoidance or reduction is the action taken to reduce th e risk exposure. Risk recovery is the corrective action taken to offset perturbations caused by a materialized peril. A risk determinant factor is the quantified attribute, or risk determinant, that serves as a measure of factors contributing to risk exposure. Risk metrics are a system of risk determinant factors quantifying risk exposure. Micheals also categorized risk as (1) obj ective or subjective, which the former can be described by statistics and the latter as a reflection of attitudes and states of mind and subject to perceptions; (2) specu lative or pure, where in the former there is uncertainty about both hazards and perils, while in the latter, there is only uncertainty as to whether a hazard leading to a peril will o ccur and not as to whether the resulting peril produces loss. Holton (2004a), in the context of financ ial and economic analysis, stated in his Contingency Evaluation website that risk has true meaning only when it refers to the possibility of incurring loss, mainly financial, directly or indirectly. In his book (Holton, 2003) and publications (Holton, 1997; Holt on, 2004b), he defined uncertainty as ignorance, a personal experience, and risk as exposure to uncer tainty. As such, risk has

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13 two components: uncertainty and exposure to it; if both are not present concurrently, there is no risk. Holton then argued that sin ce ignorance is a personal experience, risk is necessarily subjective. Proba bility is a metric of uncertainty; at best, it quantifies perceived uncertainty. There are many more efforts to define risk; a further review of definitions, however, is superfluous for the purpose of this work. In the next section, we present some frameworks of risk management. Definition of Ri sk Management There are almost as many definitions for Risk Management as there are management disciplines. Everybody makes decisi ons: scholars and anal ysts in the fields of economics, finance, and accounting, ps ychology and social sciences, biology, toxicology, and medicine, mathematics and com puter science, and engineering, all have been addressing the field of risk management What all disciplines have in common though, is the definition of RM as a feedback process consis ting of several steps. The following definitions are based on the correspond ing ones, if available, presented in the previous section. Kaplan and Garrick (1981) defined risk assessment as the process of identifying what can go wrong, the likelihood of it going wrong, and the consequences. Haimes (1991) used this definition to define risk management as a process that builds on risk assessment and finds the available management alternatives and solutions, their costs, benefits, and risk trade-offs, and the imp acts of the management decisions on future options. It requires the synthesis of the empirical and normative, the quantitative and qualitative, and the obj ective and subjective.

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14 Lave (1986) defined the risk management pr ocess as a cyclical succession of (1) risk identification, or the identification of hazard and its associated risks, (2) risk assessment, or the identification of the qua ntitative magnitudes of the hazards, (3) management options, or the identification of goals, (4) decision analysis, or the identification of alternatives, and (5) implementation and monitoring. Michaels (1996) defined RM as the execu tive function of cont rolling hazards and their consequential perils that causes some kind of loss. Its aim is to reduce risk exposure rather than recovery; hence it stresses risk av oidance as first line of defense and risk recovery as a backup. He divided RM to three concurrent processes: (1) risk identification, (2) risk quantification, and (3) risk control. The first step includes determining the scope of i nvestigation and, establishi ng the baseline model, and identifying the hazards and perils. The sec ond consists of deriving the risk hierarchy, selecting the risk metrics and formulation, establishing a risk m odel, calculating risk exposure, and estimating contingency reserv e. The third includes establishing risk organization and funding it, propagating best practices, implementing audits, initiating motivational programs, and rewarding performance. Haimes (2004) defined risk assessmen t and management as two overlapping processes; he used two perspectives, quali tative normative and quantitative empirical. In his qualitative normative perspective, Ha imes defined risk assessment as the set of five logical, systemic, and well-defined ac tivities of (1) risk identification, (2) risk modeling, quantification, and measurement, (3 ) risk evaluation, (4) risk acceptance and avoidance, and (5) risk management.

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15 Haimes distinguished risk identification, the first step of RM, as the process of identifying the sources and nature of risk and the uncertainty a ssociated with it; this stage attempts to uncover and describe all risk-bas ed events that might occur, be it natural (hydrologic, meteorological, and/or environmental,) or man-made (demographic, economic, technological, institutional, and/or political) causes. The second step, risk modeling, quantification, and measurement, involves assessing the occurrence likelihood of adverse events through obj ective or subjective probabilities and modeling the causal relationship among the different sources of risk, or adverse events, and their impacts. In other words, it involves the quantification of the input a nd output relationships of the random and decision variables and their rela tionship to the state variables, objective functions, and constraints. In the third step of RM, risk evaluation, various policy options are formulated, developed, and optimized. Risks, benefits, and costs tradeoffs are generated and evaluated. The fourth step, risk acceptance and avoidance, is the decisionmaking step where the level of acceptability of risk is determined by evaluating the considerations that fall beyond the modeling and quantification pro cess; it answers the question of “how safe is safe enough?”. The fifth and final step, risk management, is the execution step where the chosen policy option is implemented. In his quantitative empirical perspective, Haimes defined risk assessment as the set of three major, though overlapp ing, activities: (1) informa tion measurement, (2) model quantification and analysis, a nd (3) decision-making. Information measurement includes data collection and processing. Model qua ntification and anal ysis includes the quantification of risk and other objectives, the generation of Pareto-optimal solutions and their trade-offs, and the conduct of impact a nd sensitivity analysis. Finally, decision-

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16 making involves the interaction between analys ts and decision-makers and the subjective judgment for the selection of preferred policies. Haimes also defined total risk management as the process that harmonizes risk management with the overall system mana gement; it addresses hardware, software, human, and organizational failures involving all aspects of the system’s life cycle, planning, design, construction, operation, a nd management. Finally, he defined riskbased decision-making refers to a decision-m aking process that accounts for uncertainties through some process in the formulation of policy options. Our Definition For the purpose of this dissertation, we do not dwell on the philosophical concerns associated with uncertainty, probability, and risk, mainly, the concepts of their existence and the extent of their objectivity. We start with Holton’s (1997; 2003; 2004a; 2004b) distinction between the terms metric, measure, and measurement. A metric is the designation of a tool. An operation or algorithm that supports a metric is cal led a measure. The value obtained from applying a measure is a measurement. Hence, a measure is used to obtain a measurement of a metric; there may be many measures to one metric. We associate uncertainty with ignorance. We define stochasticity as a special type of uncertainty associated with randomness. Fo r practicality, we classify uncertainty into two main groups: natural and ma n-made; natural uncertainty is associated with a natural system’s components, while man-made uncerta inty is associated with a man generated system’s components. Uncertainty is quantified using probability. We adopt the classical objective theory of statistics; we assume that probability exists and can be quantified. The met hods of its generation, through historical

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17 information or mathematical algorithms, and their accuracy and precision are outside the scope of this work. Probability is a metric of uncertainty. We associate risk with loss; any type of loss, resulting from a decision-making policy or action. Loss, or risk, is hence repr esented using risk metr ics. The choice of a risk metric and measure depends on the pr oblem at hand and the decision-maker’s objectives and priorities. The description of risk is done through measurement of the risk metric and its associated statistics. We define risk management, RM, as the processes of risk identification, risk estimation, risk evaluation, and risk mon itoring. Risk identification consists of uncovering and describing natural, man-made risk -based events that might occur. Risk estimation refers to the quantification of thes e events, their probability of occurrence, and their causal relationship. In the third step of RM, risk evaluation, various policy options are formulated, developed, and optimized. Risks, benefits, and costs tradeoffs are generated and evaluated. This definition is based on Haimes (Haimes, 2004). Finally, risk monitoring is the continuous process of the first three steps. RM is a feedback process. Risk Management Techniques “To manage risk, one must measure it” is an adage that Haimes (2004) uses in his book Risk Modeling, Assessment, and Management Public interest in the field of RM has expanded significantly during the last two decades as an effective and comprehensive procedure that complements/s upplements the management of almost all aspects of our lives. As federal and state le gislators and regulatory agenci es have been addressing the importance of the assessment and manage ment of risk, industrial and government

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18 agencies in many management disciplines su ch as financial management, health care, human safety, manufacturing, the environment, and physical infrastructure (e.g. water resources, transportation, and power generation) all started incorporating risk analysis in their decision-making process. In parall el, the scholastic community witnessed an unprecedented release of articles covering th e development of theory, methodology, and practical applications (Haimes, 2004). What are the methods, how are they distinguished, and how are they categorized? This section is devoted to the presentation of the main classes of RM techniques. Note that it is not meant to be an exhaustiv e reference of every available tool, sub-tool, and application developed in every discipline, but a comprehensive categorized overview of RM tools. As we undertake the task of classificati on, we are faced with the multiplicity of possible ways with which this task may be approached, depending on our interests and objectives. For the purpose of this disserta tion, we base our clas sification, on a first level, on the concept of probability. Therefor e, we classify risk-based decision-making methodologies into probability-based and non-pr obability-based techniques. Note that the presented classes of methods are not mutu ally exclusive; in other words, in many instances, their concepts may overlap and multiple methods may be incorporated into one model, as always, depending on the problem at hand and the decision-maker’s objectives. We start this section by defining the gene ral mathematical notations we use. Following, we describe non-proba bilistic and probabilistic risk management techniques. The non-probabilistic methods presented ar e sensitivity analysis, decision-making criteria, decision matrix, multiobjective optimi zation, and game theory. The probabilistic

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19 techniques presented are scenario analys is, moments and quantiles, decision trees, stochastic optimization, downside ri sk metrics, and utility theory. Mathematical Notations An uncertain parameter Z may be continuous or discrete. A continuous parameter can assume all values in a specified interv al. A discrete parameter assumes different values with different associated probabil ities. Either way, the uncertainty of Z is described via its statistics; such statistics ar e its mean, variance or standard deviation, and, at best, its probabilit y density function, pdf, and cu mulative distribution function, cdf. The pdf and cdf are represented by the functions Z f and Z F, which are formalized differently for continuous and disc rete parameters. In the following sections, we consider only cases of discrete paramete rs; we assume that continuous ones may be discretized. The occurrence of each value, jZ, of Z with a probability js p, is represented by a scenario, js, where J j ,..., 2 1 denotes the scenario’s name or number. Each scenario js occurs with a probability js p of jZ We denote by ia the decision or action alternatives adopted by the decision-maker, where I i ,..., 2 1 is the alternative’s name or number. We also define the pair j is a, as the outcome from the combination of the decision ia and the scenario js. The risk resulting from a pair j is a, is ijr. Finally, the parameter value jZ, the corresponding scenario, js, and risk or loss ijr have the same js p, Z f, and Z F.

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20 Non-Probability-Based RM Techniques Although the management of risk, generally, connotes the quan tification of risk through reliance on probability and statistics (Haimes, 2004), several risk-based decisionmaking methodologies do not require the knowle dge of probabilities. These methods include sensitivity analysis, decision-making criteria, decision matrix, multiobjective optimization, and game theory; they ar e described in the following sections. Sensitivity analysis Sensitivity or what-if analysis is the process of varying model input parameters over a reasonable range (range of uncertain ty in values of model parameters) and observing the relative cha nge in model response. The purpose of this type of analysis is to demonstrate the sensitivity of the model simulations to uncertainty in values of model input data. The sensitivity of one model parameter relative to other parameters is also demonstrated. Sensitivity analysis is also bene ficial in determining th e direction of future data collection activities. Data for which th e model is relatively se nsitive would require future characterization, as opposed to data for which the model is relatively insensitive (Morgan and Henrion, 1990; Winston, 1994). Decision making criteria Decision making criteria are methods for handling risk and uncertainty without adhering to probability (Haimes, 2004; Winst on, 1994) in an optimization formulation. The three most common criteria are the pessimistic rule, the optimistic rule, and the Hurwitz rule. Pessimistic rule. Also called the maximin or minimax criterion because it consists of maximizing the minimum gain or minimizi ng the maximum loss; the rationale is that,

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21 by applying this rule, a decisionmaker will at least realize the minimum gain or avoid the maximum loss. Its formulation is ij J j I ir1 1max min. Optimistic rule. Also called the maximax or mini min criterion because it consists of maximizing the maximum gain or minimi ze the minimum risk. Its formulation is ij J j I ir1 1min min. Hurwitz rule. This rule is a compromise be tween the two extr eme criteria through an index, where 1 0 that specifies the degree of the decision-maker’s optimism: the smaller the the greater the optimism; it is the linear combination of the minimax and minimin criteria formulated as ij J j ij J j I i i I ir r r1 1 1 1min 1 max min min Analytic hierarchy proce ss or decision matrix The Analytic Hierarchy Process, AHP, al so called decision matrix method for the evident reason that it is based on a ranki ng matrix of decision and corresponding attributes, such as risk. It is a semi-quant itative decision making tool for situations where the attributes are not amenable to explicit quantification. The attributes are also assigned weighing factors. The decision option with th e highest weighted su m of attributes is considered the best solution. Changing th e weights of the assi gned attributes is performed for sensitivity analysis (Haimes, 2004; Winston, 1994). Utility and game theory Utility is used to represent individual pr eferences, therefore predict their choice behavior. The theory of games uses utility to model strategic interactions between competing and conflicting decision-makers (Heap, 2004; Myerson, 1991). Both theories, their limitations, and alternatives are discussed in details in Chapter 4.

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22 Multiobjective optimization Multiobjective optimization, MO, also known as Multi-Criteria Decision Making, MCDM, refers to optimization problems with several, possibly conf licting or competing, objectives. Objectives are weighted accordi ng to their priority. There are several methods for solving MO problems. Th e most commonly used method is Goal Programming, where the objectives are given goals that are ranked by weighting factors and the problem is reduced to a single objectiv e function of the wei ghted minimization of the deviations from the assigned goals. A nother method is the weighted sum approach, where the objectives are assigned weighing factors and combined into one objective forming a single optimization problem. A thir d method is the hierarchical optimization method, where the objectives are ranked in a descending order of importance and each objective is then optimized individually subject to a constraint that does not allow the optimum for the new function to exceed a prescribed fraction of a minimum of the previous function. Other methods are the trade-off, constraint, or method, the global criterion method, the distance function me thod, and min-max optimization (Haimes, 2004; Winston, 1994). Probability-Based RM Techniques This section describes the following probability-based RM technique: scenario analysis, moments and quantiles, decision trees, stochastic optimization, downside risk metrics, and utility theory. Scenario analysis Scenario analysis combines sensitivity analysis and the expected value metric, where the uncertain parameter is assigned di fferent values, corresponding to different

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23 scenarios, with associated probabilities, and th e expected value of th e scenarios results is calculated (Haimes, 2004; Winston, 1994). Moments and quantiles The expected value operator, E mean, or first central moment, is a central metric that for a discrete parameter multiplies the parameter’s values, such as loss, from different scenarios, by their corresponding probabili ty of occurrence and then sums these products. For example, in an optimization fr amework, the expected value of loss for all policy options is minimized; formally, for a discrete problem: J j ij j I ir s p1 1min. The variance, 2, or second central moment, and standard deviation, are measures of dispersion of the values of a parameter around its mean. For example, the variance of a discrete loss is 2 2 1 2 ij ij j J j ij ijr E r E s p r E r A quantile is the generic term for any fracti on that divides the values of a parameter arranged in order of magnitude into tw o specific parts. For example, the 90th percentile of the loss is the value for which the value of Z F is 90% or 0.9; in other words, 90% of the losses lay below the value of the 90th percentile. Decision trees Decision Trees are one of the most used tools in risk-based decision-making; it relies both on a graphical descriptive and an analytical probability-based representations (Haimes, 2004; Winston, 1994). The basic co mponents of a decision tree are the decision nodes, designated by squares, chance nodes, designated by circles, and consequences, designated by rectangles (F igure 2-2). Branches emanating from a decision node represent the various decisions or actions, ia, to be investigated. Branches emanating

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24 from a chance node represent the various scenarios, js, with their associated probabilities, js p; at their end are the consequences, ij associated with the scenario/action pair j is a, (Figure 2-2). Figure 2-2. Generic decision tr ee (adopted from Haimes, 2004) Stochastic optimization The theory and applications of Stochast ic Programming, SP, a ppeared in the 1950s, when authors such as Beale, Dantzig, Ch arnes, and Cooper (Beale, 1955; Charnes and Cooper, 1959; Dantzig, 1955) realized the need to incorporate uncertainty in Linear Programming, LP. Stochastic optimization, SO refers to optimization in the presence of uncertain parameters, with the uncertainty quantified statistica lly by continuous or discrete probability distributions. Depending on the way the uncertainty is expressed and modeled, SP models can be categorized as recourse problems, SPR, or chanceconstrained problems, CCP; these methods are briefly explained below (Birge and Louveaux, 1997; Di Domenica et al., 2003). Recourse optimization. Recourse optimization, RO, is also referred to as multistage optimization, MSO, or dynamic optimizatio n, DO. Recourse is the ability to take 1 s2s 1s2s 1s2s 3a 1 1 11 s a 2 1 12,s a 1 2 21 ,s a 2 2 22,s a 1 3 31,s a 2 3 32,s a

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25 corrective actions after an un certain event has taken place. An example is two-stage recourse problem; in simple terms, the pr oblem involves choosing a variable to control what happens in the present time and taki ng some recourse corrective action after an uncertain event occurs in the future. Chance-constrained optimization. Chance-constrained optimization problems, CCP, or probabilistically constrained optim ization problems, PCP, are optimization problems that involve statistical terms in their objectives and/or constraints. Bayesian analysis Bayesian analysis involves uncertainty caused by incomplete understanding or knowledge. One technique is Bayesian networ k, also known as belief networks, causal networks, Bayesian nets, quali tative Markov networks, influe nce diagrams, or constraint networks. Bayesian networks use a graphical structure to represent the complex causal chain linking decisions and consequences via a sequence of conditional relationships; variables are represented by a round node a nd the dependence between two variables is represented by an arrow. Dependence is represented by a conditional probability distribution for the node at the end of the arrow, based on Bayes formula. The graphical network constitutes a description of the proba bilistic relationships among the system’s variables (Batchelor and Cain, 1999; Borsuk et al., 2004; Bromley et al., 2005). Fuzzy sets Fuzzy set theory was suggested (Zadeh, 1965) to deal with decision situations involving risk and uncertainty without using probabilities. It d eals with situations characterized by imprecise information descri bed by membership functions (Hatfield and Hipel, 1999).

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26 Information gap Information gap models, also know as conve x models, define uncertainty to be an information gap between what is known and wh at is needed to be known for the decision making process; its aim is to quantify and re duce this information gap (Ben-Haim, 1997; Hatfield and Hipel, 1999). Downside risk metrics Under this category are the second, first, and zero order lower partial moments, LPM, value-at-risk, and conditional value-at-ris k metrics. In general, these metrics are referred to as measures in the literature (A lbrecht, 2003). Note that an LPM of order n is computed at some fixed quantile q and defined as the thn moment below q. Developed by Bawa (Bawa, 1975) and studied by Fishbur n (Fishburn, 1977), LPM measure risk by a probability weighted mean of deviati ons below a specified target level q; the higher the n, the higher the risk aversion. Second order LPM or semi-variance. The second order LPM, or SLPM, is also referred to as semi-variance, SV; it describe s the downside risk computed as the average of the squared deviations below a target loss. Formally, it is q j j js p r q SLPM1 2. First order LPM. The first order LPM, or FLPM, describes the downside risk computed as the average of the deviations belo w a target loss. Formally, it is defined as q j j js p r q FLPM1. It refers to risk neutral behavior.

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27 Zero order LPM. The zero order LPM, or ZLPM, describes the downside risk computed as the average of th e probabilities below a target loss; it coincides with the cumulative probability of q. Formally, it is defined as q F s p ZLPMq j j 1. Value-at-Risk. Value-at-Risk is denoted by VaR. In simple terms, VaR is a quantile. If the cumulative probability of q is denoted p q F then VaR is the inverse of the ZLPM, such as q p F VaR 1, and is defined as the maximum potential loss with a confidence level p A detailed and more correct discussion of VaR is presented later in this chapter. Conditional Value-at-Risk. Conditional Value-at -Risk is denoted by CVaR; it is equivalent to expected shortfa ll, ES. Generally, it measures the expected value of losses exceeding VaR. At a confidence level VaR r r E CVaR A detailed and more correct discussion of CVaR is presented later in this chapter. The Different RM Methods: A Discussion In the past, risk was investigated using a variety of ad hoc tools, such as sensitivity and scenario analysis. A lthough these techniques allow the observation of model response versus the change in an uncertain parameter using a deterministic model, they only provide some intuition of risk. Risk was also commonly quantified through the mathematical expected value concept. Alt hough an expected value provides a valuable measure of risk, it fails to highlight extr eme-event consequences, which are adverse events of high consequences and low probabi lities in advantage of events of low consequence and high probabilities, regardless of the formers’ poten tial catastrophic and

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28 irreversible impacts. From the perspectiv e of public policy, events like dam failure, floods, water contamination, or water shortage, with low probabilities cannot be ignored. This is exactly what the use of expected value would ultimately generate. The precommensuration of these low probability high damage events with high probability low damage events into one expectation function by the analyst markedly distorts the relative importance of these events as they are view ed, assessed, and evaluated by the decisionmakers (Haimes, 2004). Other methods developed to surpass the drawbacks of these quasi-deterministic techniques relied on decision tr ees and hierarchical approaches where uncertainty is introduced via discrete probabi lities of uncertain parameters. These methods fail to generate robust and efficient solutions in situ ations of highly uncertain environments with a large number of dynamic and correlated stocha stic factors and multiple types of risk exposures. The explicit introduction of statistical central moments, such as variance and standard deviation, which have well estab lished calculation methods, into stochastic simulation and/or optimization approaches, such as chance-constrained programming, allowed for some control of uncertainty and a ssociated risks. Central moments, however, do not account for fat-tailed distributions a nd penalize positive and negative deviations from the mean risks value equally (Cheng et al., 2003). The introduction of the recourse and multi-stage concepts into stochastic optimization allows for the separation of the decision-making process to accommodate info rmation, and associated uncertainties, available at different time steps. Recourse, however a very useful concept in practical

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29 decision-making applications, does not provi de means to control risk, and most importantly, downside risk, which is risk asso ciated with low probability and high losses. In the past ten years, a new method b ecame popular in industry regulations, the Value-at-risk, VaR, which was introduced by the leading bank, JP Morgan. Unlike the past methods, VaR provides a downside risk measure w ith a probability associated with it. VaR, however, has several shortcomings. The reduction of the risk information to this single number may lead to misleading interpretations of results. VaR does not provide any information about the extent a nd distribution of the losses that exceed it, where for the same VaR, we can have very different di stribution shapes with different associated maximum losses; Hence, it is incap able of distinguishing between situations where losses that are worse may be deemed only a little bit worse, and those where they could well be overwhelming. In additi on, the recent research on the axiomatic characterization of risk metrics revealed that VaR is not coherent. The coherence concept was first introduced by Artzner, Delbaen, Eber, and Heath (Artzner et al., 1999), who defined a coherent measure of risk as one that satisfies the following four axioms: 1. Subadditivity: Y X Y X 2. positive homogeneity: X X if 0 3. Monotonicity: Y X Y X if 4. Translation invariance: W X W X Where is a risk metric, W and Y X ,are different risk functions.

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30 A major drawback of VaR is that it does not satisfy axiom 1, i.e., it is not (with a few exceptions) subadditive. In practical terms, this signifies that the total risk associated with a certain project may be larger than the sum of individual risks resulting from different sources (Cheng et al., 2003). In addition, VaR is not convex; hence, may have many local extremes, which makes it unstable and difficult to handle mathematically. An alternative measure that was developed to overcome the limitations associated with VaR is the Conditional Value-at-Risk, or CVaR (Rockafellar and Uryasev, 2000; Rockafellar and Uryasev, 2002). Rockafella r and Uryasev defined expectation-bounded risk measures as satisfying axioms 1, 2, 4, and axiom 5, which is: 5. X if X E X non constant, and X if X E X constant. If axiom 3, monotonicity, is also satisfied, then the risk measure is coherent and expectation-bounded. CVaR is both coherent and exp ectation-bounded. It is a simple representation of risk that accounts for risk beyond VaR, making it more conservative than VaR. CVaR is also stable as it has integral characte ristics. It is continuous and consistent with respect to the confidence level CVaR is also a sub-a dditive convex function with respect to decision variables, allowi ng the construction of efficient optimizing algorithms; it can be optimized using linea r programming techniques, which makes it efficient. Value-at-Risk and Conditional Value-at-Risk To set the ground for formal definition of CVaR, we start with the formal definition of VaR. VaR is a quantile; it has three components: a time period, a

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31 confidence level, and a loss amount; it answer s the question: with a given confidence level, what is our maximum loss, VaR, over a specified time period, T? Note that there is always a probability, 1, that the actual loss will be larger. What does this mean? In reference to Figure 2-3, whic h presents the probability and cumulative distribution functions for two loss functions (solid and broke n lines), we can read that with confidence (60 percent), we expect that our worst loss, over time T, will not exceed VaR (5.5 loss units, for both loss func tions); there is a probability 1(40 percent), that this measure may be exceeded in the right tail of the distribution (> 5.5 loss units) (shaded area). Increasing the confid ence level will result in an increase in VaR, by moving into this right tail; for a confidence level of 95 percent, the VaR corresponds to 8 and 9.5 units for the broken and solid lines loss functions, respectively. In reference to Figure 2-3, CVaR quantifies the losses in the right tail of the distribution, the shaded area. Mo st importantly for applications, CVaR can be expressed by a minimization formula that can be incorporated into problems of optimization that are designed to minimize ri sk or shape it within bounds, such as the minimization of CVaR subject to a constraint on loss, the minimization of loss subject to a constraint on the CVaR, and the maximization of a utility function that balances CVaR against loss. But how do these concepts translate math ematically? And how is this risk measure, CVaR, calculated and controlled?

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32 0.0 0.2 0.4 0.6 0.8 1.0 024681012 LossConfidence Level, VaR0.6 CVaR0.6 CVaR+ 0.6 Figure 2-3. A visualization of VaR and CVaR concepts Note that the following mathematical formulation is limited to losses with discrete distribution functions; those w ith continuous distribution func tions can be discretized. For the mathematical formulation using losse s with continuous functions refer to the literature (Rockafellar and Uryasev, 2000; Rockafellar and Uryasev, 2002). Let ,x L be a loss function depending on a decision vector x and a stochastic vector Let ,x be its cumulative distribution func tion. Assume that the behavior of the stochastic parameter can be represented by a discrete probability distribution function, from which a scenario model can be constructed. Index the scenarios S s , 1 corresponding to the stochastic parameter s with corresponding probabilities sp, such that the losses are listed in an increasing order Sx L x L ,1 In this setting, we can define the following terms: VaR is the value of ,x L corresponding to th e confidence level Formally, , min x L x x L VaR

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33 CVaR, or upper CVaR, is the expected value of x L strictly exceeding VaR; it is also called Mean Excess Loss and Expected Shortfall. Formally, for equally probable scenarios, S s s sx L s S VaR x L x L E CVaR1, 1 , CVaR, or lower CVaR, is the expected value of x L weakly exceeding VaR, i.e., values of x L which are equal to or exceed VaR; it is also called Tail VaR. Formally, for equally probable scenarios, S s s sx L s S VaR x L x L E CVaR 1 1 ,. VaR is the probability that x L does not exceed VaR. Formally, VaR x L xVaR , max, where 1 VaR. is a weighing factor. Formally, 1VaR, where 1 VaR and 1 0 Finally, CVaR is the weighted average of VaR and CVaR. Formally, CVaR VaR CVaR1. Having defined all the terms, we can disti nguish four cases in the calculation of CVaR. These cases are demonstrated in Figur es 2-4(a) through 2-4( d). The examples provided correspond to cases where we have 6 scenarios, 6 , 1 s, and 4 scenarios, 4 , 1s, with equal probabilities 6 16 5 4 3 2 1 p p p p p p and 4 14 3 2 1 p p p p, respectively: 6. corresponds to the cumulative proba bility of one of the scenarios s: VaR, 0 sx L VaR, and CVaR CVaR CVaR VaR. 7. does not correspond to the cumulative pr obability of one of the scenarios: VaR, 0 VaRx L VaR ,, and CVaR CVaR CVaR VaR.

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34 8. does not correspond to the cumulative probabi lity of one of the scenarios and is greater than that of the last scenario: 1VaR, 0 1 VaRx L VaR,, and CVaR CVaR VaR with undefined CVaR. 9. corresponds to the cumulative pr obability of the last scenario s: 1VaR, undefined is sx L VaR, and CVaR CVaR VaR with undefined CVaR. A major issue in decision-making under unc ertainty is the representation of the underlying uncertainty. Usually, we are faced with either continuous distribution or a large amount of data, making the problems too complex or too large to solve regardless of the algorithm or computing capacity. He nce, we need to pass from the continuous distribution, or data, to a di screte distribution with a sma ll enough number of realizations, or scenarios, for the stochastic program to be solvable, and a large enough number of scenarios to represent the underlying continuous distributi on or data as close as possible (Dupacova et al., 2000b; Dupacova et al., 2003). Scenario Tree The process of creating this discrete distribution is ca lled scenario generation; it results in a scenario tree (D i Domenica et al., 2003; Dup acova et al., 2000b; Hoyland et al., 2003; Hoyland and Wallace, 2001). Forma lly, The uncertainty in the model is represented by the parameter with a probability density function, pdf, f and a corresponding cumulative dist ribution function, cdf, F. The true pdf, f, of is approximated by a discrete probability func tion, or mass distribution function, mdf, denoted P, concentrated on a finite number of scenarios S s, 1 corresponding to the stochastic parameter s with corresponding probabilities s sP p ,such that

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35 11S s sp. The scenario tree for a two-stage stochastic pr oblem with recourse is illustrated in Figure 2-5. ............................................(a) 6 4 3 2 6 4VaR,0 6 5 6 42 1 2 16 4f f CVaR CVaR ..............................................(b) 12 7 12 8VaR, 5 1 6 5 4 12 7 12 75 2 5 2 5 1 5 4 5 112 7f f f CVaR VaR CVaR Probability CVaR 1 4 1 4 1 4 1 4 1 8 1f 2f 3f 4f VaR Loss .....................................................(c) 8 7 1 8 8VaR,1 4 8 7 8 7f VaR CVaR undefined is CVaR8 7 Probability CVaR 1 6 1 6 1 6 1 6 1 12 1 6 1 6 1f 2f 3f 4f 5f 6f VaR --CVaR + 11 56 22CVaR ff Loss Probability CVaR 1 6 1 6 1 6 1 6 1 6 1 6 1f 2f 3f 4f 5f 6f VaR --CVaR +CVaR Loss

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36 .....................................................(d) 8 8 1 8 8VaR,undefined is 4 8 8 8 8f VaR CVaR undefined is CVaR8 8 Figure 2-4. Measure of CVaRscenario generation The process of scenario generation is done through the discretization of a continuous process, the aggregation of a discre te process, or internal sampling. In the following paragraphs we provide an overview of the first proc ess, i.e., the methods used for discretization or quantizati on of continuous distributions. Figure 2-5. Scenario tree for a two-st age stochastic problem with recourse Aggregation or reduction processes consist of deleting scenarios or data from an already existing large collection. An overvie w of the reduction me thods is outside the scope of this work; for examples, refer to the literature (Chen et al., 1997; Consigh and Dempster, 1996; Dupacova, 1996; Dupacova et al., 2000a; Dupacova et al., 2003; Wang, t=1 t=2 t= k t=T t=k+2 t=k+1 First Stage Second Stage 1 s 3 s s 2 s S s Probability CVaR 1 4 1 4 1 4 1 4 1 8 1f 2f 3f 4f VaR Loss

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37 1995). Internal sampling methods sample th e scenarios during the solution procedure, without using a pre-generated s cenario tree; some of these methods are stochastic and Lshaped decompositions and stochastic quasigrad ient methods. The reader is referred to the literature (Casey and Sen, 2002; Dantzig and Infa nger, 1992; Dempster and Thompson, 1999; Ermoliev and Gaivoronski, 199 2; Higle and Sen, 1991; Infanger, 1992; Infanger, 1994). Discretization Discretization or quantifica tion of a continuous distribu tion function is the process of constructing a discrete dist ribution function from this con tinuous distribution. Several methods have been developed; they can be classified into three main groups: Monte Carlo simulations, bracket methods, mome nt-matching methods, and optimization methods (Dagpunar, 1988; H oyland et al., 2003; Pfeifer et al., 1991). One of the most widely used techniques to generate discrete values from a continuous distribution are M onte Carlo simulations, which draw randomly from the distribution of a parameter. Monte Carl o simulations require a sequence of random numbers, usually provided by random number generators, R NG. RNG may be broadly classified as mixed linear congruential, mu ltiplicative linear congr uential, and general linear congruential, such as Fibonacci, Tauseworthe, shuffled, and portable generators. For univariate distributions, some general methods for generating random numbers are inversion, composition, stochastic model, e nvelope rejection, band rejection, ratio of uniforms, Forsythe, alias rejection, and pol ynomial sampling methods. The applicability and degree of suitability of each method varies with the type of distribution (Beaumont, 1986; Dagpunar, 1988; Pfeifer et al., 1991).

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38 The simplest and most trivial discrete approximations are the bracket methods. There are two traditionally used bracket methods: the bracket-median and bracket-mean methods. In the bracket-median approximation the cdf scale is divided into a number of equal intervals, or brackets, and the median of each is assigned the probability of its interval. The error in calculating the moment s can be substantial if only a few intervals are used; the most commonly used is the fi ve-point equiprobability bracket median approximation. The bracket-mean method is similar to the bracket-median method except that the intervals are re presented by their means rather than their median (Clemen, 1991; Hoyland and Wallace, 2001; Miller a nd Rice, 1983; Smith, 1993; Tagushi, 1978; Zaino and D'Errico, 1989a). Another type of approximations is the moment-matching approximations. Generally, an n-point moment-matching di screte distribution approximation, nPDDA, consists of n values and their corresponding prob abilities of occurrence chosen to approximate the pdf of a continuous parame ter (Di Domenica et al., 2003; Kaut and Wallace, 2003). Usually, the n values are specified fractile s from the cdf of the uncertain parameter with specified probabilities to work well in estimating moments of the pdf. The standard type of nPDDA is the threepoint approximations, 3PDDA, sin ce at least three points are needed to represent the underlying pdf we ll while the number of scenario tree paths increases exponentially with n. In 3PDDA, th e pdf is substituted by a three-point mdf. 3PDDA provide a convenient and simple way to approximate a pdf; in addition, it can be constructed to match the first three moment s of a pdf exactly (Keefer, 1994). Several 3PDDA have been developed based on differen t methods such as the Pearson-Tukey, P-T

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39 (Pearson and Tukey, 1965), the Extended Pear son-Tukey, E-PT (Keefer and Bodily, 1983), the Brown-Kahr-Peterson, B-K-P (Brown et al., 1974), the Swanson-Megill, S-M (Megill, 1977), the Extended Swanson-Megi ll, ES-M (Keefer and Bodily, 1983), the Miller-Rice One-Step, M-RO (Miller and Rice, 1983), the McNamee-Celona Shortcut, M-CS (McNamee and Celona, 1987), the Zain o-D’Errico Tagushi,Z-DT (D'Errico and Zaino, 1988), and the Zaino-D’Errico Im proved, Z-DI (Zaino and D'Errico, 1989b) approximations. Miller and Rice (Miller a nd Rice, 1983) introduced the use of the Gaussian quadrature technique for approxi mating n-point mdf. The resu lt is an n-point discrete distribution that matches the first 2n1 moments of the underlying continuous distribution. The values and probabilities for many distribut ions are obtained as solutions to polynomials; they are tabulated for diffe rent distributions in many references (Abramowitz, 1965; Beyer, 1978; Stroud and Secrest, 1966). Smith (Smith, 1993) developed another n-point the moment-matching approximation for approximating mdf using th e Gaussian quadratu re technique; like Miller and Rice, the result is an n-point di screte distribution that matches the first 2n-1 moments of the underlying conti nuous distribution. A charac teristic of this method is that it incorporates extrem e values of the latter. Hoyland, Kaut, and Wallace (2003) developed a new moment-matching approximation by applying a least-square m odel to minimize the distance between the generated and target first four moments and correlations for multivariate problems. Another general method to construct a discrete distribution is optimal discretization. Hoyland and Wallace (2001) suggeste d a nonconvex optimization

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40 problem by minimizing a measure of distance between the moments of the constructed distribution and the ones of th e underlying distribution. The model is rerun heuristically from different starting points until a local minimum is obtained. Other optimization techniques were deve loped by Pflug (2001). The accuracy of a discrete approximation of a probability distribution is measured by the extent to which the moments of the approximation match thos e of the original distribution. Several author s undertook the task of compar ing the performance of the previously presented methods for different underlying distributions. Miller and Rice compared their Gaussian quadrature based method to the brackets methods for uniform, normal, beta, and exponent ial distributions. Their method resulted in smaller approximation errors on the mean, variance, skew, and kurtosis; in addition, that error decreased as the number of points in the discre te approximation increased The bracket methods resulted in the underestimation of almost all moments. Keefer and Bodily compared several twopoint, three-point, five -point, and bracket approximations in estimating the mean, varian ce, and the cdf, for beta and log-normal distributions. They showed that different methods result in different approximation errors depending on the performance measures and types of distributions. The best performance was observed for the EP-T method followed by the ES-M method; these methods, however, perform poorly in cases of extremely skewed or peaked distributions. Zaino and D’Errico performed a Monte Carlo Simulation to compare different bracket and three-point appr oximations. They showed that all methods perform comparably well in estimating the mean and th e variance and that the Z-DI was superior when estimating higher order moments.

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41 Smith compared the bracket-media, bracket-mean, EP-T, and his moment-matching methods for normal, log-normal, beta, and ga mma distributions. He concluded that the bracket-mean method accurately approximates th e mean but generally underestimates all the other moments (Miller and Rice, 1983) a nd that the EP-T approximation accurately estimates the mean and the variance. Keefer (1994) compared the performance of six three-point approximations in estimating the mean, variance, and certainty equi valent, at different risk aversion, for beta and log-normal distributions. He demonstr ated that while 3PDDA methods can exactly match the first three moments of the pdf, they do not approximate the certainty equivalent accurately, except for the EP-T and Z-DI a pproximations. Keefer concluded that the choice between the different approximations should depend on the trade-off between the approximation accuracy and the reliability required. For example, these methods’ accuracy in representing extremely skew ed or peaked pdfs can be very poor. The authors argued that most of the e rrors associated w ith these discrete approximations can be reduced by taking more points; however, the tr ee branches grow in proportion to the number of points, n, rais ed to the power of th e number of uncertain parameters being discretized (Hoyland a nd Wallace, 2001; Smith, 1993). It is also argued that the errors generated from the use of these approximations are acceptable based on the premise that very little informati on is available about th e actual distribution anyways. Risk in the Water Resources Management Literature In the water resources management literature, a large number of research emphasized the need to account for the un certainties and dynamism. Authors used methods such as two stage programmi ng with recourse, chance constrained

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42 programming, dynamic and goal programming, fu zzy analysis, genetic algorithm, neural networks, Bayesian methods, probabilistic anal ysis, and scenario and sensitivity analysis, to name a few, to manage reservoir op eration, groundwater pumping, groundwater contamination, water quality, conjunctive supply, irrigation, etc. (Table 2-1). Most of these studies considered expected va lues of the issue of interest; only a few accounted for the risks of low probability ev ents. Rouhani (1985) minimized the mean square error of differences between measured and predicte d groundwater head values in the design of monitoring networks. Asef a, Kemblowski, Urroz, McKee, and Khalil (2004) used support vector machines to minimize the bound on generalized risk of the difference. Feyen and Gorelick (2004) inspec ted the effect of uncer tainty in spatially variable hydraulic conductivity on optimal groundwater production scheme via a multiple realization approach. Wang, Yuan, and Zhang (2003) applied reliability and risk analysis for reservoir operation and flood analysis us ing Lagrange multipliers. Sasikumar and Mujumdar (2000) used fuzzy sets of low water quality to manage a river system. Ziari, McCarl, and Stockle (1995) intr oduced a variance term in a two-stage stochastic model to manage irrigation scheduling and crop mix. Others used scen ario or sensitivity analysis to model hydrologic uncertainti es (refer to Table 2-1). These methods provide estimates of risk but no means of controlling or managing this risk, other than through trial and erro r (Watkins and McKinney, 1997). In the next chapter, we propose the application of CVaR to water resources management problems. Conclusion The process of Risk Management may be viewed through many lenses depending on the discipline and objectives. In this chapter we introduc ed the fundamentals of risk

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43 and its management. We reviewed the different definitions or risk and risk management and provided a definition that was adopted in this work. We then categorized and briefly described the various general risk manageme nt approaches with emphasis on value-atrisk and conditional value-at-risk concepts. The approaches were compared and the advantages and drawbacks were outlined. As representing uncer tainty is as important as modeling it, we also described and compared different scenario generation techniques. Finally, we presented a review of the various risk management modeling approaches in the field of water resources management. In the following chapter, we present an application of the recommended approaches to a case of water resources management.

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44 Table 2-1. The use of DMuU techni ques in water resources management Application Optimization technique Reservoir operation Two-stage (Ferrero et al., 1998; Huang and Loucks, 2000; Loucks, 1968; Wang and Adams, 1986) Chance-constrained (Abrishamshi et al., 1991; Askew, 1974; Azaiez et al., 2005; Ouarda and Labadie, 2001; ReVelle et al., 1969) Dynamic (Ben Alaya et al., 2003; Burt, 1964; Chaves et al., 2003; El Awar et al., 1998; Estalrich and Buras, 1991; Karamouz and Mousavi, 2003; Kelman et al., 1990; Mousavi et al., 2004b; Nandalal and Sakthivadivel, 2002; Philbrick and Kitanidis, 1999; St edinger and Loucks, 1984; Stedinger et al., 1984; Trezos and Yeh, 1987; Wang et al., 2003) Fuzzy sets (Chang et al., 1997; Chang et al., 1996; Chaves et al., 2004; Hasebe and Nagayama, 2002; Maqsood et al., 2005; Maqsood et al., 2004; Mousavi et al., 2004a) Bayesian networks (Wood, 1978) Multistage stochastic programming (Pereira and Pinto, 1985; Pereira and Pinto, 1991; Watkins et al., 2000) Optimal control (Georhakakos and Marks, 1987; Hooper et al., 1991) Neural networks and genetic algorithms (Akter and Simonovic, 2004; Hasebe and Nagayama, 2002; Ponnambalam et al., 2003) Groundwater management Two-stage stochastic (Wagner et al., 1992) Chance constrained programming (C han, 1994; Eheart and Valocchi, 1993; Hantush and Marino, 1989; Morgan et al., 1993; Tung, 1986; Wagner, 1999; Wagner and Gorelick, 1989) Dynamic(Andricevic, 1990; Andricevic and Kitanidis, 1990; McCormick and Powell, 2003; Provencher and Bu rt, 1994; Whiffen and Shoemaker, 1993) Optimal control (Georgakakos and Vlasta, 1991; Whiffen and Shoemaker, 1993) Neural networks and genetic al gorithms (Hilton and Culver, 2005; Ranjithan et al., 1993) Scenario and sensitivity analysis (Aguado et al., 1977; Burt, 1967; Feyen and Gorelick, 2004; Flores et al., 1975; Gorelick, 1982; Gorelick, 1987; Hamed et al., 1995; Kaunas and Haimes, 1985; Maddock, 1974; Mao and Ren, 2004) Bayesian networks (B atchelor and Cain, 1999) Fuzzy sets (Bogardi et al., 1983; Dou et al., 1999) Other (Asefa et al., 2004; Be ll and Binning, 2004; Rouhani, 1985; Tiedeman and Gorelick, 1993; Wagner and Gorelick, 1987)

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45 Table 2-1. Continued Application Optimization technique Water quality Chance constrained (Fujiwara et al., 1988; Huang, 1998) Goal programming (Al-Za hrani and Ahmad, 2004) Genetic algorithm (He et al., 2004) Bayesian networks (Ricci et al., 2003; Varis, 1998; Varis and Kuikka, 1999) Neural networks (Boger, 1992) Fuzzy sets (Baffaut and Chameau, 1990; Chang et al., 2001; Chaves et al., 2004; Jowitt, 1984; Julien, 1994; Koo and Shin, 1986; Lee and Chang, 2005; Lee and Wen, 1997; Liou et al., 2003; Liou and Lo, 2005; Ning and Chang, 2004; Sasikumar and Mujumdar, 2000) Scenario and sensitivity analysis (Chu et al., 2004; Kawachi and Maeda, 2004a; Kawachi and Maeda, 2004b; Mao and Ren, 2004; Vemula et al., 2004) Floodplain Eutrophication Water transfer Estuary Conjunctive management Lake/wetland Irrigation Planning Hydrologic Cycle Runoff Two stage (Lund, 2002) Fuzzy sets (Esogbue et al., 1992) Neural networks (Sahoo et al., 2005) Bayesian networks (Despic and Simonovic, 2000) Simple recourse (Somlyody and Wets, 1988) Bayesian networks (Borsuk et al., 2004) Two stage (Lund and Israel, 1995) Optimal control (Zhao and Mays, 1995) Chance constrained (Nieswand and Granstrom, 1971) Evolutionary annealing (Rozos et al., 2004) Sensitivity analysis (Escudero, 2000; Jenkins and Lund, 2000) Mean/Variance (Maddock, 1974) Two stage (Cai and Rosegrant, 2004; Watkins and McKinney, 1998; Ziari et al., 1995) Dynamic (Bergez et al., 2004) Two stage (Lund and Israel, 1995) Bayesian networks (Varis and Kuikka, 1999) Fuzzy sets (Hobbs, 1997) Chance constrained (Dupacova et al., 1991; Loucks, 1976) Bayesian networks (Batchelor and Cain, 1999) Fuzzy sets (Suresh and Mujumdar, 2004) Scenario and sensitivity analysis (Pallottino et al., 2005) Goal programming (Sutardi et al., 1995) Bayesian networks (Bromley et al., 2005) Fuzzy sets (Alley et al., 1979; Babovic et al., 2002; Bender and Simonovic, 2000; Chen and Fu, 2005; Faye et al., 2005; Slowinski, 1986; Sutardi et al., 1995; Virjee and Gaskin, 2005; Yi and Zhang, 1989) Information gap (Hipel and Ben-Haim, 1999) Fuzzy set (Cheng et al., 2002)

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46 CHAPTER 3 COMPARISON OF RISK MANAGEME NT TECHNIQUESFOR A WATER ALLOCATION PROBLEM WITH UNCERTAIN SUPPLIES A CASE STUDY: THE SAINT JOHNS RI VER WATER MANAGEMENT DISTRICT In this chapter, we applied and compared different risk management techniques to a water resources management problem, where ri sk is quantified as cost. These methods are the expected value, scenario model, twostage stochastic programming with recourse, and CVaR. They were built into a mixed in teger fixed cost li near programming framework. Five models were developed: (1 ) a deterministic expected value model, (2) a scenario analysis model, (3) a two-stage stochastic model with recourse, (4) a CVaR objective function model, and (5) a CVaR constraint model. Uncertainty was introduced via water supplies. Assumi ng continuous normal distribution for the allowable withdrawals, two disc rete distributions with equal expected values, or means, and different standard deviations were developed based on the method developed by Miller and Rice (1983). The two different central dispersion parameters were assumed for the additional assessment of extreme events effect on the results. The result is 9 models formulations. To compare the perf ormance of the different formulations, the models were applied to a case study, usi ng the Saint Johns River Water Management District, SJRWMD, Priority Water Resource Caution Areas, PWRCA, in East-Central Florida, ECF as a study area. The diagram in Figure 3-1 exhibits the plan of this chapter.

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47 Figure 3-1. Chapter 3 organizational diagram. Model Formulation Five models were developed: (1) a de terministic expected value model; (2) a stochastic single stage scenario model; (3) a two-stage stochastic model with recourse, which is the base of, (4) a CVaR objective function model and (5) a CVaR constraints model. Uncertainty was introduced via water supplies. Two discrete distributions, with equal expected values, or means, of supplie s, were developed by assuming a standard normal underlying continuous distributions; for additional assessment of extreme events effect on the results, two different central dispersion parameter, variance, were assumed for each distribution, leading to the nota tions (a) and (b). The result is 10 5 2 models formulations denoted 1(a), 1(b) 2(a), 2(b), 3(a), 3(b), 4(a), 4(b), 5(a), 5(b). Note that since the expected value of supplies is the same for both assumed distributions, models Model Formulation Scenario Definition Study Area Definition of decision vari ables, state variables, constraints, and different formulations Description of the uncertainty representation process Definition of the water allocation case study Results and Analysis Presentation and comparison of results from different models Conclusions Summary of main findings

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48 1(a) and 1(b) are equivalent and hence denoted by 1, reducing the total number of formulations to 9. This section presents th e different model formul ations, their decision variable and problem data, objec tive function, and constraints. Objective Function The problem at hand is a fixed cost problem. Given several options (i,j,k), each of which corresponds to one of m supply sources i (i=1,…,m), one of n locations j (j=1,…,n), and one of capacity levels k (1,..., k), the decision makers have to decide in which of the options to invest in which time period t, in a way to incur minimum cost over the planning horizon (t=1,…,T), while satisfying the situation’s constraints. Once an investment is made, the corresponding op tion will be available for the remainder of the planning horizon. The objective is to minimize the sum of two main terms: (1) the total fixed costs, FC, corresponding to the initial inve stments, or capital costs, CC, incurred to make the chosen options available from the chosen times and the corresponding yearly operati on and maintenance costs, OMC, and (2) the total variable costs, ContC, corresponding to the co ntinuous operational cost s of withdrawing water from each option after it is made av ailable and the total penalty costs, PC, penalizing unsatisfied demand, or alternative source cost, AC from using an alternative source: m i n j l k T t t ijkt ijkt ijkt ijktX O X C OMC CC FC11111 T t t t m i n j l k T t ijkt t ijkt T t m i n j l k ijkt t t m i n j l k T t ijkt ijktD p x p c x D p x c PC ContC VC1 1111 1111 1111 Note that the last term of VC is a constant and is irrelevant in the optimization.

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49 Decision Variables Xijkt denotes the decision variable for the investment option (i,j,k) at time t: it is a binary variable which assumes the value of 0 if investment option (i,j,k) will not be made available at time t and its corresponding investment will not be made, or 1 if option (i,j,k) will be made available and its corre sponding investment will be made. xijkt denotes how much water will be withdrawn from option (i,j,k) in period t. Problem Data Sijkt is the total capacity of option (i,j,k) in period t. Wit is the total capacity of source i in period t. Cijkt is the fixed cost of making option (i,j,k) available at time t; it is a one-time investment cost. For example, you could have 1t ijktijkCC, where Cijk is the nominal fixed cost of making option (i,j,k) available and (0,1) is a discount factor representing the time value of money. Oijkt is the O&M cost incurred every period t starting the time of making option (i,j,k) available; it is a yearly cost. cijkt is the unit cost of supplying water from option (i,j,k) in period t. For example, you could have 1t ijktijkcc, where cijk is the nominal unit cost of withdrawing water from option (i,j,k) and (0,1) is a discount factor representing the time value of money. pt or is the unit cost of not supplying water in period t. For example, you could have p pt t1, where is the nominal penalty/alternative cost and (0,1) is a discount factor representing the time value of money. The pena lty/alternative cost could either be the unit cost for acquiri ng water from an altern ative supply, or it could be a penalty cost to indicate that shortages are undesirable. Constraints The problem is subject to different sets of constraints depending on the formulation; below is a descripti on of all the used constraints: 1. For all i,j,k,t: 1 t ijktijktijk x SX, which ensures that water will be withdrawn from option (i,j,k) in period t only if option (i,j,k) was made available before or at time t.

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50 2. For all t: 0111 m i n j l k ijkt tx D, which states that the tota l water made available in period t from all options should not exceed the total water demand in period t. Note that any shortage will be penalized; or t m i n j l k ijktD x 111 3. For all i,t: 11 n ijktit jk x W, which states that for each source i, the maximum allowable withdrawal in period t should not be exceeded. 4. For all i,j: 111T ijkt ktX, which ensures that for each source i and location j, only one capacity k can be chosen; also, investment option (i,j,k) will be built at most once. 5. For all i,j,k,t: {0,1}ijktX, which are the binary constraints on the investment choice variables. 6. For all i,j,k,t: 0ijktx, which are the non-negativity c onstraints on the quantities of water withdrawn. Summarizing, the entire optimization model now reads: Deterministic Expected Value Model The deterministic expected value model, model 1, treats the uncertainty of supplies by averaging them into one number, the expect ed value of allowed withdrawals, resulting in the following formulation:

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51 m i n j l k T t ijkt t ijkt m i n j l k T t t ijkt ijkt ijkt ijktx p c X O X C1111 11111min subject to T t l k n j m i all for x T t l k n j m i all for X n j m i all for X T t m i all for W E x T t all for x D T t l k n j m i all for X S xijkt ijkt l k T t ijkt n j it l k ijkt m i n j l k ijkt t t ijk ijkt ijkt,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 1 0 ,..., 1 ; ,..., 1 1 ,..., 1 ; ,..., 1 ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 111 11 111 1 Scenario Model The scenario model, model 2, is equivalent to a single-stage deterministic model. Unlike the expected value met hod, which considers only the e xpected value of supplies, the uncertain supplies are considered by different independent scenarios; the result is the minimization of the expected value of th e objective function over the scenarios set. m i n j l k T t s ijkt t ijkt m i n j l k T t t s ijkt ijkt ijkt ijkt sx p c X O X C E1111 11111min Or, m i n j l k T t s ijkt t ijkt m i n j l k T t t s ijkt ijkt ijkt ijkt sx p c X O X C E1111 11111min subject to

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52 S s T t l k n j m i all for x T t l k n j m i all for X n j m i all for X S s T t m i all for W x S s T t all for x D S s T t l k n j m i all for X S xs ijkt s ijkt l k T t s ijkt n j s it l k s ijkt m i n j l k s ijkt t t s ijkt ijkt s ijkt,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 1 0 ,..., 1 ; ,..., 1 1 ,..., 1 ; ,..., 1 ; ,..., 1 ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 111 11 111 1 Two-Stage Stochastic Model with Recourse The two-stage stochastic model, model 3, like the scenario model, represents uncertainties by a scenarios set. Unlike the scenario model, however, in this model, the scenarios are linked by a set of variables, referr ed to as the first-stage variables; the first stage variables are the same for all the scen arios, and hence are scenario independent. The expected value operator is applied onl y to the terms involving the rest of the variables, referred to as the second stage variables; the second stage variables are scenario dependent and hence they are differe nt for each scenario. The result is a fixed cost problem with recourse. In other words, the model allows the decision-maker to make two sets of decision: (1) a first stage decision, an un certainty independent decision, consisting, in this case, of the fixed costs of making supplies availabl e, and (2) a second stage decision, an uncertainty dependent decision consisting, in this case also, of th e variable costs of allocating the resources from the supplies ma de available by the first decision. This

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53 process allows the decision maker to postpone his allocation decisions until information about the uncertainties is revealed. The model is formulated below. m i n j l k T t s ijkt t ijkt s m i n j l k T t t ijkt ijkt ijkt ijktx p c E X O X C1111 11111min subject to S s T t l k n j m i all for x T t l k n j m i all for X n j m i all for X S s T t m i all for W x S s T t all for x D S s T t l k n j m i all for X S xs ijkt ijkt l k T t ijkt n j s it l k s ijkt m i n j l k s ijkt t t ijk ijkt s ijkt,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 1 0 ,..., 1 ; ,..., 1 1 ,..., 1 ; ,..., 1 ; ,..., 1 ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 111 11 111 1 CVaR Objective Function Model The CVaR objective function model, model 4, is based on the two-stage recourse model. In this case however, the CVaR operator is applied to the extreme events of high costs scenarios only, correspond ing to a cumulative probability ; high costs events are associated with high risk events of water shortage. m i n j l k T t s ijkt t ijkt m i n j l k T t t ijkt ijkt ijkt ijktx p c X O X C CVaR1111 11111min subject to

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54 S s T t l k n j m i all for x T t l k n j m i all for X n j m i all for X S s T t m i all for W x S s T t all for x D S s T t l k n j m i all for X S xs ijkt ijkt l k T t ijkt n j s it l k s ijkt m i n j l k s ijkt t t ijk ijkt s ijkt,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 1 0 ,..., 1 ; ,..., 1 1 ,..., 1 ; ,..., 1 ; ,..., 1 ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 111 11 111 1 CVaR Constraint Model. The CVaR constraint model, model 5, is al so based on the two-stage recourse model. In this case however, the CVaR of high risk events are restrained to a value rather than minimized, while the to tal two-stages costs are minimized. m i n j l k T t s ijkt t ijkt s m i n j l k T t t ijkt ijkt ijkt ijktx p c E X O X C1111 11111min subject to

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55 S s T t l k n j m i all for x T t l k n j m i all for X n j m i all for X S s T t m i all for W x S s T t all for x D S s T t l k n j m i all for X S x x p c X O X C CVaRs ijkt ijkt l k T t ijkt n j s it l k s ijkt m i n j l k s ijkt t t ijk ijkt s ijkt m i n j l k T t s ijkt t ijkt m i n j l k T t t ijkt ijkt ijkt ijkt,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 1 0 ,..., 1 ; ,..., 1 1 ,..., 1 ; ,..., 1 ; ,..., 1 ,..., 1 ; ,..., 1 0 ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 1 ; ,..., 111 11 111 1 1111 11111 Scenario Generation As noted in Chapter 2, a major issue in stochastic optimization is the accurate representation of underlying un certainties. In the case wh ere those uncertainties are represented by continuous dist ributions, this may be done by scenario generation through discretization. The various discretizati on methods were discussed in Chapter 2. Discretization allows for the approxim ation of the uncertain parameter, underlying continuous distribution, f, by a discrete probability function, or mass distribution function, mdf, denoted P, concentrated on a finite number of scenarios S s , 1 corresponding to the stochastic parameter s with corresponding probabilities s sP p ,such that 11 S s sp In this case, corresponds to the available water supply from different projects, denoted itW in the previously presented model formulation section.

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56 We chose to use the an optimization fram ework of the method developed by Miller and Rice (Miller and Rice, 1983), who suggest ed a moment-matching approximation that allows the constructi on of an n-point moment-matchi ng discrete distribution, which consists on n pairs of probabil ity-value for the uncertain parameter, chosen to represent the pdf of a continuous parameter that matche s 2n-1 moments of the latter. There method is based on the Gaussian quadrature tec hnique of numerical integration, which approximates functions integrals by a linear set of polynomials summation. The values, sz for different total number of pairs (i.e. order of polynomial approximation), n, and many functions can be easily obtained from tables in published literature (Abramowitz, 1965; Beyer, 1978; Stroud and Secrest, 1966), as the solutions to the polynomials. For example, for a standard normal distributi on we used Table 5, page 218 in Stroud and Secrest. The table gives values of the integral dx x f ex2, szI; hence to obtain the values, sz, for a standard normal distribution, the table values were multiplied by 2. The values corresponding probabilities ar e then obtained as a solution to N linear equations, by substituting these values into the set of equations for the moments of the approximate discrete distribution. Follo wing the model formulat ion notation, these equations are of the form N s q s s qp1 where 1 2 ,..., 2 1 N q is the moment order. For example q is the sum of probabilities, the mean, and the variance for q equal to 0, 1, and 2, respectively; they are equal to 1, 0, and 1, respectively, for a standard normal distribution.

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57 We chose to represent our uncertain parame ter by a continuous normal distribution, with equal mean and different standard de viations, to simulate different cases of parameters scattering and extreme events. The normal distribution is ap plicable to a very wide range of phenomena that occur in nature, i ndustry, and research and is the most widely used in statistics. Physical meas urements in meteorolog ical and hydrological experiments as well as manufact ured parts are often adequate ly represented using normal distributions. In addition, the normal distribution is in many conditions a good approximation to other distributi ons. It is also the asympto tic form of the sum of random variables or parameters under a wide range of conditions, if the underlying phenomena are additive (DeGroot, 2002; Evans et al., 2000; Walpole, 1989). We present here the results for standard normal distributions based on 10 pairs, corresponding to 10 scenarios (Tables 3-1 and 3-2). The values, s can be transformed for specific normal distributions by as simple transformation using the specific mean, and standard deviation, For normal distribution, this is obtained using the formula s sz, where sz are the values obtained for a standard normal distribution. Note that for a standard normal distribution, symmetric around the mean, 0, i.e., with s n s 2, if N is even, the sum is zero for all odd q. In addition, its symmetry allows the reduction the number of linear equations to 2n, as s n sp p 2. In the case of 10 scenarios, 10 N, nineteen moments may be matched. Since N in this case is even, all odd moments are equa l to zero, and the linear system reduces to 10 equations with 10 unknowns, corresponding to the 0th, 2nd, 4th, 6th, 8th, 10th, 12th, 14th, 16th, and 18th moments. As s sp p 5, the unknown probabilities ar e reduced to 5 and for all practical reasons we restrict oursel ves to the first even moments equations.

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58 2 / 105 2 / 15 2 / 3 2 / 1 2 / 18 5 5 8 4 4 8 3 3 8 2 2 8 1 1 6 5 5 6 4 4 6 3 3 6 2 2 6 1 1 4 5 5 4 4 4 4 3 3 4 2 2 4 1 1 2 5 5 2 4 4 2 3 3 2 2 2 2 1 1 5 4 3 2 1 p p p p p p p p p p p p p p p p p p p p p p p p p The thq moments were calculated from th e distributions’ moments generating function, M; 2 22 1 e M for for a normal distribution, where the thq moments equals the thq derivative of M at zero, 0qM; for details refer to the literature (Beaumont, 1986; DeGroot, 2002). This system was solved using the excel What’s Best Optimization package, where the sum of probabilities was minimized subject to the set of the first two, three, four, and five linear equations as constr aints. Additional constraints were imposed setting higher probabilities for values closer to the mean The resulting sets of probabilities for matching up to the 2nd, 4th, 6th, and 8th moments, a total of f our minimization problems, are presented in Table 3-3. Note that matching the 4th moment was a source of infeasibility in the minimization problems; hence, its corresponding constraint was not satisfied, i.e., it was relaxed, in the four minimizations. Matching up to the 8th moment resulted in violation of the 6th moment too. The moments from each minimization in comparison with the continuous standard nor mal distribution moment s are presented in Table 3-2. The resulting histograms and smoot h distributions are compared in Figures 32 and 3-3. The visual comparison of the results reve als that constraining all the moments results in the best discrete approximation. To confirm this observation, a least-square regression analysis was run on the results. Analysis of Table 3-3 and Figure 3-4

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59 regression findings verified our obser vation. Constraining up to the 8th moments resulted in the best fits, with the highest r2, slope closest to unity, and in tercept closest to zero, and smallest corresponding errors. Case Study Area Groundwater serves as the primary water sour ce for most of the State of Florida. Faced with continuous grow th, local utilities throughout the state are working on identifying alternative viable and ec onomic sources of potable water. Recognizing the need to develop new sour ces and plan well in advance, the Saint Johns River Water Management District, SJRWMD, one of five water management districts in Florida (Figure 35), initiated in the year 2000 several water supply plans that (1) identified limit of fresh groundwater in Priority Wate r Resources Caution Areas, PWRCA (Figure 3-5), which are areas where existing and anticipated sources of water and conservation efforts are not adequate; (2) identified alternative water resources options and development projects with cost data and likely project users; (3) initiated the Alternative Water Supply Construction Co st Sharing Program in 1996 to provide cooperative funding for the construction of alte rnative water supply f acilities (Vergara, 2004; Wilkening, 2004; Wycoff and Parks, 2005). Water Demand The total water demand of the SJRWMD PWRCA is projected to linearly reach 830 MGD in the year 2025 (Table 3-4). This demand consists of public supply, domestic, agriculture and recreational irrigati on, commercial, industrial, institutional, and power generation water needs (Wilkening, 2004; Wycoff, 2005).

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60Table 3-1. Standard normal discrete dist ribution approximation for N=10 and up to the 2nd, 4th, 6th, and 8th moments constraints q 0th, 2nd 0th, 2nd, 4th 0th, 2nd, 4th, 6th 0th, 2nd, 4, 6, a 8th szI sz pdf cdf pdf cdf pdf cdf pdf cdf -3.43616 -4.85946 0.001534 0.001534 0.02464 0.02464 0.000000 0.000000 0.000000 0.000000 -2.53273 -3.58182 0.001534 0.003067 0.02464 0.04928 0.031523 0.031523 0.016867 0.016867 -1.75668 -2.48433 0.001534 0.004601 0.02464 0.07392 0.053653 0.085176 0.068881 0.085748 -1.03661 -1.46599 0.247699 0.252301 0.02464 0.098559 0.053653 0.138829 0.068881 0.154629 -0.3429 -0.48494 0.247699 0.500000 0.401441 0.500000 0.361171 0.500000 0.345371 0.500000 0.3429 0.48494 0.247699 0.747699 0.401441 0.901441 0.361171 0.861171 0.345371 0.845371 1.03661 1.46599 0.247699 0.995399 0.02464 0.92608 0.053653 0.914824 0.068881 0.914252 1.75668 2.48433 0.001534 0.996933 0.02464 0.95072 0.053653 0.968477 0.068881 0.983133 2.53273 3.58182 0.001534 0.998466 0.02464 0.97536 0.031523 1.000000 0.016867 1.000000 3.43616 4.85946 0.001534 1.000000 0.02464 1.000000 0.000000 1.000000 0.000000 1.000000 Table 3-2. Moments constraints Discrete Approximation Moments 0th, 2nd 0th, 2nd, 4th 0th, 2nd, 4th, 6th 0th, 2nd, 4, 6, a 8th q Moment Constraints Moment Constraints Moment Constraints Moment Constraints Moment 0 0.5 = 0.500 = 0.500 = 0.500 = 0.500 2 0.5 = 0.500 = 0.500 = 0.500 = 0.500 4 1.5 0.656 Not= 1.197 Not= 0.936 Not= 0.871 6 7.5 2.324 18.870 = 7.500 Not= 5.737 8 52.5 26.258 382.538 79.716 = 52.500

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61Table 3-3. Least square regression analysis results Regression Analysis Results Moments Constraints 0th, 2nd 0th, 2nd, 4th 0th, 2nd, 4th, 6th 0th, 2nd, 4, 6, a 8th Slope 0.501 7.464 1.538 1.008 Intercept -0.218 -12.580 -1.398 -0.575 Standard error of slope 0.0179 0.361 0.0397 0.0192 Standard error of intercept 0.425 8.558 0.943 0.454 Correlation coefficient r2 0.996 0.993 0.998 0.999 Standard error on ordinate 0.808 16.266 1.793 0.864

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62 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a) 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a)Moments Matched: 0th, 1st, and 2nd Moments Matched: 0th, 1st, 2nd, and (4th) 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a) 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a)Moments Matched: 0th, 1st, 2nd, (4th), and 6th Moments Matched: 0th, 1st, 2nd, (4th), 6th, and (8th) Figure 3-2. Discretized standard normal distribution (m oments in parentheses, i.e., (qth), are the unmatched moments)

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63 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a) 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a)Moments Matched: 0th, 1st, and 2nd Moments Matched: 0th, 1st, 2nd, and (4th) 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a) 0 0.1 0.2 0.3 0.4 0.5 -4.86-3.58-2.48-1.47-0.480.481.472.483.584.86 0 0.2 0.4 0.6 0.8 1 pdf (a) cdf (a) Moments Matched: 0th, 1st, 2nd, (4th), and 6th Moments Matched: 0th, 1st, 2nd, (4th), 6th, and (8th) Figure 3-3. Discretized standard normal distribution (m oments in parentheses, i.e., (qth), are the unmatched moments)

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64 y = 1.0077x 0.5751 R2 = 0.9989 y = 7.4641x 12.58 R2 = 0.993 y = 0.5012x 0.2177 R2 = 0.9962 y = 1.5383x 1.398 R2 = 0.998 0 100 200 300 400 0100200300400 Continuous pdf MomentsDiscrete mdf Moments 2 4 6 8 Figure 3-4. Moments least-squa re regression analysis plots Figure 3-5. Priority water resource cau tion areas in the SJRWMD, Florida, USA (Vergara, 2004; Wilkening, 2004)

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65 Table 3-4. SJWRMD caution area wate r demand projections (Wilkening, 2004) Year Demand 2010 676 2011 686 2012 697 2013 707 2014 717 2015 727 2016 738 2017 748 2018 758 2019 768 2020 779 2021 789 2022 799 2023 809 2024 820 2025 830 Water Supply The demand is currently supplied from th e Floridan Aquifer groundwater. It is projected that the aquifer’s capacity for the caution areas, 670 mgd, will be reached before the year 2010 (Wilken ing, 2004; Wycoff, 2005). With that in mind, the SJRWMD identified three main potential alternative sources for water, with a total capac ity of 335 mgd: (1) 175 mgd fr om the Saint Johns River basin (SJR) at seven locations; (2) 100 mgd fr om the Lower Oklawaha River (LOR),3 at one location; and (3) 60 mgd from Collocated seawat er (CSW) at three locations. These rates correspond to the maximum allowable withdrawal s from the sources; they are assumed to 3 Note that although 100 mgd may be withdrawn from the Oklawaha River, a project capacity of 21.5 mgd has been suggested since this source is at a remote loca tion with respect to the priority caution areas of the SJRWMD, rendering the transmission costs from this source prohibitive.

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66 be the source of uncertainty in the mode ls (Vergara, 2004; Wilk ening, 2004; Wycoff and Parks, 2005). In sum, the SJRWMD identified 11 a lternative water s upply projects; the approximate locations of these projects ar e shown in Figure 3-6. Various project development scenarios were examined by the district to provide examples of various water supply quantities and costs associated wi th each project. The projects details are presented in Table 3-5. Each of these projects has a maximum allowed withdrawal, which is the total water that can be withdrawn from this source while subject to constraints such as upstream and downstream water levels, sea and riverine ecology, and aesthetics. Cost estimates were provided fo r several possible average capacities at each location; only one of these capacities may be chosen at each location. The maximum capacities are only design capacities and not demand capacities. The design, permitting and construction of new source facilities wi ll likely take 5 to 10 years (Vergara, 2004; Wycoff and Parks, 2005). Figure 3-6. Approximate locations of pot ential alternative water supply projects (Vergara, 2004; Wilkening, 2004)

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67 Water Cost Table 3-5 presents a summary of the estimated total capital, O&M, and unit costs of water from different projec ts locations and capacities. The method followed for these estimates are briefly summarized in this secti on. Note that the last row in Table 3-5 stands for existing groundwater supplies, he nce, they are not part of the proposed projects, have no associated fixed cost, a one significant figure unit cost of $1/1000 gallons, and an O&M cost of 0.2/100 gallons based on year 2000 Dollar (Wycoff, 2005). Total capital cost is the sum of construction cost, nonconstruction capital cost, land cost, and land acquisition cost. The Oper ation and Maintenance, O&M, Cost is the estimated annual cost of operating and ma intaining the water supply project when operated at average day capacity. The Equiva lent Annual Cost is the total annual life cycle cost of the water suppl y project based on facility se rvice life and time value of money. Equivalent annual cost, expressed in dollars per year, acc ounts for total capital cost and O&M costs with facility operating at average day design capacity. Finally, the Unit Production Cost is Equiva lent annual cost divided by annual water production. The unit production cost is expre ssed in terms of dollars per 1,000 gallons produced (Wycoff and Parks, 2005). These costs are in year 2003 Dollar. They were convert ed to future years Dollar values using the Construction Cost Index (M ichaels, 1996), or CCI; results are tabulated in Appendix A. CCI is estimated by Engineering News R ecords, ENR, on a monthly basis, and represents the underlying trends of construction costs in the USA. It is determined by several factors such as labor, materials, and others (ENR, 2005). Table A-1 lists historical yearly averages of CCI for the years 1908 – 2005. Figure A-1 is a plot of these

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68 values to obtain a best fit of the year – CCI relationship. Using the Equation of the best fit, projections of CCI were calculated fo r the years 2000 – 2025; CCI was also estimated from the equation for the years 2000 – 2005 for consistency. But what is the significance of CCI and how is it used? Actually, CCI is used as a measure of change of costs between different years. This change, CCI for consecutive time periods, years, is estimated using Equation 2. The change in CCI can also be calculated for non-consecutive year s using Equation 3, such as t t '. 1001 1 t t t tCCI CCI CCI CCI 2 100' t t t tCCI CCI CCI CCI 3 To estimate the value of costs at time t, tC, the cost at time 't, 'tC is multiplied by tCCI with t t', Equation 4. 't t tC CCI C 4 Scenario Generation The scenarios were defined around the uncer tainty in supply. The previously described projects are assigned deterministic ex pected values of wa ter supply designated as average withdrawals cap acities. Assuming a norma l distribution, two mass distribution functions, (a) and (b), each with 10 scenarios, were defined, assuming two different standard deviations, at 5 and 10 pe rcent of the mean of each supply, for each supply (Table 3-6). This was based on the Miller and Rice (1983) moment matching method in an optimal discretization framework. Both methods were discussed earlier in this chapter.

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69Table 3-5. Supply sources, capacities, and costs4 System Capacity (mgd) Source (Maximum Allowed Withdrawals) (i) Location ( j ) AverageMaximum k Capital Cost ($M) O&M Cost ($M/yr) Unit Production Cost ($/1,000 gallons) Near SR 520/528 (1 j) 20 30 1 k 189 7.56 3.03 Near SR 50 (2 j) 10 15 1 k 91 3.81 3.00 50 75 1 k 457 18.71 2.93 30 45 2 k 238 11.29 2.74 20 30 3 k 217 7.56 3.27 Near Lake Monroe (3 j) 9.6 14.4 4 k 84 3.67 2.94 10 15 1 k 81 3.80 2.80 50 75 2 k 372 18.80 2.63 Near Lake Monroe (4 j) 100 150 3 k 714 37.20 2.55 20 30 1 k 210 7.56 3.22 10 15 2 k 105 3.80 3.25 50 75 3 k 447 18.80 2.91 Near DeLand (5 j) 100 150 4 k 871 37.20 2.84 Saint Johns River Near Lake George (6 j) 33 49.5 1 k 386 12.40 3.41 10 15 1 k 55 2.20 1.66 Saint Johns River Basin (175 MGD) 1i Taylor Creek Reservoir Near Cocoa (7 j) 25 37.5 2 k 134 6.00 1.68 Lower Ocklawaha River (100 MGD) 2i Lower Ocklawaha River Putnam (1 j) 21.5 32.25 1 k 255 5.45 2.94 4 (Vergara, 2004; Wilkening, 2004)

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70Table 3-5 Continued Source (Maximum Allowed Withdrawals) (i) Location ( j ) System Capacity (mgd) Capital Cost ($M) O&M Cost ($M/yr) Unit Production Cost ($/1,000 gallons) 10 15 1 k 90 5.00 3.33 20 30 2 k 180 9.40 3.23 FP&L Cape Canaveral Power Plant (1 j) 30 45 3 k 274 13.60 3.20 10 15 1 k 90 4.50 3.20 20 30 2 k 177 8.40 3.07 Indian River Lagoon Reliant Power Plant (2 j) 30 45 3 k 268 12.10 3.28 5 7.5 1 k 83 3.10 5.06 10 15 2 k 121 5.20 3.99 Collocated Seawater (60 MGD) 3i Intracoastal Waterway Near New Smyrna Beach (3 j) 15 22.5 3 k 159 7.60 3.61 Floridan Aquifer5 670 0 48.91 1 5 Wycoff, R. (2005). "Phone Interview." Consultant, Saint Johns River Water Management District.

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71Table 3-6. Scenarios of suppl y capacities at 5% and 10% STD Supply Source (Mean) 1 (175.0) 2 (21.5) 3 (60.0) GW (670) Scenario, s 75 8 (a) 5 17 (b)07 1 (a) 15 2 (b) 3 (a) 6 (b) 5 33 (a) 67 (b) 1 132.4797 89.9594 16.27608 11.05215 45.42161 30.84322 507.208 344.416 2 143.659 112.3181 17.64954 13.79908 49.25453 38.50906 550.0089 430.0178 3 153.2621 131.5243 18.82935 16.1587 52.54702 45.09404 586.7751 503.5502 4 162.1726 149.3452 19.92406 18.34812 55.60203 51.20407 620.8894 571.7787 5 170.7568 166.5136 20.97869 20.45739 58.54519 57.09039 653.7547 637.5093 6 179.2432 183.4864 22.02131 22.54261 61.45481 62.90961 686.2453 702.4907 7 187.8274 200.6548 23.07594 24.65188 64.39797 68.79593 719.1106 768.2213 8 196.7379 218.4757 24.17065 26.8413 67.45298 74.90596 753.2249 836.4498 9 206.341 237.6819 25.35046 29.20092 70.74547 81.49094 789.9911 909.9822 10 217.5203 260.0406 26.72392 31.94785 74.57839 89.15678 832.792 995.584

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72 Results and Discussion As explained in the previous sections, fi ve model formulations were considered: the expected value of supplies formulation, m odel 1, the scenario or expected value of costs model, model 2, the tw o-stage stochastic model w ith recourse, model 3, the CVaR minimization model, model 4, and the CVaR constraint model, model 5. As the objective of this work was to demonstrate the tradeoffs between costs and risk, as CVaR, the next sections focus on the Pareto e fficient frontier. Th e significance of this method is to provide for a given CVaR/cost value, the minimum cost/CVaR that can be obtained without exceeding that CVaR/cost value. Focusing on model 5, model 5 was run for three constraints co rresponding to three confidence levels 50, 75, 80, 85, 90, 95, and 99 per cent; the models are designated as 5 5-50, 5-75, and 5-95, respectively. In addition, each of the models 5 was run at different values of designated as B b ,... ,..., 1 in an increasing order; the values of ranged between the smallest feasible value to the value at and after which no change was observed. Different runs are referred to by the model number, the confidence level of the constraint, and the number of the value used in the constraint, i.e., b 5 For the confidence level the Pareto efficient frontier, for a given confidence level was generated by plotting a point, correspondi ng to total cost on the abscissa and CVaR on the ordinates, for each model run, or value of ; there exists a different frontier for each confidence level and a set of frontiers for each confidence level To compare solutions of models 1, 2, and 3, the costs solutions from these models were added as

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73 points to the efficient frontiers plots; this a llows to see whether these models solutions are efficient in the Pareto sense, i.e., undominated. Note that model 3 can be obtained from model 5 by deleting the CVaR constraint or setting ; hence, the solution to model 3 corr esponds to one of the endpoints of the Pareto efficient frontier. Also note that model 4 minimizes CVaR of the total cost and does not control costs that are below the confidence level at which CVaR is minimized; as a result, the two-stage total co st result of this m odel does not possess any practical significance; this model, however, finds the smallest for which a feasible solution can be obtained, provi ding the other endpoint of th e Pareto efficient frontier, hence, model 4 is used to find this point and not as a model by itself. The models were ran for two normal dist ributions of allowable withdrawals, itW, with equal means and different standard devi ations, namely 5 and 10 percent. In this section, we present and compare the diffe rent model results w ithin and across both distributions, (a) and (b). 5% Standard Deviation The results for the distribution (a), corres ponding to 5 percent st andard deviations are summarized in Figures 3-7 to 3-14. Figure 3-7 presents plots of the change in cost with CVaR where is the confidence level, of the constraint; these plots correspond to the efficient frontiers at different The figure demonstrates that, for all values of the cost increases as decreases; in other words, a tighter constraint results in an increase in cost. In ad dition, intuitively, a higher value of results in a higher range of The range of consists of a lower minimum below which the model

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74 can no longer be feasible and an upper limit beyond which the solution is independent of the constraint. The lower limit coincide with the minimized CVaR, for of model 4; the upper limit corresponds to CVaR, for calculated from the solution of model 3. The cost solutions obtained fr om model 1, model 2, and model 3, i.e., expected value solution, individual scenario s, and two-stage stoc hastic solutions, are plotted in the graphs at values of equal to the cost for m odel 1 and model 2 scenarios and at CVaR for corresponding to for model 3. Note that the efficient frontier delineates a risk-return space correspon ding to the lowest risk for a given level of return or –cost or the best possible return or minimum cost for a given level of risk; points below the concave frontier line correspo nd to inefficient or suboptimal solutions and points above the frontie r are infeasible. Figure 3-7 shows that, for all (1) the expected value solution cannot be achieved fo r any level of risk, (2) only two of the scenarios of model 2, corresponding to high risk values, are feasible but inefficient, and (3) the two-stage stochastic solution is an efficient solution corres ponding to the lowest possible cost and high level of risk. Figures 3-8 to 3-14 demons trate the dependency of CVaR, calculated at different confidence levels, on (A) (B) and (C) cost, for each model 5 simulated, 50, 75, 80, 85, 90, 95, and 99 percent, respectively. For each for different values, the model’s CVaR were recalculated and plotted; for example, if the model was run for 10 different constraints values 10 lines were plotted, mon itoring the change of the recalculated solution CVaR with the change in the confidence level, the constraint, and the cost solution. The results were consistent with each other and the theory for all plots;

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75 a higher confidence level corr esponds to a higher solution CVaR, a tighter constraint corresponds to a lower solution CVaR, and a higher cost corresponds to solution a higher CVaR. -18 0 1.091.17 -Cost -18 0 0.604.40 -Cost 1 2 3 50%50% -18 0 1.171.29 -Cost -18 0 0.604.40 -Cost 1 2 3 75%75% -18 0 1.201.35 -Cost -18 0 0.604.40 -Cost 1 2 3 80%80% Figure 3-7. Efficient frontier for 50, 75, 80, 85, 90, 95, and 99 percent, 5% STD.

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76 -29 0 1.251.46 -Cost -29 0 0.604.40 -Cost 1 2 3 85%85% -29 0 1.341.46 -Cost -29 0 0.604.40 -Cost 1 2 3 90%90% -31 0 1.501.91 -Cost -31 0 0.604.40 -Cost 1 2 3 95%95% Figure 3-7 Continued

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77 -34 0 2.072.77 -Cost -34 0 0.604.40 -Cost 1 2 3 99%99% Figure 3-7 Continued.

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78 0 34 0100 CVaR 3 5-50-1 5-50-2 5-50-3 5-50-4 5-50-5 5-50-6 5-50-7 5-50-8 5-50-9 5-50-10 increasing(A) 0 34 1.101.16 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -170 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-8. Change of CVaR with (A) (B) and (C) cost for 50%, 5% STD.

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79 0 34 0100 CVaR 3 5-75-1 5-75-2 5-75-3 5-75-4 5-75-5 5-75-6 increasing(A) 0 34 1.171.29 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -170 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-9. Change of CVaR with (A) (B) and (C) cost for 75%, 5% STD.

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80 0 34 0100 CVaR 3 5-80-1 5-80-2 5-80-3 5-80-4 5-80-5 5-80-6 5-80-7 5-80-8 5-80-9 5-80-10 5-80-11 5-80-12 5-80-13 5-80-14 increasing(A) 0 34 1.201.35 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -170 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-10. Change of CVaR with (A) (B) and (C) cost for 80%, 5% STD.

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81 0 34 0100 CVaR 3 5-85-1 5-85-2 5-85-3 5-85-4 5-85-5 5-85-6 5-85-7 5-85-8 5-85-9 5-85-10 5-85-11 5-85-12 5-85-13 5-85-14 increasing(A) 0 34 1.251.46 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -280 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-11. Change of CVaR with (A) (B) and (C) cost for 85%, 5% STD.

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82 0 34 0100 CVaR 3 5-90-1 5-90-2 5-90-3 5-90-4 5-90-5 5-90-6 5-90-7 5-90-8 5-90-9 increasing(A) 0 34 1.341.46 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -280 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-12. Change of CVaR with (A) (B) and (C) cost for 90%, 5% STD.

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83 0 34 0100 CVaR 3 5-95-1 5-95-2 5-95-3 5-95-4 5-95-5 5-95-6 5-95-7 5-95-8 5-95-9 5-95-10 5-95-11 5-95-12 increasing(A) 0 34 1.501.91 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -300 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-13. Change of CVaR with (A) (B) and (C) cost for 95%, 5% STD.

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84 0 34 0100 CVaR 3 5-99-1 5-99-2 5-99-3 5-99-4 5-99-5 5-99-6 5-99-7 5-99-8 5-99-9 5-99-10 5-99-11 5-99-12 5-99-13 5-99-14 5-99-15 increasing(A) 0 34 2.072.77 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(B) 0 34 -330 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-14. Change of CVaR with (A) (B) and (C) cost for 99%, 5% STD.

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85 10% Standard Deviation The results for the distribution (a), corres ponding to 5 percent st andard deviations are summarized in Figures 3-15 to 3-22. Figure 3-15 presents plots of the change in cost with CVaR where is the confidence level, of the constraint; these plots correspond to the efficient frontiers at different The figure demonstrates that, for all values of the cost increases as decreases; in other words, a tighter constraint results in an increase in cost. In ad dition, intuitively, a higher value of results in a higher range of The range of consists of a lower minimum below which the model can no longer be feasible and an upper limit beyond which the solution is independent of the constraint. The lower limit coincide with the minimized CVaR, for of model 4; the upper limit corresponds to CVaR, for calculated from the solution of model 3. The cost solutions obtained fr om model 1, model 2, and model 3, i.e., expected value solution, individual scenario s, and two-stage stoc hastic solutions, are plotted in the graphs at values of equal to the cost for m odel 1 and model 2 scenarios and at CVaR for corresponding to for model 3. Note that the efficient frontier delineates a risk-return space correspon ding to the lowest risk for a given level of return or –cost or the best possible return or minimum cost for a given level of risk; points below the concave frontier line correspo nd to inefficient or suboptimal solutions and points above the frontie r are infeasible. Figure 3-15 shows that, for all (1) the expected value solution cannot be achieved fo r any level of risk, (2) only two of the scenarios of model 2, corresponding to high risk values, are feasible but inefficient, and (3) the two-stage stochastic solution is an efficient solution corres ponding to the lowest possible cost and high level of risk.

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86 Figures 3-16 to 3-22 dem onstrate the dependency of CVaR, calculated at different confidence levels, on (A) (B) and (C) cost, for each model 5 simulated, 50, 75, 80, 85, 90, 95, and 99 percent, respectively. For each for different values, the model’s CVaR were recalculated and plotted; for example, if the model was run for 10 different constraints values 10 lines were plotted, mon itoring the change of the recalculated solution CVaR with the change in the confidence level, the constraint, and the cost solution. The results were consistent with each other and the theory for all plots; a higher confidence level corres ponds to a higher solution CVaR, a tighter constraint corresponds to a lower solution CVaR, and a higher cost corresponds to solution a higher CVaR. -11 0 1.921.98 -Cost -17 0 0.0018.00 -Cost 1 2 3 50%50% Figure 3-15. Efficient frontier for 50, 75, 80, 85, 90, 95, and 99 percent, 10% STD.

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87 -17 0 2.773.01 -Cost -17 0 0.0018.00 -Cost 1 2 3 75%75% -17 0 3.203.32 -Cost -17 0 018 -Cost 1 2 3 80%80% -27 0 3.884.04 -Cost -27 0 0.0018.00 -Cost 1 2 3 85%85% Figure 3-15 Continued

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88 -29 0 5.075.16 -Cost -29 0 0.0018.00 -Cost 1 2 3 90%90% -31 0 6.436.58 -Cost -31 0 0.0018.00 -Cost 1 2 3 95%95% -34 0 10.1810.35 -Cost -34 0 0.0018.00 -Cost 1 2 3 99%99% Figure 3-15 Continued

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89 0 34 1.921.98 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-50-1 5-50-2 5-50-3 5-50-4 5-50-5 5-50-6 5-50-7 5-50-8 increasing(B) 0 34 -170 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-16. Change of CVaR with (A) (B) and (C) cost for 50%, 10% STD.

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90 0 34 2.773.01CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-75-1 5-75-2 5-75-3 5-75-4 increasing(B) 0 34 -170 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-17. Change of CVaR with (A) (B) and (C) cost for 75%, 10% STD.

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91 0 34 3.203.32CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-80-1 5-80-2 5-80-3 5-80-4 5-80-5 5-80-6 5-80-7 5-80-8 5-80-9 5-80-10 increasing(B) 0 34 -170 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-18. Change of CVaR with (A) (B) and (C) cost for 80%, 10% STD.

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92 0 34 3.884.04CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-85-1 5-85-2 5-85-3 5-85-4 5-85-5 5-85-6 increasing(B) 0 34 -270 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-19. Change of CVaR with (A) (B) and (C) cost for 85%, 10% STD.

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93 0 34 5.075.16CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-90-1 5-90-2 5-90-3 5-90-4 5-90-5 5-90-6 5-90-7 5-90-8 increasing(B) 0 34 -290 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-20. Change of CVaR with (A) (B) and (C) cost for 90%, 10% STD.

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94 0 34 6.436.58 CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-95-1 5-95-2 5-95-3 5-95-4 5-95-5 5-95-6 5-95-7 increasing(B) 0 34 -310 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-21. Change of CVaR with (A) (B) and (C) cost for 95%, 10% STD.

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95 0 34 10.1810.35CVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 Cost increasing(A) 0 34 0100 CVaR 3 5-99-1 5-99-2 5-99-3 5-99-4 5-99-5 5-99-6 5-99-7 5-99-8 5-99-9 5-99-10 increasing(B) 0 34 -340 -CostCVaR 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 99 1 2 3 increasing(C) Figure 3-22. Change of CVaR with (A) (B) and (C) cost for 99%, 10% STD.

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96 Analysis This section’s purpose is to demonstrat e and emphasize the importance of using risk management techniques, such as CVaR, versus traditional expected value, scenario analysis, and two stage modeli ng. This is accomplished by comparing the cost and CVaR results obtained for the different models and the two distributions. The examination of Figures 3.7 to 3-23 re vealed the following general results, for both standard deviations: In model 5, keeping constant, as increases, CVaR increases; in other words, an increase in confidence level results in higher CVaR value. In model 5, keeping constant, as increases, cost increases; this is consistent with the theory and logic that imposing hi gher confidence levels results in incurring higher costs. In model 5, keeping constant, as increases, CVaR decreases; in other words, the less tight the CVaR constraint imposed on model 5, the smaller the CVaR values obtained. In model 5, keeping constant, as increases, cost decreases; in other words, the less tight the CVaR constraint imposed on model 5, the smaller the costs values obtained. In model 5, for each confidence level at which 'CVaR is constrained, there exists a minimum value below which the model is infeasible; this value corresponds to the minimum 'CVaR solution generated by model 4. In model 5, for each confidence level at which 'CVaR is constrained, there exists a maximum value above which the model yields the same solution; this solution corresponds to that obtained by running model 3. In model 5, the higher the confidence level at which 'CVaR is constrained, the higher the values minimum and maximum limits are. The expected value model, model 1, yiel ded the lowest cost solution, followed by the two-stage mixed integer fixed cost mode l, model 3; model 3 solution laid at the lower limit of model 5 solutions which was higher for smaller values, i.e.,

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97 tighter constraints. The solutions of model 2 varied with each scenario, from values lower than model 1 and greater than model 3. Hence, including CVaR in the formulation resulted in higher costs than the expected value and two-stage stochastic models; this is attributed to the fact that model 5 specifically considers high cost ev ents by minimizing them or constraining them. Model 4 does not allow for the control of all the scenarios, hence, it has no practical significance. Both the expected value a nd scenario models, models 1 and 2, underestimated costs, when compared with models 3 and 5. As expected, model 1 run for both distributions, i.e. mode ls 1(a) and 1(b), resulted in the same cost for the different distri butions; as mentioned earlier, these models are identical as both distribution have the same expected value of allowable withdrawals, which is used in the expected value model, i.e. model 1. A higher standard deviation resulted in hi gher scattering of costs in the scenario analysis formulation. A higher standard deviation resulted in hi gher costs for the two-stage stochastic model, model 3(a) versus model 3(b). A higher standard deviat ion resulted in higherCVaR calculated at different confidence levels in the two-stage stochastic model, model 3. -34 0 1.082.77 -Cost (A) -34 0 1.0011.00 -Cost 50% 75% 80% 85% 90% 95% 99%(B) Figure 3-23. Efficient frontiers for 50, 75, 80, 85, 90, 95, and 99 percent, (A) 5% STD and (B) 10% STD.

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98 Conclusions and recommendations The purpose of this chapter was to compare the performance of different risk quantifying techniques for a wa ter allocation model. We compared five models: 1. expected value, 2. scenario analysis, 3. mixed integer two-stage stochastic with recourse, 4. CVaR minimization, and 5. CVaR constraints. Uncertainties in the model parameters were represented via two normal distributions with equal mean and different standard deviations; they were derived based on a standard normal probability distribution function discretization technique developed by Miller and Rice (1983). The case study was based on data from the Saint Johns Water Management District; the uncertainty was assumed to be in the allowable withdrawals from the different supplies. The main findings were: 1. The larger the number of scenarios and th e moments matched used to form a mass distribution function, mdf, by discretizati on a probability distri bution function, pdf, the more representative the mdf is of the pdf. 2. Using the expected value of the uncertain parameter underestimates cost, and hence shortage. 3. Using the expected value method resulted in identical cost estimates for different standard deviations distribu tions with identical mean. 4. Using the expected value of scenario s costs underestimates cost, and hence shortage. 5. Using a two-stage stochastic mixed intege r formulation with recourse allowed for an improved representation of uncertainti es and real life decision-making and higher estimates of costs. 6. The use of CVaR in the two-stage stochastic re course mixed integer formulation allowed for the optimization and control of high risk events.

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99 7. Minimizing CVaR allowed for the control of high risk, or high cost – high shortage events; however, it did not control the lower risk events. 8. Constraining CVaR while minimizing cost allowed th e control of high risk events while minimizing the costs of all events. 9. A higher standard deviation of the un derlying distribution of the uncertain parameter resulted in, generally, higher associated costs and CVaR. 10. CVaR exhibited continuous and consistent behavior with respect to the confidence level when compared to VaR (Figure 3.24). for a given confidence level, the associated risk as CVaR was higher when the standard deviation was higher (Figure 3-25). 0.9 2.8 0100 CVaR(A) 0.7 10.6 0100 CVaR VaR CVaRCVaR+ CVaR(B) Figure 3-24. Comparison of CVaR, CVaR, CVaR, and VaR values calculated using model 3, (A) 5% STD and (B) 10% STD 0 11 0100 CVaR 10% 5% Figure 3-25. Comparison of CVaR values calculated using model 5% and 10% STD

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100 We suggest that future resear ch considers the following: 1. Assess the performance the different mode ls for non-symmetrical, i.e. skewed, distribution functions to represent uncertainties. 2. Study the effect of different sources a nd distributions of uncertainties on the different models performance, their intera ctions, and their subadditivity properties. 3. Investigate the applicability of the differe nt methods on non-linear systems such as contaminant transport and gr oundwater drawdown problems. 4. Apply the CVaR method on water shortage, contam inant transport, or aquifer drawdown, rather than cost; i.e. tr eat risk as a non-monetary concern!

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101 CHAPTER 4 UTILITY, GAME, AND WATER: A REVIEW Decision making under situations of conflict are referred to as conflict theories; they stem from the theories of economic, political, and social behavior. Although these theories date back as far as Plato and Aristo tle, the integration of quantitative elements into them did not start until the end of th e nineteenth century; the rigorous use of quantitative data, statistical analysis, mathem atical modelling, and computer simulations in political and social sciences did not star t until the 1950s with the theoretical advances in conflict models, the theory of games and co alitions, systems theory and other theories (Alker, 1973). The resulting formal theories are generally classified based on the extent of (a) rationality and (b) strategy assumed in the definition and pred iction of individual’s preferences and choices in a decision making situation. In terms of the first assumption, rationali ty, a formal model of choice in decision theory may be broadly distinguished as strong ly rational or weakly rational. A rational model is also referred to as normative, prescr iptive, and recommendatory. It is concerned with criteria of cohere nce, consistency, and rationalit y in preference pa tterns that are represented by axioms. On the other hand, a we akly rational model is also referred to as descriptive, behavioral, psyc hological, predictive, positive, and explanatory. It is interested in actual choice behavior rather than in criteria for “right” decisions; it seeks to identify patterns in preferences and actual choices, develop models to characterize them, and, eventually, predict them (Fishburn 1988).In terms of the second assumption,

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102 strategy, a formal model of choice in decisi on theory under risk and uncertainty may be an individual behavior model, game theoretic model, or social choice model, depending on the extent of individuals’ interactions and type of goa ls involved in the decision making process. A cross classification of th e models using the above explained criteria results in five groups of models presented in Table 4-1. Table 4-1. Choice model classification Assumption about aggregation and strategic levels Individual Behavior Models Game Theoretic Models Aggregate Behavior Models Normative Strong Rationality Expected Utility Theory Standard Game Theory Assumption about choice Descriptive Weak Rationality Non-Expected Utility Theory Behavioral Game Theory Social Theory The next sections present a review of the development of the different classes of models, moving by column and then by row in Table 4.1. In other words, the next section starts with a summary of the formal conception of expected utility theory (column 1, row 1), followed by its critique as a hum an choice predictive tool in decision making situations, and an overview of its alternatives (column 1, row 2). These theories of

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103 individual choice are at the core of strate gic decision making, mainly, game theory. Hence, the discussion is continued with the examination of standa rd game theory, its main taxonomy, and solution concepts, setting th e stage for behavioral game theory. The theory of social choice is beyond the scope of this work, and thus was not reviewed. We finally conclude this chapter with an extensiv e review of the applic ation of game theory to water resources decision making problems. The diagram in Figure 4-1 exhibits the plan of this chapter. Figure 4-1. Chapter 4 organizational diagram Theory of Preference Preferences are expressed via a utility function and ordered using a preference comparison relationship; in other words, a knowledge of an indivi dual’s utility function would allow the prediction of that individual’s preferen ces, and hence, behavior (Machina, 1982). In the following sections, th e preference relationship is defined. Using Linear Expected Utility Theory Critique of Expected Utility Theory Preference relation, rationality axioms, and expected utility Framing, preference reversal, common consequence, reference point, etc. Non Expected Utility Models Methods review (linear ge neralizations, weighted utility theory, prospect theory, etc.) Game Theory Formulations, classifications, solution concepts, extensions, and alternatives Application of game to water resources Review

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104 this definition, the different variants of preference theories are distinguished and compared in the context of choi ce behavior and utility theories. Preference Comparison Relationship As mentioned, preferences are assessed vi a a comparison relation. Formally, define a binary relationship of preference comparisons, on a convex set P of probability measures. obeys the following properties, for all P r q p ,: Property 1. obeys preference / indifference ~: q p ~ if neither q p nor p q, q p if either q p or q p ~ Property 2: obeys comparability iff: q p or q p or q p ~ Property 3. obeys asymmetry iff: p q not q p Property 4. ~ obeys reflex ivity and symmetry iff: p p ~ and p q q p ~ ~ Property 5. obeys transitivity iff: if q p and r q r p Property 6. obeys monotonicity and convexity iff: s q r p s r q p 1 1 s q r p s r q p 1 ~ 1 ~ ~. Monotonicity reflects the property of stoc hastic dominance preference, where one prospect dominates another if it can be obtained from it by shifting probability from lower to higher; in other words, more is better (Machina, 1987a; Machina, 1987b). Whether the preference relati onship observes or violates one or another of these properties was, and still is, the object of concern and the impetus that lead to the development of a wide range of utility theories and underlying choice behavior assumptions. The following sections elucidate the development of the different utility or preference theories, their basi c assumptions, and th eir limitations. There is a tremendous

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105 number of literature providing reviews on thes e topics with different emphasis (Fishburn, 1988; Fishburn, 1989; Kahneman et al., 1997; Laville, 2000; Mach ina, 1982; Machina, 1987a; Machina, 1987b; Schoemaker, 1982; Starmer, 2000). Expected Utility Theory The first expression of preferences in te rms of expected u tility was provided by Bernoulli. Almost two centuries later, the concept of expect ed utility was formalized in the context of strategic decision theory under risk and uncertainty, as a basic element of game theory by von Neumann and Morgenster n. Savage extended this theory to situations of uncertainty, as distinguished from risk by Knight (1921), or Bayesian subjective uncertainty. The latter two linea r expected utility models for preference comparisons under risk and uncertainty are the main paradigms of rational decision making. Bernoulli’s expected utility th eory, von Neumann and Morgenstern’s linear expected utility theory (LEUT), and Savage’s linear expected utility theory (SEUT) are discussed herein. Bernoulli’s utility theory In the early years of development of the probability theory, risky decisions, mostly monetary ones, were evaluated usi ng the expected value criterion, X xx xp p x E,, on a set X of monetary gains (0x) and losses (0 x), with x p being the probability distribution on X Early in the eighteenth century, Bernoulli and others observed that individuals often violate the principle of maximizing expe cted return (Machina, 1987). In 1973, Bernoulli evoked the pr incipal of diminishing marginal utility: a person’s subjective value of wealth does not increase linearly with wealth, but rather at a decreasing rate, with the value of wealth bei ng inversely proportional to wealth, hence, a

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106 logarithmic function of the latter. In addi tion, Bernoulli, argued that a risky decision should be evaluated based on the subjective expected value – the first statement of expected utility – taking into account present wealth. Ber noulli’s theory presupposed the existence of a cardinal utility scale (Mach ina, 1987; Fishburn, 1988; Starmer, 2000). Formally, defining X x as a state of wealth, w, in X 0w as its initial state, w as its value, and x p as the probability distributi on of its states, Bernoulli’s expected utility theory is defined using the expected subjective value: X xx p x w p E0, 4.1 with x p preferred to x q in the probability set XPwhen q E p E, Note that Bernoulli viewed utility as an intensively measurable quantity that has nothing to do with probability or risk; his utility was interchangeably used with subjective value, moral worth, or psychic satis faction. The assumption of cardinality or measurability was a matter of debate; generally, it was perceived as an ordinal method of preference assessment that can be used to or der preference over alternatives and cannot be measured using an objective scale (Edgeworth, 1887; Fishburn, 1982a; Machina, 1987b; Starmer, 2000). In the 1920 – 1930 period, with the emergence of axiomatic theories of mathematics, interest in measuring utility, cardinal utility, was reincited by several authors (Frisch, 1927; Lange, 1934) who provided an axioma tic basis to Bernoulli’s expected utility theory, rendering the value function unique up to a positive linear transformation that change the origin and/or scale of the u tility but not the shape of its function; i.e., for two value functions and on X for all p and q,

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107 X x all for a b x a x iff q E p E q E p E 0 ' , (Machina, 1987; Fishburn, 1988). Utility was st ill, however, perceived as an ordinal tool of preference comparisons. What was missing was the formalization of the notion of choice in risky or uncertain situations. Already acholars (Hic ks, 1931; Marschak, 1938) had a sense that people should form preferences over dist ributions. Alternative hypothesis included ordering random ventures via their means, variances, etc. Linear expected utility theory In 1944, Von Neumann and Morgenstern (von Neumann and Morgenstern, 1944), VNM, developed an alternative expected utility theory, the linear expected utility theory, LEUT. Although LEUT preserved the mathemati cal form, individual’s preference order, and uniqueness up to positive linear transfor mations of Bernoulli’s expected utility theory, it differed radically from the latter (Fishburn, 1988). The difference is depicted by defining the concept of lottery or random prospect. Let x be an outcome from a set X of outcomes. Let p be a simple probability measure on X nx p x p x p p,..., ,2 1, where ix p is the probability of outcome X xi. Define XP as the set of simple probability measures on X A particular random prospect or lottery p is a point in XP, such as XP p In Bernoulli’s utility theory, an individual’s utility are from the outcome or consequence, X x In VNM, an individual’s utility is from the lottery or probability distribution. In other words, preferences are formed over lo tteries and from these preferen ces over lotteries, combined with objective probabiliti es, an individual deduces the underlying preferences on outcomes. Thus, in VNM’s theory, unlike Bern oulli's, preferences over lotteries logically

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108 precede preferences over outcomes, i.e., pr eferences over altern ative probability distributions over weal th (Machina, 1982). In addition, VNM major contributions to economics more generally was to show that if an agent has preferences defined ove r lotteries, then there is a utility function U, assigning a utility, u, to every lottery XP p that represents these preferences: X xx p x u p u E U) ( 4.1 This is the linear expected utility theory, LEU T. Note that this form does not assume linearity in the outcomes, x ; however, it maintains lineari ty in the probabilities. Graphically, this is illustrated in Figure 4-2, using Marschak’s triangle. Consider a set of lotteries 3 2 1, ,x p x p x p P over outcomes 3 2 1x x x with 13 2 1 x p x p x p or 3 1 21x p x p x p ; the lotteries are represented in the unit triangle in the plane of 3 1,x p x p. The individual indifference curves are the solutions to the linear equation: constant ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) (3 3 3 1 2 1 1 3 1 i i i ix p x u x p x p x u x p x u x p x u U, consisting of parallel st raight line with slope 2 3 1 2x u x u x u x u Note that upward movement in the triangle increases 3x p at the expense of 2x p and leftward movement decreases 1x p to the benefit of 2x p; these northwest movements generate stochastically dominating lotteries, i.e., mo re preferred indifference curves (Machina, 1987).

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109 0 1 01 p(x1)p(x3) Figure 4-2. Expected utility indifference curves The authors showed that U can be derived from a set of axioms defining rational choice of fully informed individuals. Th e rationality assumption is the principal conjecture of LEUT. Rational choice theory assumes that individuals are rational, i.e., they are perceived as homo economicus with preferences over outcomes from which they choose more preferred alternatives over less preferred alternatives, knowing that other individuals are also rational. For a discussion the reader is referred to the literature (Kuhberger, 2002; Vriend, 1996). The theory was first axiomatized by VNM as the motivation of human behavior in decision ma king situations. It is formally based on three principal axioms: (1) order, (2) inde pendence, and (3) con tinuity, which are the necessary conditions for the exis tence of the LEUT function. A formal analysis follows. A real-valued function U preserves the order of on P and is linear in the convexity operation if and only if it obeys the following axioms, for all P p, P q and 1 , 0 : Axiom 1: Order: on P is a weak order: q U p U q p

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110 Axiom 2: Independence: r q r p q p 1 1 Axiom 3: Continuity: r p q and q r p r q q p 1 1 Preferences are defined as rati onal if they are complete and transitive. That is, that the decision maker is able to compare all of the alternatives, and th at these comparisons are consistent. If uncertainty is involved, then the independence axiom is often assumed in addition to rational preferences. The first axiom, order preserving axiom, stat es that the decision maker must be able to state preferences for all outcomes of a lottery. The unstated assumption is that more is always preferred to less. Orderability implie s comparability and transitivity, or properties 2 and 5 of the preference rela tionship described earlier. Axiom 2, independence or linearity assu mption, imposes a restriction on the functional form of the preference function, constraining U to be linear over the set of distributions, i.e., linear in probabilities, where q U p U q p U 1 1. This axiom implies consistency a nd coherence by requiring that if p is preferred to q, then a non trivial convex combination of p and r is preferred to the similar combination of q and r (Machina, 1982, 1987; Fishburn, 1988). The third axiom, continuity, implies that for r q p then there are 1 0 such that q r p 1 and r p q 1. It states that given any three lotteries strictly preferre d to each other, r q p it is possible to form the combination of the most and least preferred lotteries, p and r such that their compound is strictly preferred to the middling lottery q and the combination p and r so that the middling lottery q is strictly preferred to th eir compound. Substitutability in this case implies that the certainty

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111 equivalent can always be substituted for the lott ery and vice versa, i.e. indifferent options can be substituted by one another without ch anging their preference. Axiom 3 is also known as the Archimedean axiom; it prevents one measure from being infinitely preferred to another. These axioms define what is known as th e rationality assumption of behavior, the fundamental hypothesis of LEUT and SEUT, desc ribed next. Alternative restatements and reaxiomatizations of these axioms have been developed by seve ral authors (Herstein and Milnor, 1953; Marschak, 1950; Samuelson, 1952). Subjective linear expected utility theory In the VNM LEUT, probabilities were assumed to be "objective". In this respect, they followed the "classical" view that randomness and probabilities. However, many statisticians and philosophers, tracking back to Bayes (1763) had long objected to this view of probability, arguing that randomness is not an objectively measurable phenomenon but rather a knowledge phenomenon (Barnard and Bayes, 1958). In this view, probabilities are r eally a measure of the lack of knowledge about the conditions and thus merely represent our beliefs about the experiment. Ramsey (1926) asserted that probabilities are related to the knowledge possess ed by a particular individual alone; it is governed by personal belief, hence, it is subj ective. He suggested a theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities, providi ng the first attempt at an axiomatization of choice under uncertainty, more than a decade before von Neumann and Morgenstern's attempt (Barnard and Bayes, 1958).6 Independently of Ramsey, de Finetti (de Finetti, 1931) also 6 Note that Ramsey's paper was published posthumously in 1931.

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112 suggested a similar derivation of subjective probability. Ramsey and de Finetti argued that subjective probabilities can be in ferred from observation of people's actions. The Ramsey-de Finetti view was famously axiomatized and developed into a full theory by Savage in his revolutionary Foundatio ns of Statistics (Savage, 1954). Savage followed a decade after von Neumann and Mo rgenstern’s LEUT with the axiomatization of subjective linear expected utility, SEUT in which subjective probabilities for uncertain events as well as von Neuman-M orgenstern utilities are derived from preferences between uncer tain prospects, setti ng Bayesian statistical decision theory on a firm foundation (Fishburn, 1989; Fishburn, 1991). Savage’s SEUT, was followed by Anscombe and Aumann (Anscombe and Auma nn, 1963) simpler axiomatization which incorporated both subjective and objective pr obabilities into a single theory. Another subjectivist approach was initiated by Arrow’s (Arrow, 1951; Arrow, 1958) statepreference approach to uncertainty, wh ich did not involve the assignment of mathematical probabilities, wh ether objective or subjective. Multiattribute expected utility Multidimensionality refers to situations in which preferences depend on many factors, including time sequences. A multiattribute utility function, nx x x U,..., ,2 1, represents preferences in decisions under ri sk involving multiattribute outcomes of the form nx x x x,..., ,2 1 in nX X X X,..., ,2 1 The assessment of such a utility function is simplified by decomposing it into algebraic combinations of functi ons of the individual variables and on interactive te chniques that allow utility maximization without having to assess the utility function (Farquhar, 1975; Farquhar a nd Fishburn, 1981; Fishburn, 1971a; Fishburn, 1979; Fishburn, 1980; Fishburn, 1984c; Fishburn and Keeney, 1974).

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113 The two simplest decompositions of U are the additive and multiplicative forms, equations 4.6 and 4.7, re spectively (Fishburn, 1988): n n nx U x U x U x x x U ... ,..., ,2 2 1 1 2 1 4.6 1 ... 1 1 ,..., ,2 2 1 1 2 1 n n nx kU x kU x kU x x x kU 4.7 where constant 0k. Additive decompositions were first sugge sted by Debreu (Debreu, 1960), Luce and Tukey (Luce, 1966; Luce and Tukey, 1964) and Pollak (Pollak, 1967). Fishburn (Fishburn, 1965) showed that additive decompositions of LEUT are possible iff preferences between prob ability distributions p and q depend only on their marginal distributions for the different outcomes, i.e., ip and iq on iX, or q p q q p pn n~ ,..., ,...,1 1 Fishburn (1967) reviewed the different methods of estimating additive utilities. Multiplicative decompositions were studied by Keeney and Raifa (Keeney, 1974; Keeney and Raiffa, 1976). Both decompositions have been axiomatized by several authors (Farquhar, 1975; Farquhar and Fis hburn, 1981; Fishburn, 1965; Keeney, 1971; Keeney, 1973; Keeney, 1981). Descriptive Limitation s of LEUT and SEUT While the previously presented linear exp ected utility theory is mathematically elegant and computationally convenient, num erous empirical and experimental studies have demonstrated systematic and predictabl e violations of its basic axioms, raising concerns with its descriptive merits, i.e., wh ether the expected utility theory provides a sufficiently accurate representation of actual ch oice behavior. Empirical studies from the early 1950s revealed a variety of patterns in ch oice behavior that appe ar inconsistent with LEUT, namely, its underlying axioms. Shackle (Shackle, 1949), A llais (Allais, 1953b),

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114 Edwards (Edwards, 1962a), Ellsberg (Ellsberg, 1961) were among the first to empirically challenge the expected utility theory and to suggest substant ial modifications. Influential experimental studies have reinforced the need to rethink much of the theory, leading to the development of the theory of bounded ra tionality. Bounded rationality theory was formulated by Simon (Simon, 1955); its ma in hypothesis are th e limitations of individuals capabilities, which bound them away from following the optimal rational choice and lead them to display satisficing behavior, i.e., choose options that are good enough given the limitations of the task a nd their cognition (Kuhberger, 2002; Vriend, 1996) The following paragraphs summarize the main detected discrepancies of LEUT as a predictor of human choice. Interested r eaders are referred to excellent reviews by Schoemaker (1982), Machina (1987), and Fishburn (1988). Violation of independence The most denied, relaxed, or abandoned a nd investigated as both descriptive and normative principle of choice is the independe nce axiom, or equivalently, linearity in probabilities assumption. Many scholars have shown persistent system atic violations of the independence axiom. Detailed reviews of these violations are present in Machina (1982, 1987). It is noteworthy that authors such as Mach ina (1982) argued that LEUT basic concepts and predictions are independent of this axiom as they may be obtained by only assuming smooth preferences. Common Consequence. Allais (1953a; 1953b; 1979) pl ayed a central role in stimulating and shaping theoretical devel opments in non expected utility theory by providing the earliest example of criticism of the linear expected utility theories as a representation of human prefer ences (Starmer, 2000). Using empirical examples, Allais

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115 showed that human preference should be represented by fa nning-out indifference curves, rather than parallel ones (Figure 4-3). This is known as Allais’ Paradox. 0 1 01 p(x1)p(x3) Figure 4-3. Fanning-out effect Following Allais, several experimental st udies confirmed Alla is observations; it was later known as a special case of th e common consequence effect (Hagen, 1979; Kagel et al., 1990; Kahneman and Tversky, 1979; Loomes, 1998; Loomes and Sugden, 1987; MacCrimmon and Larsson, 1979; M acCrimmon and Toda, 1969; Raiffa, 1968; Tversky and Kahneman, 1981). Consider the distributions 1p, 2p, 3p, and 4p, with 1p denoting the prospect leading x with certainty, 2pgreater and smaller than x and 3p stochastically dominating 4p. When choosing between probability mixtures, M: 3 1) 1 ( 1p p M versus 3 2) 1 ( 2p p M and 4 1) 1 ( 3p p M versus 4 2) 1 ( 4p p M individuals tend to choose M1 in the first and M4 in the second, even though the independence axiom implies choice s of either M1 and M3, for 2 1p p, or M2 and M4, for 1 2p p; thus, the common consequence effect. This observation states that if the

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116 distribution 3p involves very high outcomes, the indi vidual may prefer not to risk not receiving it and prefer the sure outcome x in 1p over 2p; but, if 4p involves very low outcomes, the individual may be more willing to take risk and prefer 2p over 1p (Machina, 1987). Common Ratio effects. This violation was termed the common ration effect by MacCrimmon and Larsson (1979); the certainty effect of Ka hneman and Tversky (1979) and the Bergen Paradox of Hagen (1979) are sp ecial cases of the common ratio effect (Machina, 1987). It states that if an individual is i ndifferent between a p chance gain x and a p q chance of gain y, then a p q r chance of y will be weakly preferred to a pr chance of x (Cubitt et al., 1998; Machina, 1982). In other words, individuals underweight outcomes that are merely probabl e in comparison with outcomes that are obtained with certainty (Ka hneman and Tversky, 1979). Violation of transitivity Intransitivity in choice under risk has b een observed by several scholars (Edwards, 1962a; Fishburn, 1970; Flood, 1980; Luce, 1956; May, 1954; Tversky and Edward Russo, 1969). In a risky prospects setting, with a, b, and c being outcomes in X probabilities in XP, or both, the indifference relation is intransitive when, b a~, c b~, and c a; the preference relation is intransitive when b a, c b, and c a ~ or c a When > is assumed asymmetric and transitive while ~ is assumed intransitive, > is a partial order, rather than a weak order. Two main types of tr ansitivity violation phenomena are preference reversal and framing effect, discussed below. Preference Reversal. Preference reversal is second to independence failure in the extent to which it has been investigated. Formally, let p and q be risky prospects for

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117 gain outcomes with certainty equivalents p c and q c; the preference reversal phenomenon occurs if for q p q c p c i.e., the individual prefers p to q but would sell p for less than selling q. Given that more gain is preferred to less, > cannot be a weak order when q p and q c p c Edwards (1962a; 1953; 1954), focusing on hu man information processing in the assessment of utility and probabilities, uncovered that people’s behavior reveals preferences among probabilities th at were reversed in the loss domain. Lichtenstein and Slovic (1971; 1973) and Lindman (1971) were the first to experimentally observe the phenomenon of preference reversal. Other experiments were performed (Grether and Plott, 1979; Pommerehne et al., 1997; Pommerehne et al., 1982). In a review of this phenomenon, Slovic and Lichtenstein (1983) and Grether and Plo tt (1979) emphasized information processing in preference judgment. Preference reversal was also observed in several more recent studies (Ben-Dak, 1972; Fishburn and Keeney, 1975; Hausman, 1997; Kahneman, 2003a; Karni and Safra, 1987; Knez et al., 1985; Lichtenstein and Slovic, 1971; Lichtenstein and Slovic, 1973; Loomes et al., 1989; Loomes et al., 1991; Loomes and Sugden, 1983; Lovallo and Kahneman, 2000; Payne et al., 1980; Quattrone and Tversky, 1988; Slovic and Lichtens tein, 1983; Tversky and Kahneman, 1981; Tversky and Kahneman, 1991; Tversky et al., 1990; Tversky and Thaler, 1990). Preference reversal has been interpreted as a failure of procedure invariance, which states that preferences are independent of the method used to elicit them (Starmer, 2000), failure of the transitivity axiom (Loomes and Sugden, 1983), or as a consequence of the invocation of different mental processes for choice and valuation (Slovic, 1995). Several authors attribute this violat ion to the failure of both, procedure invariance and the

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118 transitivity axiom (Loomes et al., 1989; Tversky et al., 1990). Alternativ e explanations were also suggested (Karni and Safra, 1987). This phenomenon is reviewed in several works (Ben-Dak, 1972; Fishburn and Keeney, 1975; Hershey and Schoemaker, 1985; Kahneman, 2003a; Karni and Safra, 1987; Loom es et al., 1989; Loomes et al., 1991; Loomes and Sugden, 1983; Lovallo and Kahneman, 2000; MacCrimmon and Wehrung, 1984; Muney and Deutsch, 1968; Payne et al., 1980; Quattrone and Tversky, 1988; Slovic and Lichtenstein, 1983; Starmer, 2000; Tversky and Kahneman, 1981; Tversky and Kahneman, 1991; Tversky et al ., 1990; Tversky and Thaler, 1990). Framing effect. Theories of choice invoke the ax iom of descriptive invariance, which holds that differences in descriptions that don’t alte r the actual choices should not alter behavior, i.e., preferences are function only of probability distributions of the consequences and not how thes e distributions are described. A framing effect occurs when different descriptions of the problem l ead to different behavi or. Framing violates asymmetry; by placing p and q comparison in different frames, it may be possible to induce a preference for p over q in one, and a preference for q over p in another, i.e. q p or p q depending on the frame of the decision, leading to choosing stochastically dominated prospects. One of the earliest examples of this phenomenon was observed by Slovic (1969) and Payne and Braunstein (1971). Kahne man and Tversky (1979) and Tversky and Kahneman (1981; 1986) provided the first experimental evidence of failure of description invariance; in their experiment, the authors s howed that variation in the presentation of the same problem resulted in variation in the preference ordering. In general, people are more willing to take risk when outcomes ar e described as losses than when the same

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119 outcomes are described as gains (Camer er, 1999; Camerer, 2005; Kahneman, 2003a; Kahneman, 2003b; Kahneman, 2003d; Kahnema n and Tversky, 2003); avoidance of losses may act as a focal point leading to coordination in certain games (Cachon and Camerer, 1996). Other studies c onfirm the framing effect (Hershey et al., 1982; Hershey and Schoemaker, 1980; Moskowitz, 1974). Reflection and reference point effects. In 1948, Friedman and Savage (1948) offered an LEUT function that is locally conc ave about low outcome levels, locally linear at an inflection point, and convex about high out come levels, leading to an S-shaped () function. In a later articl e, the authors (Friedman and Savage, 1952) modified this function to include a terminal concave s ection (). Mosteller and Nogee (1951) and Markowitz (1952) observed that individuals prefer positively skewed distributions over negatively skewed ones. Kahneman and Tv ersky (Kahneman, 1992; Kahneman, 2003c; Kahneman et al., 1991; Kahneman and Tversky, 1979) observed prevalence of risk aversion in gains and risk seek ing in losses as individuals were more likely to take risks when outcomes were described as losses than when the same outcomes were described as gains. This observation imp lies that the utility function U is concave in gains and convex in losses, with an incr easing S-shaped pattern that is steeper for losses than for gains. This is the reflection effect. Mach ina (1987a) observed this effect as a special type of framing effect, the reference point effect. Mar kowitz (1952) and Simon (1955) observed that risk attitudes over gain and lo sses should be explained as changes from a reference point of wealth. Accordingly, authors such as Fishburn, Kahneman, and Tversky (Fishburn, 1988; Kahneman, 1992; Tver sky and Kahneman, 1991) affirmed that people’s utility for wealth depended mainly on changes from present wealth rather than

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120 absolute level of wealth. Other studies (Cohen et al., 1985) concluded that subjects exhibit consistent risk attitudes in gains and lo sses. For an overview refer to the literature (Budescu and Weiss, 1987; Camerer, 1997; Camerer, 2003b; Fishburn and Gehrlein, 1977; Machina, 1982; Neale and Bazerman, 1985). Probability judgment Preston and Baratta (1948) were the firs t to explore whether individuals accounts for chance events at their true mathematical probabilities or whether they systematically distorted probabilities in their presumed expectation maximizing choices. They found that subjects tend to overvalue small proba bilities and undervalue large ones. Marks, Irwin, and Slovic (Irwin, 1953; Marks, 1951; Slovic, 1966) found that subjective probabilities tend to be higher for more de sirable outcomes. Edwards (1953) also revealed preferences among probabilities. C ohen (1985) concluded that probabilities are accounted for precisely in the gains region bu t coarsely in the loss region. Another probability judgment type of bias is repres entativeness, where individuals use heuristic shortcuts to make difficult judgments easie r (Svenson, 1981). Kahneman, Tversky, BarHillel, and Grether (Bar-H illel, 1973; Bar-Hillel, 1974; Grether, 1978; Kahneman and Tversky, 1979) detected a systematic devi ation of probability updating from Bayes law by underweighting prior information and ove rweighting current ones. Kahneman and Tversky also observed that individuals tend to overvalue small probabilities and undervalue large ones; they noted that probabili stic judgments lead to probabilities that violate the mathematical pr operties of probabilities. Additional evidence on the psychology of probabilistic information pro cessing and its biases distortions, and illusions are available in the literature (Edwards, 1962a; Edwards, 1954; Edwards, 1961; Edwards, 1968; Karmarkar, 1978; Machina, 1987a; Machina, 1989; Machina, 2005;

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121 Machina and Schmeidler, 1992; Schoemak er, 1982; Slovic and Lichtenstein, 1971) Kahneman et al. (1982), (Arkes et al., 1986; Tversky and Fox, 1995; Tversky and Kahneman, 1973; Tversky and Kahneman, 1974; Tversky and Wakker, 1995). Non-Archimedean preferences This axiom is regarded more as a technica l convenience than a rationality postulate and has a different formal standing in logic th an the other axioms, where its failure leaves the notion of LEUT intact (Fishburn, 1988; Fi shburn, 1989). Examples of failure of axiom 3 were suggested by GeorgescuRoegen, and Chipman (Chipman, 1960; Georgescu-Roegen, 1954a; Georgescu-Ro egen, 1954b); however, no experimental evidence of its failure have b een presented (Fishburn, 1988). Alternatives to Expected Utility Theory Non expected utility theories, NEUT, were suggested as alterna tives to the linear expected utility theories of von Neumann and Morgenstern and Savage. They were developed in an effort to account for the viol ations of rationality observed in real human choice behavior, which LEUT di d not account for, by relaxing some of the axioms of LEUT. These theories are referred to as non-rational or with bounded rationality. These theories assume that individuals don’t ne cessarily make choices that maximize their utility due to cognitive limitations affecti ng their ability to pr ocess all available information; thus, individuals make decisions th at are less than fully rational. In NEUT, we refer to the preference ordering function as (.)V, rather than (.)U, the utility function in LEUT. The following sections present some of th e main classes of a lternative theories, namely, linear generalizations, intensity th eory, moments theory, generalized EUT, weighted EUT, expectation quotient, prospect theory, rank-dependent expected utility,

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122 disappointment theory, state-de pendent preferences, skew-s ymmetric bilinear theory, regret theory, alpha utility, lo cal linearizati on, and others. Linear generalizations These are generalizations of the von Ne umann and Morgenstern theory that preserve linearity and at least strict prefer ence transitivity. The simplest generalization was obtained by dropping axiom 3, the Archim edean axiom, leading to lexicographic preference ordering (Blume et al., 1991; Chipman, 1960; Fishburn, 1971b; Fishburn, 1980; Fishburn and Graham, 1993; Fishburn an d LaValle, 1998a; Fishburn and LaValle, 1998b; Hausner, 1954; Hull et al., 1973; Park, 1978). Anot her generalization, first axiomatized by Aumann (Aumann, 1962), wa s obtained for Archimedean partially ordered preferences. A third axiomati zation, combining both non-Archimedean and partially ordered preferences was suggest ed by Kannai (Kannai, 1963).This type of LEUT alternatives ran their course by 1970 as it did not accommodate violations of the independence axiom (Fishburn, 1988). Non-linear generalizations With the emergence of new experimental evidence supporting Allais’ (Allais, 1953a; Allais, 1953b) findings of independence ax iom violations, inte rest in non-linear theories emerged, though still only descrip tive, with authors such as Hagen Handa, Karmakar, Kehneman, and Tversky (Hagen, 1979; Handa, 1977; Kahneman and Tversky, 1979; Karmarkar, 1978). They were followe d by other descriptive theories (Chew and MacCrimmon, 1979; Fishburn, 1982b; Machina, 1982; Quiggin, 1982); these theories were normatively refined by seve ral authors (Chew, 1983; Chew et al., 1987; Chew and Nishimura, 1992; Chew and Waller, 1986; De kel, 1989; Yaari, 1987). These and other theories are described below.

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123 Intensity theory. The first non-linear LEUT genera lizations were based on Allais’ theory of non-linear preference theory, or intensity theory. Allais considered present levels of wealth, 0w and changes in wealth, x with the psychological value of the final position, x w 0, being a function of 0w x, which is the same for all individuals; this riskless value function is approximated by 01 logw x His theory is based on his viewpoint on psychological value, the re duction principle, weak order preferences defined by the > relation (axiom 1), w eak first-degree stochastic dominance (if q p or q p then q p ), which is a weakened form of the LEUT independence axiom, and the Archimedean axiom to ensure the exis tence of an ordered value function, V. Allais strongly rejected Axiom 2 and the expectation for of LEUT. Moments theory. Allais (1979) proposed that the preference function V of a lottery is not only a function of the expected utility but also incor porates the variance of the elementary utilities. Based on Allais ’ theory, Hagen (1979) suggested the three moments of the psychologi cal value of risky prospects as determinants of V. The AllaisHagen incorporation of second and higher moment s of lotteries into the utility function account for the fanning-out effect, however, in the Bernoulli’s utility framework. Sugden (Loomes and Sugden, 1982; Loomes and Sugden, 1986; Loomes and Sugden, 1987) showed that the properties of the disappointment term, imposed in the disappointment theory, presented later, are a restriction on Hagen’s model of moments. Generalized expected utility theory. Machina (1982) form ulated a non-linear alternative to the LEUT, the ge neralized expected utility th eory, GEUT, adopting Allais’s axioms and using regular first-degree stocha stic dominance instead of weak; Machina, however, did not adopt Allais’ Bernoullian viewpoint on psychological value. Machina’s

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124 approach was distinct in his assumption that V is smooth over P i.e., V changes continuously as p changes continuously, with p V linear around p Others proposed alternative smooth preferences over P (Allen, 1987; Harles s and Camerer, 1994). Machina kept the preference ordering axioms and the Archimedean axiom, but dropped the independence axiom, with a sp ecific hypothesis on the shape of NEUT’s indifference curves. Machina hypothesized that individuals become mo re risk averse as the net outcome they face is be tter. Formally, he defined the indifference curves by the preference function ) ( ) ( ) (i p x U q Vi, with ) ( / ) ( ) (i p q V x Ui being a utility index, where q is the net outcome, fu nction of the different i issues’ outcomes, ix, and their respective probabilities, ) (i p. The theory assumes that at any given distribution p there is a local utility index (.)U which possesses all the VNM properties; it is determined by a linear approximation of the true preference function (.)V around p and can be thought of as a local utility function of (.)V. In other words, the concavity of (.)U at every q is equivalent to the global risk aversion of (.)V; greater risk aversion implies steeper, or “fanning-out”, indiffere nce curves, non-parallel, and not necessarily linear. The “fanning-out” process is also referred to as “betweenness. Several models have been developed ba sed on Machina’s generalized expected utility model with weakened independence ax iom, with the additional restriction of linear, non parallel, indifference curves. On e example is weighted utility theory’s expectation quotient method, described in a following paragraph. Other models have been proposed that do not impose fanning-out, but conserves it, such as the quadratic utility theory. Chew (Chew, 1989) proposed th e quadratic utility theory which relies on a weakened form of betweenne ss called mixture symmetry, in which indifference curves

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125 may switch from concave to convex, or vice versa. Becker and Sarin (1987) proposed the lottery-dependent utility model, with even weaker restrictions; the model is based on an exponential form of preference function. Weighted utility theory. Another type of models us es probability transformation functions which convert objectiv e probabilities into subjecti ve decision weights, leading to the decision weights utility theo ry, DWUT, with the general form ) ( ) ( ) (ix u i w q V. These models assume that individuals tran sform the known set of objective probabilities n ip p p P,..., ,...,1 of a risky prospect into thei r corresponding subjec tive probabilities called decision weights, n ip w p w p w P W,..., ,..., ) (1 (Kahneman and Tversky, 1979), such that ip w is a strictly increasing function from 1 0 to 0 1 with 0 0 w and 1 1 w; ) (ix u are the utilities of individuals outcomes ix, for all X xi. The idea behind this type of models stat ed with Edwards (Edwards, 1954; Edwards, 1961; Edwards, 1962b), who suggested a subjective expected value i ix p V) ( for all i. Another author, Handa (1977), pr oposed that individuals maximize ix i p q V) ( ) (, where ip is a probability weighing function; his model is called the simple decision weighted utility (Starm er, 2000). Karmarkar (1978) used the form i i ip p x u with a power function 1. These models, however, do not generally satisfy monotonicity, unle ss the decision weights equal objective probabilities, reducing ) (q V to LEUT (Fishburn, 1978), which makes it an unacceptable descriptive model of behavior (Machina, 1982; Tver sky and Kahneman, 1986). As a result, decision weighting models th at ensure the monotonicity of ) (q V were designed,

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126 such as expectation quotient, prospect th eory, rank-dependent theory, and others, described in following sections. The first Chew and McCrimmon (Chew and MacCrimmon, 1979) assumed a linear preference function of the form ) ( ) ( ) ( ) ( ) ( ) (i i ix g i p x u x g i p q V where (.)u and (.)g are two different wei ghing functions for the outcomes, with LEUT being the special case in which (.)g is the same for all outcomes. Expectations quotients. This theory represents the value of a prospect by the ratio of two linear functionals. The first set of non-linear Archimedean weak order theory that incorporates the fanning out hypothesis for ar bitrary outcomes was introduced by Chew and MacCrimmon (1979). The authors sugge sted a linear weighted utility function .V composed of a linear positive nonconstant weighting function, w, and a linear elementary utility function, u, with U uw the LEUT. The weighted linearity property is q w p w q u q w p u p w q p V 1 1 1, such that p w q v q w p v q p ; for a constant w, V reduces to LEUT, as presented in Fishburn (Fishburn, 1988). The main point of Chew and MacCrimmon is that all indifference curves are linear, fanning-out, a nd intersect at the sa me point outside the triangle, hence not violating transitivity. Th is theory was later re fined by several authors such as Chew, Fishburn, Nakamura, and De kel (Chew, 1983; Chew, 1989; Dekel, 1986; Fishburn, 1981a; Fishburn, 1981b; Fishburn, 1983b; Fishburn, 1983c). Tversky and Fox (1995) suggested a non linear transformation of probability using an S-shaped weighing function exhibiting diminishing sensitivity as probability increases, i.e., that overweights small probabilities and underweights modera te and high ones. Tversky and Wakker

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127 (1995), building on Tversky and Kahneman (1992) cumulative prospect theory proposed a similar weighing function. Such a function accounts for the fourfold risk attitudes of individuals for gains and losses explained later in the “risk attitudes” section. Prospect theory. Prospect theory was developed by Tversky and Kahneman (1979) to accommodate stochastic dominan ce and invariance viol ations. It is a descriptive theory that expresses the overall value of a prospect, V, in terms of (1) the decision weight p associated with each probability p such as 1 1 p p which reflects the impact of p on the overall value of the pr ospect, and (2) the subjective value x v of the outcome x as the deviation of that outcome from a reference point. It assumes that values are attached to changes rather than states a nd that decision weights do not coincide with probabilities. Formally, i ip x V ; for a prospect defined over probabilities p and q of outcomes x and y, respectively, y v q x v p q y p x V ; ,; it is concave for gains, c onvex for losses, and steeper for losses than for gains, bei ng steepest. In 1992, the author s extended their theory to the cumulative prospect theory. Rank-Dependent Expected Utility Theory. REUT was developed by Quiggin, Yaari, and Schmeidler (Quiggin, 1982; Qui ggin, 1993; Schmeidler, 1989; Yaari, 1987). It assumes that the weight assigned to an individual outcome depends on the true probability of the latter as we ll as its ranking relative to the other outcomes. This method ranks outcomes ix from worst 1x to best nx with corresponding decision weights iw, such that i ip w for n i and n i n i ip p p p w ... ...1 otherwise; n ip p ... is the subjective weight attached to the probability of outcome ix or

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128 better and n ip p ...1 is that attached to the better outcomes, with being a transformation on cumulative probabilities. Note that decision weights w and probability weights are distinct. This theory is a type of nonlinear Archimedean weak order theory that the function satisfies fi rst-degree stochastic dominance and basic ordering. This theory was generalized (Lattimore et al., 1992) using a probability weighing function of the form n k k i i ip p p p1 such that 0 and nis the number of outcomes. Tversky, Kahnema n, and Prelec (Prelec, 1998; Tversky and Kahneman, 1992) suggested a singl e parameter weighting function / 11p p p p for 1 0 and p pln exp for 1 0 respectively. Other theories were also suggested (Abdellaoui, 2002; Bell and Fishburn, 2003; Blechrodt H. and Qu iggin J., 1998; Bleichrodt et al., 2004; Chateauneuf et al., 2005; Chew et al., 1987; Chew and Epstein, 1989; Courtault and Gayant, 1998; Fishburn, 1994; Fishburn and Falmagne, 1989; Green and Jullien, 1988; Kendall, 1975; Luce, 1991; Nakamura, 1995a; Nakamura, 1995b; Naka mura, 1996; Roell, 1987; Safra and Segal, 2001; Segal, 1993; Starmer and R., 1989). Disappointment Theory. Bell (1985) and Loomes and Sugden (1986) developed the theory of disappointment, DT, where pr eferences are represented by the function U x u D x u i p q Vi i) ( ) ( ) (, where (.)u is a measure of “basic” utility of ix when considered alone, U is a measure of the “prior expectation” of the utility from an outcome, and .D reflects individual’s dislike and av oidance of disappointment. With 0 D, the model reduces to EUT; it is assumed that individuals are disappointment averse, h D concave, for 0 h, and elation prone, h D convex for 0 h.

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129 State-preference model. This theory was developed independently by Arrow, Debreu, and Hirshleifer (Arrow, 1953; Debreu 1959; Hirshleifer, 1966) to account for probability judgments of indivi duals by representing uncertainty by a fixed finite set of mutually exclusive and e xhaustive set of states, is S The objects of choice consist of alternative stateis-payoffix bundles, defined over the re spective states with no reference to probability. If these bundles ar e assigned probabilities, indifference curves can be generated from individual behavior (Evans and Viscusi, 1991; Fishburn and Lavalle, 1987; Fishburn and LaVa lle, 1988b; Hirshleifer, 1966; Karni, 1983; Karni, 1987; Karni et al., 1983; Kelsey and Nordquist, 1991; La valle and Fishburn, 1987; Yaari, 1969). Skew-symmetric bilinear theory. Fishburn (Fishburn, 1981a; Fishburn, 1984a; Fishburn, 1984d; Fishburn, 1984e; Fishbur n, 1990; Fishburn and LaValle, 1988a) suggested the skew-symmetric bilinear theo ry, SSB, to represent preferences between probability distributions in a set P by the positive part of a skew-symmetric bilinear, SSB, functional, q p, on P P such that 0 q p q p for P P q p ,; q p, does not imply the independence, or dering, or transitivity axioms. q p, measures the expectation of the difference s in preference between two outcomes, say X x x2 1,, for the two distributions P P such that 122 1, ,xxx x pq q p with 2 1, x x being a quantification of the quali tative preference difference between 1x and 2x. Loomes and Sugden (1982) associate 2 1, x x with the regret at getting 1x when 2x might have occurred, with 1 2 x x. This is elaborated further next as the expected regret theory and Alpha utility theory.

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130 Expected regret theory. This theory was developed independently by Bell, Fishburn, and Loomes and Sugden (Bell, 1983; Bell, 1985; Fishburn, 1982b; Loomes and Sugden, 1982) as the leading example in m odeling non-transitive bi variate preferences (Machina, 1987). It replaces LEUT with a regret/rejoice function, ) (y x R, that represents the level of satisf action from receiving an outcome x over y, such that ) ( ) (x y R y x R Let np p P,...,1 and np p P' ,..., '1 be two statistically independent lotteries over a common outcome set nx x X,...,1 ; an individual will choose P if the expectation ij j i j ip p x x R ) ( is positive, and P otherwise. Note that the indifference curves in this model cr oss at one point formi ng an intransitive cycle and dividing the area of the triangle into lotteries strictly preferred to lottery. Alpha utility theory. Chew, MacCrimmon, and Fishburn (Chew and MacCrimmon, 1979; Fishburn, 1983a) extended the SSB theory by adding the transitivity constraint. This theory, the Alpha utility th eory, represents preferences with two linear functions, and on P such that p q q p q p ' and p q q p q p ' , allowing to be decomposed using and In addition, x x p p ' and x x p p " Local Linearization. Machina (1987a; 1987b) sugge sted considering non-linear functions in general and using calculus to extend them to LEUT, such as taking the first order Taylor expansion of a diffe rentiable preference function about P Consider the non expected utility function P X V that is differentiable with respect to X and P such that ) ( P X U x p P X V with P X U as the local utility at P Note that a

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131 differentiable function exhibits a specific gl obal property, such as being non-decreasing, iff that property is exhibited by its lin ear approximation at each point. Machina ascertained the applicability of this property to stochastic dominance and risk aversion properties of a non-expected utility function. Strategic Decision Making Game Theory Game theory was the first mathematical t ool to conceptualize conflict as a decision problem, capitalizing on the structural sim ilarities between a decision problem and a conflict of interest. Its foundation is ac knowledged to be von Neumann's minimax theorem (von Neumann, 1928).7 The authors first formalized the subject in 1944 in their book Theory of Games and Economic Behavior (von Neumann and Morgenstern, 1944). Game Theory is a misnomer for multi-person decision theory as it seeks to find rational strategies in situations where the outcome depends not only on one's own strategy and market conditions but upon the strategies c hosen by other players with possibly competing or conflicting goals. In ot her words, game theory considers situations where instead of individuals making decisions in response to “dead” variables, such as prices, their decisions are st rategic reactions to other individuals actions or “live” variables (Myerson, 199 1; Vega-Redondo, 2003). The next sections first describe the co mponents, representations, and types of games, then the solution concepts, and finally the available extensions to game theory. 7 mile Borel, however, had already published three papers on "the theory of play" between 1921and 1927 in which he introduced the concepts of strategic game and mixed strategy. Ernst Zermelo is credited with game theory's first theorem (1913) by showing that a solution algorithm for chess exists (even though it is not computable in practice). Morgenstern did most of the economic analysis in Theory of Games and Economic Behavior.

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132 Mathematical formulations To model a conflict, the part icipants, individuals or groups must be first identified; they are considered as individual decision make rs and referred to as players. Players have available individual decision possibilities, actions, or strategies ; strategies may be pure or mixed. The individual players’ choi ce of actions, jointly, determines the final outcome of the conflict and th eir individual preferences or objectives over these possible outcomes. The possible actions and their relati onship to the final outcome are modeled in different ways by the different game forms. There are three main forms to represent a game: the extensive, dynamic, or tree form; normal, strategy, static, or matrix form; and characteristic function, cooperative, or coal itional form (von Neumann and Morgenstern, 1944, 1953) differing in the amount of details represented in the model; extensive and normal forms are usually used to repr esent noncooperative games, while the characteristic function is used to repres ent cooperative games (Myerson, 1991; VegaRedondo, 2003). Other forms were developed such as the option form (Howard, 1971) and the graph form (Kilgour et al., 1987), pres ented later. More information about games representations is abundantly available in the literature (Heap, 2004; Myerson, 1991; Rasmusen, 2001; Romp, 1997; Vega-Redondo, 2003). Extensive form. It is the most detailed, where the structure captures the actual rules of the game; it specifies: (1) the play ers of a game, (2) their individual potential moves and resulting positions, (3) informa tion possessed by individual players at every move, and (4) the payoffs received by every player for every possible combination of moves. The game is represented by a ga me tree. Each node, called a decision node, represents every possible stage of the game as it is played. There is a unique node called the initial node that represents the start of the game. Any node that has only one edge

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133 connected to it is a terminal node and represents the end of the game and also a strategy profile. Every non-terminal node belongs to a player in the sense that it represents a stage in the game in which it is that player 's move. Every edge represents a possible action that a player can ta ke. Every terminal node has a payoff for every player associated with it. These are the payoffs fo r every player if the combination of actions required to reach that termin al node are actually played. Normal form. In games of perfect information, a normal form is a summary representation of a game; it sp ecifies players' strategy spa ces, or action spaces, and payoff functions. A strategy space for a player is the set of all strategies available to that player. A strategy is a complete plan of action for ev ery stage of the game, regardless of whether that stage actually arises in play. A strategy profile is a specificat ion of strategies for every player A payoff function for a player is a the crossproduct of players' strategy spaces to that player's set of payoffs, i.e., it takes as its input a strategy profile and yields a representation of payoff as its output. Play ers consider their and other players possible strategies and make their indi vidual choices simultaneously; choices are revealed at the end of the game. Note that the final outcom e depends on the choices of all the players. Formally, let 2 n denote the number of play ers in the game, numbered n,..., 1, and n N ,..., 1 be the set of these players; the no rmal form specifies the sequence of strategy sets, nA A ,...,1, and the sequence of real -valued payoff functions nf f ,...,1 of each of these players, such that n ia a f ,...,1 with i iA a Characteristic form. In this form, the notion of strategy disappears; it is defined by its coalitions and their corresponding valu es or utilities. This form describes cooperative game theory with tr ansferable utility. Coalitions of players are formed to

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134 favor common interests of those players; the grand coalition c onsists of all the players. Formally, let 2 n denote the number of play ers in the game, numbered n ,..., 1, and let n N ,..., 1 be the set of these players. A coalition S is defined as a subset of N i.e., N S with N S designated as the grand coalition and S as the empty coalition. The set of all coalitions consists of N2 coalitions. The coalition form of an n-person game is given by the pair N, where N is the set of players and is a real valued function, namely, the characteri stic function of the game. satisfies the following properties: Superadditivity: if S and T are disjoint coalitions such that T S then T S T S where S and T are the values of coalitions S and T respectively. Group rationality or efficiency: for a payoff vector, nx x ,...,1 x, N xn i i1; in other words, no player could be expected to receiv e less than that player could obtain alone, which implies individual rationality: i xi A payoff vector th at satisfies both group and individual rationali ty is called an imputation. The set of stable imputations is the core, C, such that N S all for S x and N x x x x CS i i N i i n i1 1: ,..., The core of a cooperative game consists of al l undominated allocations in the game. In other words, the core consists of all allocations with the pr operty that no subgroup within the coalition can do better by deserting the coali tion. Notice that an allocation in the core of a game will always be an efficient allocation.

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135 Note that a normal form game can be tr ansformed to its characteristic form by specifying the value of each of its coalitions, such that S i n i ia a f S ,..., Value; this reduction however loses important features of the game and for a given characteristic function there are usually many strategic forms. Classification of games Games may be classified as zero and non-zero sum games, the level of cooperation between the players, and the level of information available to the players. Zero-sum and non-zero-sum games. In zero-sum games the total benefit to all players in the game, for every combination of st rategies, always adds to zero i.e. a player wins exactly the amount one's opponents loses (chess and poker). Formally, 0 ,...,N i n i ia a f and for two complimentary coalitions, S and S N S S i n i i S i n i ia a f a a f ,..., ,..., or 0 S S N A constant-sum game, of which zero-sum game is a special case, has constant a a fN i n i i ,..., and constant S S N Most real-world examples in business and politics, as well as the famous prisoner’s dilemma are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, a gain by one player does not necessarily correspond with a loss by another. Note that one can more easily analyze a zero-sum game; and it turns out that one can transform any game into a zero-sum game by adding an additi onal dummy player. Cooperative versus noncooperative games. In a cooperative or coalitional game, players are allowed to form binding agreements or work together to receive the largest

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136 possible outcome. It studies games in characteristic form, specifying the utilities for coalitions; it is further classifi ed into transferable and non-tran sferable utility games, such that in the former there is a mean of utility exchange. The latter is not discussed in this section; interested reader s may refer to the literature (Myerson, 1991; Vega-Redondo, 2003). It is to the joint bene fit of all the players, by supe radditivity, to form the grand coalition, N, with value N The main issue is to agree on how N is to be split among the players. Noncooperative games are usually represented by the extensive and normal forms. In noncooperative or stra tegic games, players may not form binding agreements. Games with complete/perfect versus incomplete/imperfect information. In games of complete information each player has the same game-relevant information as every other player. Each decision maker ha s perfect knowledge of the game and of his opposition; that is, he knows in full detail the rules of the game as well as the payoffs of all other players; hence, players cannot move simultaneously. In addition, it is assumed that all decision makers are ra tional. These assumptions rest rict the application of game theory in real-world conflict situations as complete information games occur only rarely in the real world, and game theorists usua lly use them only as approximations of the actual game played. Games of incomplete info rmation, also called Bayesian games, arise when information is not commonly or is asy mmetrically shared between players; this type of games was introduced by Harsanyi (1968b; 1978). These games presume that players begin with a common prior probability distribution over the different features of the game, which are not common knowledge; before play start, each player is aware of his private information, his type but is ignorant, or probabili stically aware, of others’.

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137 Static versus dynamic games. In dynamic games, play proceeds in a sequence of stages. We say a player has the move when it is his or her turn to choose an action. In a dynamic game, there is often scope for one pl ayer to observe another before deciding how to play. This allows dynamic games to desc ribe more sophisticated situations than is possible using static analysis. In a special class of dynamic games called repeated games, the same static game is played several times in a row. One-shot versus repeated games. Repeated games are static games which are played several times over (Luce and Raiffa, 1957 ). A repeated game consists of several stage games over a finite or infinite time horizon; the resulting payoff or outcome is usually accordingly discounted (Vega-Redondo, 2003). Solutions concepts The stability analysis of a conflict is carri ed out by analyzing the stability of each state of every decision maker under the particular behavior model, referred to as stability definition or concept. A state is stable fo r a decision maker, under the specified solution concept, if and only if he has no incentive to deviate from it; a state is an equilibrium, constituting a possible resolution, under that concept if and only if all decision makers’ states are stable (Fang et al. 2003a; Kilgour et al. 1987). There is a broad range of stability concepts embodying different attitudes to strategic risk and levels of insight about the decision makers’ pattern of conflict behavior, decision styl es, and the resulting movement of the conflict (Fang et al., 2003a). The next paragraphs list and differentiate different solution concepts under the fram ework of cooperative and noncooperative games, starting with the latter. The two main solution concepts in n oncooperative game theory are iterated dominance and Nash equilibrium. They are obtained by one of three processes: minimax

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138 solution, backward induction, or forward i nduction. The minimax solution concept was derived by von Neumann (1928) as the minima x theorem and later used by von Neumann and Morgenstern (1944) to solve simple zero-s um normal games. In backward induction, a sequence of nested optimizations is solved from the future to the present; it evaluates current choice possibilities in terms of the pr edicted future courses of action. Forward induction interprets the actions of players al ong the game in terms of what they could have done in previous actions but did not do (Kohlberg a nd Mertens, 1986). Forward induction is so called because just as backwa rd induction assumes future play will be rational, forward induction assumes past play was rational; that player forms a belief of what type that player is by obser ving that player's past actions. Iterated dominance elimination. Iterated dominance was first suggested by Luce and Raiffa (1957) who suggested that a normal game’s solution may be reached by eliminating dominated strategies, iteratively. For an individual pl ayer choosing how to play, one strategy dominates another if it de livers higher payoffs regardless of the strategies chosen by his opponents. A rati onal player will never choose a dominated strategy. If a player knows his opponent is rational, and one if his opponent's strategies is dominated by another, he can safely a ssume his opponent will never choose it. The dominated strategy can effectively be deleted from the game. This may leave one of the player's own strategies dominated, so in turn it can also be deleted. Because it is assumed rationality is common knowledge (Aumann, 1976), also known as rationalizability (Vega-Redondo, 2003), this iterated deletion may proceed indefinitely, until there are no further dominated strategies. In some circumst ances, this will leave just one way to play the game, in others, the games may not be dominance solvable.

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139 Nash equilibrium. Although this concept stretches back to Cournot (1838), it was formally introduced by Nash (1950). The idea is to look for a set of strategies, such that every player's strategy is a best response to the strategies of thei r opponents. Put another way, when a set of strategies constitute a Nash equilibrium, no player has an incentive to unilaterally deviate by playing differently. If before the game, players could agree to play a set of Nash equilibrium strategies, each would find it in his interest to keep his part of the agreement. This property has led so me to describe Nash equilibria as selfenforcing. A game can have no or more th an one Nash equilibrium, suggesting the need for additional strategic stability criteria. Se veral solution concepts have been suggested as generalizations or refinements of the Nash equilibrium. These refinements include strong equilibrium (Aumann, 1959) and coalition-proof equilibrium (Bernheim et al. 1987), where no subset of players may have a joint beneficial deviation; correlated equilibrium (Aumann, 1974), which allows incen tive-compatible stochastic coordination; subgame perfect equilibrium (Selten, 1965), wh ere a Nash equilibria is induced at every subgame, even off-equilibrium ones, of the game, each starting with an information set; perfect or trembling-hand perfect equ ilibrium (Selten, 1975), which is the -perfect equilibrium or the Nash equilibrium of the perturbed game in which players cannot avoid deviating, or making a mistake, from the intended play with a probability ; proper equilibrium (Myerson, 1978), in which different deviations are give n different weights; persistent equilibrium (Kalai and Samet, 1984), which is a Nash equilibria under a selfenforcing set of rules or retract; essential e quilibrium (Wu and Jaing, 1962) in which for every small perturbation of the game ther e exists an equilibrium is close to the component ; Bayesian Nash equilibrium (Harsanyi, 1967; Harsanyi, 1968a; Harsanyi,

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140 1968b) where players choose optimal strategies in view of th eir explicit subj ective beliefs of their opponents’strategies; sequential or perfect Bayesian equilibrium (Fudenberg and Tirole, 1991; Kreps and Wilson, 1982), which ac counts for change in players’ beliefs in subsequent plays; weak perfect Bayesian equ ilibrium such that beliefs are updated at each information set in accordance (Vega-Redondo, 2003), which is a variant of perfect Bayesian equilibrium s in which updating of beliefs probabilities is performed using Bayes rules; sequential bargaining (Rubinstein 1982) which is a transformation of the cooperative Nash bargaining solution into a noncooperative strategic extensive game of sequential bargaining; perfect folk theorem (Fudenberg and Maskin, 1986) which proposes a solution for infinitely repeated ga mes; limited-move stability (Kilgour et al., 1987; Zagare, 1984) where the behavior of a pl ayer depends on the horizon of foresight using horizontal distance as a parameter; nonm yopic stability (Brams and Wittman, 1981) is the limiting case of limited move stabilit y as the horizon increases without bound; and others. Solution concepts in cooperative game theo ry aim to distribute the value generated by cooperation while accounting for issues such as fairness and stability (Driessen, 1991); the main ones are the core solution, Sh apley value, nucleolus, nonseparable costs; Nash bargaining solution, and others. They are all briefly explained in the following sections. Core solution. The core solution was one of the very first proposed set of solutions introduced by Shapley (Driessen, 1991; Lucas, 1971; Shapley, 1953). The core was defined in n earlier section. Aumann (A umann, 1959) introduced a similar concept,

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141 the bargaining sets. Davis and Maschler (1965) suggested the kernel solution concept as a non-empty set of the bargaining sets. Th e nucleolus, discussed later, is a unique outcome of the kernel (Lucas, 1972). Shapley value. Shapley (Shapley, 1953; Shapley and Shubik, 1954) introduced the concepts of the Shapley value and the core as solutions to cooperative games. The Shapley value defines a way to split the pa yoff from the grand coalition by defining a value function that assigns an n-tuple of real numbers to each possible characteristic function of an n-person game, such that n ,...,1, where i represents the value of player i in the game with characteristic function is unique; it obeys the following axioms of fairness: Efficiency or group rationality: the total value of the player s is the value of the grand coalition, Nn i i 1. Symmetry: if the characteristic function is symmetric in i and j then the values assigned to both should be equal, or, if j S i S for every coalition S such that S i then j i Dummy: if a player i is a dummy in the sense that the player neither benefits nor harms a coalition, then the playe r’s value is null; or, if i S S for every coalition S, S i then 0 i. Additivity: this axiom reflects that the arbitr ated value of two games played at the same time should be the sum of the arbitrated values of the games if they are played at different times; i.e., if and are characteristic functions, then ' Note that if and are characteristic functions, then is one too.

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142 The Shapley value is given by the vector n i ,..., ,...,1 where S i N S ii S S n S n S,! ! 1 is the average amount player i contributes to the grand coali tion if the players sequentially form this coalition in a random order. Other variants of the Shaple y value exist, such as the average of the selector vector, suggested by Derks and Pete rs (1997; 1998). Others variants extend the Shapley value solution concept to random payoff; Timmer, Borm, and Tijs (2000) defined three forms called marginal value, dividend value, and selector value. The Shapley-Shubik power index is base d on the Shapley value (Shapley and Shubik, 1954). Defining a simple game such that a game in which for every coalition is either a losing coalition with 0 S or a winning coalition with 1 S the Shapley value reduces to the Shapley-Shubik index g lo i S winning S in S n Ssin! ! 1. Note that the Shapley value falls in th e core for convex games, but might fall outside the core for nonconvex ones ((Driessen and Tijs, 1985; Heaney and Dickinson, 1982). Nucleolus. Schmeidler (1969) develope d another value function for n-person cooperative games. Instead of defining an axio matization of fairness to a value function, this method suggests to search for an imputa tion that minimizes the worst inequity using a fixed value function. Formally, the nucleolus is defined as the excess, which measures the amount a coalition S falls short of its potential S in the imputation x: S j jx S S e, x from the definition of the core, an imputation x is in the core iff all its excesses are 0 It can be proven that the nucleolus of a game in coalitional form

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143 exists and is unique; it satisfies the four axioms of the Shapley value and is in the core if the core is not empty. Several variants of th e nucleolus have also been defined, such as, the weak nucleolus (Shapley and Shubik, 1966) and Disrupti on nucleolus (Littlechild and Vaidya, 1976). Nonseparable cost Defining a game by the pair ,N, where N is the set of players and is a real valued cost function, cost allocated to any player i should not be less than the marginal cost or the separable cost of including that player in the game defined as i N N SCi The nonseparable costs ar e the costs remaining after the allocation of the sepa rable costs, such that n i iSC N NSC1. Generally, these latter are allocated costs among players using a ratio i Each player obtains the sum n NSC SCi i. The egalitarian nonseparable cost, ENSC, method is the simplest way as it assumes 1i for all players, hence proportioni ng costs equally among players, such that NSC n SC ENSCi i1 ; this method may violate the individual rationality principle. Another method is the separable cost s-remaining benefits method, SCRB, (Eckstein, 1958) with i i iSC i b min where ib is the benefit to player i by acting independently in the game; this met hod allocates nonseparable costs in proportion to each player’s willingness to pay adjusted by the player’s separable cost, such as a player is not willing to pay more than the his benefit ib or alternate cost i in order to participate in a joint project. For i bi then i iSC i which is the alternate cost avoided by including the player in the project used in the a lternate cost avoided

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144 method, ACA. Generally, SCRB costs are expressed for each player i as NSC SC j SC i SC SCRBn j j i i i 1 1 SCRB satisfies individual rationality but is generally unstable. A third method is the nonseparable cost gap, NSCG, method which uses the cost gap function that assigns to every coalition its remaining alternate costs such that S i iSC S S g; the cost gap of an empty coalition is 0 g, and the cost gap of the grand coalition is NSC N g If T is a coalition such that T i i will not be willing to pay more than T g SC T g SCT i T i i T i T ; ;min min; in addition, i will reject any allocation that charges him more than T g SCi or equivalently i T j iSC T. T gT i T i ;min is the concession amount of player i or his maximum contribution to nonseparable cost in the game. Finally, NSC SC NSCGn j j i i i 1 1 This method obeys efficiency, individual rationality, sy mmetry, continuity, dummy property, and monotonicity. A fourth method is minimum cost-remain ing saving MCRS method; it is based on the sharpest possible lower and upper bounds of the core; they are the minimization and maximization solution of the n 2 iy for l i iy SC and i i u iSC y for all i, respectively (Heaney and Dickinson, 1982). Denoting those solutions by l iy and u iy.

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145 The cost allocated to player i is then n j l j n j l j u j l i u i l i iy N y y y y y MCRS1 1 1. This method is generally stable. Nash bargaining solution It was developed by Nash (1950) as an extension to the Nash equilibrium for coalitional games; it corresponds to the solution at which players make equal proportional sacrifices. This solution corresponds to the outcome of maximizing the product of utilities percei ved by the players, modified by the disagreement point; which corresponds to the payoff implied by the non-cooperative solution, the Nash equilibrium strategy; it obeys the Pareto optimality, invariance, symmetry, and independence axioms. The be st known variation of Nash bargaining solution, the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) in which Nash’s fourth axiom was replaced by a monotonicity axiom. Others Other developed solution concepts include reasonable outcomes (Luce and Raiffa, 1957), -stability (Luce and Raiffa, 1957), and partition function games solution (Lucas, 1965), and equilibrium se lection (Harsanyi and Selten, 1988). Extensions to Standard Game Theory Extensions of normative game theory were designed to resolve five main shortcomings (Dacey and Carlson, 1996): (1) differing perceptions, (2) conflict dynamics, (3) combinatorial complexity, (4) li nked multilevel games, and (5) deliberate game changing. Differing perceptions refers to the assumption that all players see the same game (Bennett, 1995); this was firs t discussed by Harsanyi (Harsanyi, 1967; Harsanyi, 1968a; Harsanyi, 1968b) Conflict dynamics refers to the sequence of moves and change of parameters; the former has been addressed via multistage games, repeated

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146 play (Axelrod, 1984; Taylor, 1976), and the th eory of moves (Brams., 1994), while the latter has been addressed by game changi ng where players purposefully substitute preferences leading to what is known as bu lly games or called bluff games (Dacey and Carlson, 1996). Combinatorial complexity linked multilevel in games emerges in cases of games with large numbers of players, choices, and decision stages. Subsequent research in decision and conflict analysis a dopted systematic mathematical approaches to find rational a nd stable solutions. These include metagame analysis, hypergame analysis, analysis of options, conflict analysis, drama theory, graph theory, and theory of moves. All are essentially game theo ry variants designed to yield better decision advice (Fang et al., 2003a; Hamouda et al., 2004; Obeidi et al., 2003). Metagame analysis Metagame analysis was developed by Ho ward (1971) to descriptively model the strategic relationships of negotiations. Ba sed on game theory, it emphasizes the process of building and using the model. The method consists on building a set of scenarios of players’ chosen options of actions; the numbe r of scenarios equals that of feasible combinations of options. Analysis generally proceeds by considering the nature of the various scenarios, the preferences actors have for them, and the strategic implications of these preferences (Klein, 2000). Developments and applications to Metagame have been suggested by several research (Dixon, 1986; Dutta and King, 1980; Evans and Harris, 1982; Fraser and Hipel, 1980; Howard, 1986; Richelson, 1979; Walker, 1977); Howard (Howard, 1993; Howard, 1998) suggested a so ft game theory of emotions in the framework of metagames, drama theory.

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147 Hypergame theory First suggested by Bennett (1977; 1980), a hypergame instead of modeling one game, models each player's perception of the c onflict as a game or a series of games. Analyzing a hypergame involves analyzing each of the games for stability and then comparing the results to find stable equilibri ums for the hypergame. It is noted by Wang, Hipel and Fraser (1989) that solutions to a hypergame may not necessarily be created by outcomes that are stable for all players and th at it is possible that an outcome that is unstable individually for players may actually be an equilibrium for the hypergame. Many hypergame analyses have been published, showing its use in modeling conflicts and their resolutions (Bennett, 1985; Bennett and Dando, 1979; Bennett et al., 1980; Bennett and Huxham, 1982; Bryant, 1984; Hipel et al., 1988; Inohara et al., 1997; Laing, 1979; Said and Hartley, 1982; Shupe et al., 1980; Takahashi et al., 1984; Zwicker, 1987). Analysis of options Analysis of options was suggested by Howard (1971, 1987); it employs combinatorial complexity which uses a bina ry simple preference comparisons (yes/no) representation of complex games, as opposed to the previously pr esented extensive and normal forms. The elements of the analysis are the actors, their possible options, the scenarios or futures of different combinati ons of options, and the actors corresponding preferences; these elements are summarized in a tableau. confr ontation analysis was originally proposed as a ``quick and dirty'' form of analys is of options, which does not necessitate a complete initial preferences set, since at each stage, actors are only searching for reachable s cenarios (Bennett, 1998).

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148 Conflict analysis Conflict analysis is an extension and refo rmulation of metagame analysis suggested by Fraser and Hipel (1984). This method comb ines the metagame analysis tables into a single tableau, incorporates players’ k nowledge, and outputs possi ble strategies and stable equilibria. Drama theory Drama Theory was developed by Howard and Bennett (Bennett and Howard, 1996; Bennett, 1995; Howard et al., 1993) as soft game theory. It is a direct descendent of metagame analysis for complex situations wi th emotional responses; its main components are confrontation analysis and immersive briefings. Many of the concepts of game theory have direct parallels in drama theory with different terminology: players, outcomes, preferences, and strategies are replaced by characters, scenarios, preferences, and combinations of options. A drama unfolds in episodes in which characters interact; an episode consists of scene-setting; build-up, where characters exchange ideas, advocate preferred positions and fallbacks, and influence each other via threats and promises; climax, where characters succumb, react, or reevaluate; and denouement, leading to cooperation, trust, deterrence, inducement, threat, or positioning, each presenting characters with different options and ep isodes (Bennett, 1995; Bennett, 1996; Howard, 1993; Howard, 1998; Howard et al., 1993). Graph model for conflict resolution The graph model for conflict resolution was developed by Kilgour, Hipel, Fang, and others in a series of papers (Fang et al., 1993; Kilgour et al., 1987); it constitutes an extension and reformulation of conflict anal ysis, which in turn is an extension of metagame analysis. It is a representation of conflict via a series of directed graphs,

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149 initiating from a common vertex; with differe nt preferences. Several extensions and applications have been developed using th e graph model such as the decision support system suggested by (Fang et al., 2003a; Fang et al., 2003b; Li et al., 2004; Li et al., 2005). Theory of moves The theory of moves explicates the dynami cs of play, based on the assumption that players think not just about the immediate consequences of their actions but their repercussions for future play as well; the game ends when player passes instead of making a move or makes a move into a pr eviously encountered state (Brams, 1994); Willson (1998) suggested an extension to this theory where a given state may be visited many times, with a limit on the total number of moves, and both pl ayers must agree to stay at a state before the game is complete. Alternatives to Standard Game Theory In addition to the models of individual be havior preferences, summarized in earlier section, that account for cogni tive limitations, standard game theory, as a theory of individuals interactions, has its own alternatives that account for individual s’ strategic limitations. These alternatives are broadly classified under behavioral game theory. Behavioral game theory uses experimental evidence to inform mathematical models of cognitive limits, learning rule s, and social utility. Th e main studied elements of behavioral game theory are theories of limited strategic thinking, theo ries of learning, and social preference functions (Camerer, 2003), explained below. An additional, yet unaccounted for phenomenon, is the mental representation of complex problems (Camerer, 2003).

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150 Limited thinking models An important element of game theory is the ad infinitum iterated reasoning on what players think other players will do. However, in real decision-making situations, players use a limited number of steps of iterated reas oning. To account for this fact, models of limited thinking were developed (Binmore, 1988; Goeree and Holt, ; Nagel, 1995; Stahl, 1993). An example is the cognitive hierarc hy model, which assumes that players have partially rational expectations: k-step players believe that other players use only 0 to 1 k steps due to working memory constraints and doubts about the ra tionality of others which imply that more thinking steps are increasingly rare, such that !k e k fk with 2 1 as the mean and variance of the number of thinking steps (Camerer et al., 2003; Camerer, 2003a; Camerer, 2003b; Camerer et al., 2004). Learning models Even though strategic thinking is limited, behavior can accurately approximate equilibrium predictions from learning, evolut ion, adaptation, or imitation. Two main types of learning models have been develope d, namely, reinforcement and belief learning models; these models fit behavior better than equilibrium theory. In the former, players repeat previous strategies that yielded desirable outcomes. An example of the latter is fictitious play, where players form beliefs ba sed on a weighted average of the history of other players’ behaviors. A hybrid model is one that encompasses both reinforcement and belief learning. An example is the e xperienced weighted attr action learning model, EWA, in which players put a partial weight on counterfactual imagination of forgone outcomes from strategies they did not choos e. Another example is the rule learning model where players shift weight towards lear ning rules associated w ith higher outcomes.

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151 Finally, A fourth type of models is the general adaptive models, which assume that players have the ability to account for future effects, in addition to past ones (Camerer, 2003a; Camerer, 2003b; Camerer et al., 2004). Social preferences models This type of behavioral game theoretic models studies how social motives affect strategic interactions; social motives include altruism, fairness, trust, vengeance, hatred, reciprocity, and spite. It has been observed that players who cooperate typically expect others to cooperate, i.e., cooperation is recipr ocal or conditional rather than altruistic. Another observation is that givi ng a player a chance to punish lo w contributors, even if at a cost to themselves, raises group contributi ons closer to the optimal level at which everyone contributes. Cooperation was also observed to rise sharply with communication. Another example stems from ultimatum games where a proposer makes a one-time offer to a responder, where if the latter rejects the o ffer, both players get nothing; the assumption of a se lfish player implies the player as a proposer offers the least possible and as a responde r takes anything offered, which is contrary to observed behavior. In addition, rej ection by responders shows recipr ocity or vengeance, where the player is willing to sacrifi ce personal outcome to punish othe rs who were unfair. Other examples are dictator games, which measur e the altruism of the proposer, since the responder cannot reject the offer whatever it is ; trust games, which measure the trust of an investor and the trustworthiness of a r eceiver in the absence of legal protection. Camerer (2003b) suggests as an explanation to observed social behavior Edgeworth’s (1881) conjecture that a player ’s utility for allocations incl udes both the player’s outcome and the weighted outcomes of others, with the weight varying systematically across situations.

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152 There are three main types of social pr eference models that were proposed to account for these observations. One is the inequality-aversion theory where players prefer a higher outcome, namely money, and e qual allocations, even if at the expense of the former; this theory fails to explain why players reject unfair offers more than uneven ones (Fehr and Schmidt, 1999; Fehr and Schmidt, 2000; Fehr and Schmidt, 2004). A second theory is me-min-us Rawlsitarian theo ry, in which players care about their own payoff, minimum payoff, and the total pa yoff; this model does not account for the ultimatum game rejection (Charness and Rabin, 2002). A third model type is reciprocity theory, in which a player forms a judgment about other players’ kindness; the judgment is scaled positive or negative for improving and deteriorating the player’s situation, respectively, and is reciprocated (Rabin, 1993). Game Theory in Water Resources Management The application of GT to water resources projects dates from the 1930’s, when the Tennessee Valley Authority, TVA, considered the problem of fair allocation of joint costs of dam systems (Parker, 1943). The model developed by the TVA was later addressed by several authors (Ransmeier, 1942; Straffin and Heany, 1981; Tijs and Driessen, 1986). Rogers (Rogers, 1969) applied a game approach to the Ganges-Brahmaputra sub basin involving two water users, India a nd Pakistan. In following papers, Rogers (Rogers, 1993) discussed coopera tive GT application to wate r sharing in the Columbia basin (US and Canada), the Ganges-Brahmaput ra basin (Nepal, India, and Bangladesh) and the Nile basin (Ethiopia, Sudan, and E gypt). Degen (1972) cons idered the design of an irrigation system as a game against natu re. Giglio and Wrightington (1972) developed two bargaining methods to solve a cost-s haring problem for wastewater treatment facilities. Hipel, Ragade, and Unny (Hipel et al., 1974; Hipel et al., 1976) applied a

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153 metagame theory to analyze political non-rational conflicts in water resources problems. Suzuki and Nakayama (1976) illustrated an application of cooperative game using the Kanagawa prefecture in Japan. Loehman and colleagues (Loehman et al., 1979) formulated an economic allocation model in the framework of GT to the Meramec river basin for Missouri wastewater treatment system. Bogardi and Szidarovsky (1976) demonstrated the applicability of olig opoly games in different areas of water management such as environmental protec tion, irrigation systems, water quality management, and multipurpose management systems using a hypothetical problem. Okada and Yoshikawa (1977) applied a coopera tive game to the Kakogawa River basin and the southern part of the Hyogo Prefecture, Japan. Fraser and Hipel (Fraser and Hipel, 1980) used metagame analysis for the Poplar river conflict between Canada and the US. Guariso et al. (1981) and Ben Shachar et al. (1989) developed an economic water trade game model for trading Nile water for ir rigation technology between Egypt and Israel, and exchanging relatively cheap Nile water against expensive locally harvested water between Israel and the West Bank and the Gaza strip. Sheehan and Kogiku (1981) and Heaney and Dickinson (1982) studied the a pplication of cooperative GT to apportion costs and benefits in large wa ter resources proble ms. Young et al. (1982) developed and applied a GT method to a hypothetical water re sources allocation problem. Ratner (1983) and Yaron and Ratner (1985, 1990) dealt with regional interfarm cooperation in water use for irrigation and the determination of the optimal water quantity-quality (salinity) mix for each water user. Dinar (1984), Dinar et al. (1986), and Dinar and Yaron (1986) applied a cooperative game to wastewater trea tment and reuse projects; it considered the water scarcity problem faced by agricultural fa rms in the coastal plai n region of Israel.

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154 Szidarovsky et al. (1984) applied GT to the multiobjective management of the transDanubian karstic region in Hungary. Kraw czaka and Zilkowskia (1985) developed a Nash solution for reservoir water pollution. Kilgour et al. (1987) developed a game theoretic graph model to assi st decision making in water resources problems. Tecle et al. (1988) used three different multicriterion deci sion-making techniques, one of which was cooperative GT, to analyze a multiobjective wa stewater management problem. The case study was the Nogales Interna tional Wastewater Manageme nt Project, which treats wastewater coming from the twin cities of Nogales, Arizona and Nogales, Sonora, Mexico. Kilgour et al. (1988) formulated a cost-sharing ga me to manage the load control system for regulating chemical oxygen dema nd in water bodies. Rausser and Simon (1991) extended the Stahl-Rubinstein game (Rubinstein, 1982) to develop a noncooperative model of multilateral bargaining. The model was later applied in Adams et al. (1996) and reviewed by Simon et al. (2003). Thoyer et al. (2001) applied this model to the Adour River Basin in south we st of France. Tisdell and Harrison (1992) analyzed the distributive conse quences of alternative methods of allocating transferable water licenses in Queensland, Austra lia through the use of GT. Dinar et al. (1992) evaluated a cooperative game in water reso urces management by using two empirical applications in wastewater treatment and re use for irrigation. Rosen and Sexton (1993) applied cooperative game theory to assess orga nizational reactions to rural to urban water transfer. Becker and Easter (1995; 1997; 1999) compared a central planning solution and GT solution to the Great Lakes region betw een different US states and The US and Canada. Dinar and Wolf (1994a) developed a cooperative game that incorporates economic as well as political considerations The model was applied to the western

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155 Middle East region. In anot her paper, Dinar and Wolf ( 1994b) identified and applied alternative allocation schemes using the same model. Loehman and Dinar (1994) applied game theory to an irrigation problem in Ce ntral Valley, California. Lejano and Davos (1995; 1999) discussed and app lied a cooperative game to a water reuse and irrigation project in southern California. Ozelkan and Duckstein (1996) used GT in a multicriterion decision making framework to anal yze the projects designed for the Austrian part of the Danube river. Dinar and Howitt (1997) evaluated cost allocation schemes, including a game model, for a drainage po llution problem in the San Joaquin Valley of California. Frisvold and Caswell (2000) anal yzed the application game approaches to US–Mexico water negotiations. Barreteau et al. (2001) applied a game theoretic model to the irrigation system of the Senegal River Valley. Fisher et al. (2002) developed a game theoretic model for the allocation of water in the conflict of the Israeli, Jordanian, Palestinian region. Loaiciga (2004) used a game formulation to quantify the roles of cooperation and non-cooperation in the sust ainable exploitation of a jointly used groundwater resource. Table 4-2 lists the previously described studies with respect to their area of application in water resources management. Conclusion The task of this chapter was to comprehe nsively present the normative theories of individual and group behavior under risk and uncertainty, star ting with the origination of the expected utility theory, its failure as a descriptive choice theory, its non-expected utility alternatives, then movi ng to the group choice theory of games, its representations, solutions, extensions, and alternatives. The chapter ended with an overview of application of utility and game theory to water resources management. The reader at this

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156 stage would have readily obser ved that there is a lack of unified model capable of simultaneously handling all of the phenomena described in this chapter (Machina, 1987); while normative formulations and axiomatic re ductionism are useful and practical in modeling individual and group d ecision situations under risk and uncertainty, they do not adequately consider all the cognitive and interpersonal dynamics (Obeidi et al., 2003). Table 4-2. Applications of GT in water resources management grouped into areas of application Subject Author (year of publication) Wastewater treatment and reuse / Irrigation (Barreteau et al., 2001; Bogardi and Szidarovsky, 1976; Degen, 1972; Dinar et al., 1984; Dinar et al., 1992; Dinar and Yaron, 1986; Dinar et al., 1986; Giglio and Wrightington, 1972; Lejano and Davos, 1995; Lejano and Davos, 1999; Loehman and Dinar, 1994; Rosen and Sexton, 1993; Sheehan and Kogiku, 1981; Tecle et al., 1988; Yaron et al., 1986; Yaron and Ratner, 1985; Yaron and Ratner, 1990) Fresh Water Resources Allocation (Adams et al., 1996; Becker and Easter, 1995; Becker and Easter, 1997; Becker and Easter, 1999; Ben Shachar et al., 1989; Bogardi and Szidarovsky, 1976; Dinar and Wolf, 1994a; Dinar and Wolf, 1994b; Fisher et al., 2002; Fraser and Hipel, 1980; Guariso et al., 1981; Heaney and Dickinson, 1982; Hipel et al., 1974; Hi pel et al., 1976; Kilgour et al., 1987; Loaiciga, 2004; Loehman et al., 1979; Okada, 1977; Ozelkan and Duckstein, 1996; Ransmeier, 1942; Rausser and Simon, 1991; Rogers, 1969; Rogers, 1993; Sheehan and Kogiku, 1981; Simon et al., 2003; Straffin and Heany, 1981; Suzuki and Nakayama, 1976; Szidarovszky et al., 1984; Thoyer et al., 2001; Tijs and Driesse n, 1986; Tisdell and Harrison, 1992; Tisdell and Harrison, 2002; Young et al., 1982) Water Quality (Bogardi and Szidarovsky, 1976; Dinar and Howitt, 1997; Frisvold and Caswell, 2000; Kilgour et al., 1988; Krawczaka and Zilkowskia, 1985)

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157 CHAPTER 5 A MULTIAGENT MULTIATTRIBUTE WA TER ALLOCATION GAME MODEL The allocation of scarce resources between conflicting parties exhibits complex interdependences between a multitude of factors, such as preferences and priorities, risk attitudes, equity, economic efficiency, pa rties’ capabilities and interactions, and uncertainty, to mention a few. As a pub lic resource, water conflicts have existed throughout recorded history; several are curren tly apparent at various local and international levels, and the risk for more grows as population and degradation pressures accelerate. Water is unique, essential, and sc arce; its value is not limited to survival and economics, but also has social and religious components. The mathematical quantification and analysis of conflicts over a single or multiple resources is an active area of economic, so cial, political, and psychological research. Many models and tools have been suggested, most of which within a utility and game theoretic framework, to deal with the intr icacies of natural and human behaviors. Motivated by this research, this chapter proposes an applied m odelling tool for common pool resources conflict resolution that combines essential concepts in an n parties game theoretic framework. Specifically, these concep ts are utility, ideal position, issue linkage, equity, salience, risk propensity, conflict leve l, and political uncertainty; they are based mainly on works by Morrow (1986), de Mes quita (1980, 1985), Maoz (1995), CioffiRivella and Starr (1995), Mulford and Berejikian ( 2002), and Abdollahian and Alsharabati (2003).In the next se ctions, we clarify the main c oncepts of research in the

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158 allocation of resources between conflicting parties. Subsequently, we present the mathematical formulation of the suggested m odel, followed by the illustration of this model using a hypothetical conflict situation over water, land, and financial resources between three different parties. Figure 5-1 exhibits the plan of this chapter. Figure 5-1. Chapter 5 organizational diagram Power and Preferences A key controversy in the field of multiple parties – multiple variables decision making under risk and uncertainty is the id entification of the goals motivating parties behaviors (Maoz, 1995), namely, their prefer ences and attitudes to wards other decision makers. It is argued that decision makers are motivated by power; power has been defined as control over resources, other de cision makers, or outcomes (Hart, 1976); the relationship between power and each of these variables is co ntroversial. It has been argued that these variables share a positive, negative or even no relationship (Maoz, Basic Concepts Model Formulation Model Application Definition of main concepts: preferences, power, linkage, probability, risk, equity, etc. Definition of variables and c onstraints, linearization of utility function, and defin ition of political uncertainty Definition of the multiparty multiattribute hypothetical case study Results and Analysis Presentation and comparison of results from different models Conclusions Summary of main findings

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159 1989; Maoz, 1995). An example of a negative relationship occurs when an increase in resources results in loss of control over the ou tcome, a situation, which, if not avoided, contradicts with the assumption of rationality of decision makers; he nce, preferences are not always monotonically increasin g in resources (Maoz, 1995). Literature has been devoted to discer n the principle factor governing such decisions; these factors include preference similarity, issues salience, importance, or priority (de Mesquita, 1974; de Mesqu ita, 1990; de Mesquita, 1997; Gamson, 1961b; Kuhn et al., 1983; Maoz, 1995; Meiste r et al., 1991; Stokman et al., 2000; van Assen et al., 2003), issue linkage (Lohmann, 1997; Morrow, 1986; van Assen et al., 2003), capabilities or power asym metry and distribution (de Mesquita, 1997; Lawler and Youngs, 1975; Stokman et al., 2000; van Assen et al., 2003; Vinacke and Arkoff, 1957), historical relationships, ideological similar ities, quality of communication (Kelley, 1968; Meister and Fraser, 1993), coalition size (de Mesquita, 1974; Ke lley, 1968), skills, strategy, representation, and competitivene ss (de Mesquita, 1974), influence on outcome (Kilgour et al., 1996), resources or reward divisibili ty and distributi on (Kelley, 1968; Kilgour et al., 1996; Lawler and Youngs 1975; Niou and Ordeshook, 1994; Ofshe and Ofshe, 1969; Wagner, 1986), resources scar city (de Mesquita, 1974; Wagner, 1986), resources initial distribution or position (Gamson, 1961a), ideal position or aspirations (Morrow, 1986), probability of reciprocati on (Gamson, 1961b; Ofsh e and Ofshe, 1969), ideological and attitudina l differences (Gamson, 1961a; Lawler and Youngs, 1975), resources for survival (Waltz, 1979), risk attitude (de Mesquita, 1974; de Mesquita, 1980; de Mesquita, 1985; de Mesquita, 1997; Ka hneman et al., 1991; Machina, 1987a), and equity consideration, ratio of gains, and tolerance limits (de Me squita, 1974; Morrow,

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160 1997; Mulford and Berejikian, 2002; Ofshe and Ofshe, 1969). Coalition formation investigates studies one or mo re of these factors among deci sion makers taking part in a conflict to determines which coalitions are more likely to form and the resulting payoffs distribution among these parties (Meister et al., 1991). Salience Theories The salience of an issue to a decision ma ker is defined as the importance of that issue to the decision maker or his willingness to spend infl uence on the issue in question depending on his priorities. A decision make r will exchange between issues when the relative importance of the issue to be compro mised is lower than the issue to be gained (Abdollahian and Alsharabati, 2003; de Me squita, 1997; Morrow, 1986; Stokman and van Oosten, 1994). Morrow (1986) introduced salience into his spatial expected utility game theoretic model developed as a matrix of weights that accounts for dependency between issues; hence, his model does not assume separability of issues. The utility of a decision maker or player i over an N dimensional issues space is represented as a matrix product such that 2 /ir T i i i ix x A x x x U where x is an N dimensional vector of outcomes for each issue, ix is i’ ideal positions vector over the outcomes, iA is a positive definite salience matrix for the issues, and i r represents the risk attitude of i. Graphically, a player’s the utility function is represented by a set of N dimensions indifference curves; in case of two issues, these curv es are elliptical with their center coordinates representing the ideal positions of that players over each i ssue. The eccentricity and orientation of the major axis of the indifference curves is function of the matrix of saliences iA. In the two issues case, an identity matrix representing separable issues of e qual salience results in

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161 circular curves; a matrix representing separa ble issues of different salience results in elliptical curves with axis pa rallel to the coordinates axis such that the major axis is parallel to the more salient issue coordinate; a matrix repr esenting non separable issues of equal salience results in ellipti cal curves with the major axis as the 45 degree angle of the coordinate axis; etc. The curvature or spaci ng of indifference curves with equal drops in utility between each pair of curves is function of the risk attitude of the player; equally spaced curves depict a risk neutral player, a fast drop in utility represented by spatially close curves around the ideal point depicts a risk acceptant player, and a slow drop in utility represented by a spatially distant curves around the ideal point depicts a risk averse player. Another example was developed by de Mes quita (de Mesquita, 1990; de Mesquita, 1997; de Mesquita and Stokman, 1994). The mo del is a utility based model that accounts for salience of players to the different issues, such that k i j i i i jk iax u x u s c which states that for two alternatives, jx and kx, the power mobilized by a player i in a comparison of these two alternatives equals the potential capabilities, ic, of i discounted by how important one issue is, is, and by how much i prefers one to the other, k i j ix u x u where j ix u is the utility of jx for i, equal to ir j i j ix x x u 1, where ir represents the risk taki ng propensity of player i, described in a following section. Note that in the context of th is model, capability is the power a player can bring to bear on an issue, defined as a frac tion of total capabilities of all players on that issue.

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162 Issue Linkage Theories Issues may be separable / non separable, where the outcome of one issue is irrelevant / relevant to the desired outcome of another issue, and hence the salience a coalition gives it. The separability of issues results in different orientation of the utility function as described previously (Morrow, 1986) For a more detailed analysis refer to Stokman and Van Oosten (1994), Morrow ( 1986), and Abdollahian and Alsharabati (2003). Equity Theories Several experimental studies have shown that individuals resist unfairness, are willing to sacrifice th eir own well-being to help or puni sh those who are being kind or unkind (Kahneman et al., 1986a; Kahneman et al., 1986b; Rabin, 1993). Fairness or equity are an essential component of behavi or affecting decisions and should be an integrative part of decisi on making models (Kahneman et al., 1986a; Kahneman et al., 1986b). Camerer (2003) suggested as an explan ation to such observe d social behavior over issues Edgeworth’s (1881) c onjecture that a player’s util ity for allocations includes both the player’s outcome and the weighted ou tcomes of others, with the weight varying systematically across situations. Other auth ors suggested the inequa lity-aversion theory where players prefer a higher outcome, namely money, and equal allocations, even if at the expense of the former (F ehr and Schmidt, 1999; McClin tock, 1972). Another theory is me-min-us Rawlsitarian th eory, in which players care about their payoff, minimum payoff, and the total payoff (Charness a nd Rabin, 2002). Fish burn (1984) suggested axioms for equity as equal utili ties, as fair division of probabilities of consequences, and fair distribution of sufferance in outcom es between coalitions (Fishburn, 1984b). Experiments, however, observed that accounting for absolute payoffs or even absolute

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163 differences in payoffs as coalitions tend to refuse outcomes that result in substantially higher gains to other coalitions, even if this outcome results in an improvement in its situation; the authors suggested that models should accounts for the ratio of utility gain between coalitions, rather than the absolute or difference in utility gains, in reaching conflict resolution (Mulford and Berejikian, 2002). Coalition Formation Theories Whenever two or more parties are involved in a conflict, the opportunity exists for a subset of the parties to form a coalition; these parties, motivated by several factors, as discussed previously, communicate and agree to pool their resources for the pursuance of common articulated goals and the consequent distribution of payoffs (Gamson, 1961a; Kelley, 1968). Methods that have been s uggested include minimum resource, balance, minimum power, bargaining, equal surp lus, minimum policy-distance, outcome grouping, option preference, ordi nal deductive selection system graph model, and triads theories; these techniques are summarized in the next paragraphs with examples; note that the examples were chosen for illustrati ve purpose and not as a comprehensive review of available methods which are a bundant in the literature (Aknine et al., 2004; Arnold and Schwalbe, 2002; Barbera and Gerber 2003; Browne and Rice, 1979; Kirman et al., 1986; Konishi and Ray, 2003; La wler and Youngs, 1975; Leimar et al., 1991; Michener and Myers, 1998; Sened, 1996; Slikker, 2001; Zinnes et al., 1978). Minimum Resource Theory Equity theory predicts the distribution of shares in proportion to a coalition’s members’ inputs. Suggested by Gamson ( 1961a, b), minimum resource theory, also known as proportionality proposition or pari ty norm (Browne and Rice, 1979), is a special case of equity theory (Komorita et al., 1989). It incorporat es initial resources

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164 distribution, non-utilit arian strategy preferences based on ideologies, attractions, etc., effective decision point to cont rol the decision, coalitions payoffs, such that a member’s payoff in a coalition is proportional to his in itial contribution, a nd reciprocal strategy choices, which necessitates that members recipr ocate the desire to form a coalition. Given the motivation to maximize reward, each member will seek the cheapest winning coalition, which is the coalition with fewest resources and in which his proportion of the resources are the greatest. Other authors us ed the minimum resource theory (Chertkoff, 1971; Stryke and Psathas, 1960; Vinacke and Arkoff, 1957). Balance Theory Balance or consistency theory (Mazur, 1968; Taylor, 1970), was suggested as a replacement for the minimum resources theory ; it argues that coalition situations are instances of the general class of social situati ons, in which a state of balance, consistency, or cognitive harmony is necessary for coalition formation a nd balance is a preferential state within a set of parties, or parties with similar characteristics, independently of the distribution of resources. It is argued that balance theory is a more accurate predictor of coalition formation than minimum resource (Crosbie and Kullberg, 1973); in addition, the latter may not apply to situation of competition over scarce resources (Davis, 1966). Minimum Power Theory This theory assumes that members of a co alition divide payoff in proportion to their pivotal power (Shapley and Shubik, 1954) A party’s pivotal power is found by examining all possible winning coalitions in wh ich this party is member and determining which of them would no longer be a winning co alition if that part y withdraws from it; it is based on the Shapley value, described in Ch apter 4. This theory predicts that the

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165 winning coalition that will form is the one with the minimum combined pivotal power; payoffs are distributed in proporti on to members’ pivotal power. Bargaining Theory Payoffs are divided halfway between what th ey would be if they were divided in proportion to initial resources, equity theory, and if they were divided equally among members, equality theory (Komorita et al., 1989). This is the sp lit-the-difference or asymptotic division of payoffs within a coalition. The equal excess model is a bargaining theory model (Komorita, 1979). Several e xperiments compare this theory to the minimum power and minimum resource theori es (Komorita and Nagao, 1983; Miller, 1980; Murnighan et al., 1977). Equal Surplus Theory In this method, each member of a coaliti on receives his initia l contribution and the surplus is divided equally among members (Komorita and Leung, 1985). Policy-Distance Minimization Theory Formally, denote the set of outcomes n iN N N N,..., ,...,1 and, for each decision maker, let k ir represent the ranking of a decision maker k to outcome i, now denoted k ir iN, such that the most and least preferred outcomes are ranked 1 and n; the preference vector of k is then n i j i i kN N N DM,..., ,...,1, such that the position of outcome i in the vector equals its rank k ir by this decision maker. Decision makers with more similar outcome orderings have greater goal compatib ility. The same logic applies to option ordering, with the set of options being n iM M M M,..., ,...,1 The Pearson product-moment correlation co efficient, PPMCC, is a parametric statistical rank correlation tec hnique that measures of linearity between two variables, in

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166 our case ranks. Formally, for a decision makers pair B A ,, 11 n r r rn i B i A i AB; this measure is the square root of the correlation coefficient, 2 r The closer r is to 1, the stronger the likely positive correlation. In the contex t of coalition formation, the greater the value of ABr, the higher the lik elihood of formation of the coalition B A (Kendall, 1975). Another statistical correlation coeffici ent is the Spearman’s rank correlation coefficient, SRCC; this method, however, is a nonparametric outcome based correlation metric. Formally, for each pair of decision makers, B A ,, define id as the difference in ranks for the thi outcome between these two decision makers, such that B i A i ir r d The SRCC is then obtained as 1 6 12 1 2 n n d rn i i AB; the closer r is to 1 (1), the stronger the likely positive (negative) correlation. In the context of coalition formation, the greater the value of ABr, the higher the likelihood of formation of the coalition B A ,. Note that, for both PPMCC and SRCC, all outco mes ranks and differences, respectively, are weighted equally irrespective of the salie nce of each outcome to each decision maker. A more recent policy-distance minimizat ion theory is the Minimal connected winning coalition, which is a coalition form ed from adjacent parties on the ideological continuum (Axelrod, 1970; Barry, 1971). Outcome Grouping Theory Kuhn et al. (1983) suggested an outcome base d metric using outcome grouping to determine the relative likelihood of coalition formation, where for a pair of decision makers B A ,, an outcome group if defined as a set of outcomes occupying consecutive

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167 ordinal positions in the preference vectors, ADM and BDM, being compared. The number of ordinal groups is di rectly proportional to preference s similarities. This method can be used for coalitions with more than three members, unlike the PPMCC and SRCC methods (Kuhn et al., 1983). Option Preferences Theory Fraser and Hipel (1989) develo ped an option based metric, kC, which measures the degree of similarities between d ecision makers of a coalition k, based on their options preferences, such that m i I k i kk kr C1 ,,2 1, where m is the number of available options, 1 1, k ir the correlation or level of agreement for option i in the coalition, and k iI, the coalition importance for option i obtained as the averag e of levels of option i in the preference tree. The greater kC corresponds to a greater likelihood of coalition k formation. Ordinal Deduction Selection System Theory Another model, developed by Meister and Fraser (1993), is th e ordinal deduction selection system, ODSS; this model incorporat es, in addition to ordinal preferences, issue salience. Formally, define a set of m alternatives or options, n iM M M M,..., ,...,1 and n issues, n jN N N N,..., ,...,1, such that j iv, represents the level of satisfaction that alternative iM provides for issue, or outcomes, jN. For each decision maker, the set of issues are ranked in accordance with its importance to that decision maker; then, for each issue, the alternatives are ordered according to the level of satisfaction they provide for that issue and an overall alternatives ranking is obtained (Meister and Fraser, 1993). The authors applied this method for two member coalitions using an international

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168 trade example, where the issues are preferen ce similarity, power asymmetry, historical relationship, ideological similarity, and quality of communication. Graph Model Theory Kilgour et al. (1996) studied coalition formati on in the framework of the graph model for conflict resolution (Fang et al., 1993; Kilgour et al., 1996) The authors argued that a higher preferences similarity between the parties of a possible coalition does not necessarily imply a higher like lihood of that coalition forma tion and that the impact of that coalition on the outcome to achieve a mutu ally preferred equilibrium plays a stronger role in defining this likelihood. Triads Theory Analysis triads, or three-parties, interactions lends itself as the simplest way to study coalition formation which results may be generalized into larger groups behavior (Caplow, 1956; Caplow, 1959). Several experi mental analyses have been performed using triads (Caplow, 1956; Caplow, 1959; Ch ertkoff, 1971; Mills, 1953; Mills, 1954); these experiments confirmed Simmel (Wolff, 195 0) reports that a triad tends to segregate into a pair and a third part y, where that coalition form ation is function of small differences in power and other characteristic s of the triad members. Chertkoff (1967) modified Caplow’s theory to include probabi lities of parties’ choices (Chertkoff, 1967). This theory does not predict payoffs di stribution between members (Miller, 1980). Probability of Coalition Formation In research on coalition formation, bot h the probability of occurrence of each possible coalition and the reward division am ong the coalitions’ members are essential in the comparison of different coalitions (Kom orita, 1978). Coalition formation theories, however, do not predict the like lihood of a coalitional struct ure formation, or, in other

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169 words, the probability of forming different coalitions (Michener and Myers, 1998). Few of these methods have been proposed, such as the size-probability m odel, the Johansen-C model, and the central union methods, comp ared below. For that purpose, following Michener and Myers (1998), let’ s define a coalition structure zS, with Z z ,..., 1 as a partition of the n decision makers into m coalitions, such that n m 1; different coalition structures combinations form different coalition sets. Size-Probability Model Theory Michener and Myers (1998) suggest that su ch a theory may be used as a submodel of coalition structures soluti ons that only provide insight into the dist ribution of payoffs in coalitions, without any insight into the likelihood of forma tion of that latter, such as the Shapley, equal surplus, and nucleolus methods. This theory assigns to each coalition structure in a game a probability of form ation proportional to the size of the largest coalition in that coalition structure; coali tion structures including relatively larger coalitions have a higher probability over ones including smaller coalitions. A score is assigned to each coalition structure in the ga me equal to the number of players in its largest coalition; these scores are then summe d over all coalition structures in the game. The probability of formation of a coalition stru cture is then the ratio of its score to the total scores (Michene r and Myers, 1998). Johansen-C Probability Model Theory This model assigns equal probability of fo rmation to all coalition structures that satisfy the condition that a coalition structur e is not formed if their exists another combination of its coalitions that results in higher payoffs to its member players (Michener and Myers, 1998). Like the previ ously presented model, this method may be

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170 used as a submodel of techniques that only provide solutions to payoff distribution in coalition structures. Central Union Theory One theory that, in addition to predicting coalitions’ payoffs distribution, predicts the probability of formation of the various coalitions is the central union theory (Michener, 1992; Michener and Au, 1994). The authors define a sidepayment cooperative game with n decision makers or players. Payoff satisfaction of a player i in the coalition structure zS, zS i PS ,, is defined as the ratio of payoffs that players obtains in zS over that player’s payoffs aspirations or best alternat ive. Acceptance, zS A of a coalition structure zS is defined as the reciprocal of the sum over players of the pairwise differences in payoff satisfaction, zS i PS ,, among all players within it, such that a player’s i payoff satisfaction in zS. If, in zS, the differences in zS i PS between players is low, zS A is high. Formally, defining a binary variable 1 0 I that indicates whether zS is a member of the coalition set or not, the probability zS P of formation of a coalition structure zS is Z z z z z z zS I S A S I S A S P1, n i n j z z zS j PS S i PS S A11, 1 Willingness and Opportunity Theory The authors of this method analyze the o ccurrence of a politi cal event, or its probability, such as coalition formation, as a function of the willingness of the involved parties to be part of it and the opportunity for those parties to do so (Cioffi-Revilla and

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171 Starr, 1995). The authors pr esented a first order and sec ond order formulation of the dependency of the probability on willingness and opportunity. A formal presentation of the first order treatment of this theory is presented in a following section. Cohesion Theory Maoz (1995) developed a measure of cohesion between a coalition’s members based on the members’ ranking of each others ’ preferences; the measure is obtained as the ratio of disagreement be tween those parties over the maximum possible disagreement between them. Although Maoz does not develop probabilities of coalit ions formation, he defines homogeneous and heterogeneous coalit ions as ones with high and low cohesion between its members, respectively. The formal presentation of this theory follows in another section. Stochastic Communication Structures This model defined the probability of co alition formation as a function of the probability that players communicate, such that abp is the probability that a player a communicates with player b; it is assumed constant for all players pairs, taken as the worst probability that two players in a game communicate. The authors (Kirman et al., 1986), using graph theory, define the probability of a graph formation as m n n mp p p 2 / 11, where m and n are the arcs and vertices of the graph, respectively. Risk Attitudes The analysis of risk attitude s depends on the preferences of the decision makers and the properties of the function representing these preferences, or the utility function (Karni, 1979). The shape of the utility func tion determines risk attitudes. A concave

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172 utility function corresponds to risk aversion, where the expected value of a lottery is preferred to the lottery itself, with x E U u E which implies that the individual prefers a sure gain of x E with a utility x E U over the lottery. On the other hand, a convex utility function corresponds to risk ac ceptance, with preference for bearing the risk rather than receiving the expected value, i.e., x E U u E (Machina, 1987a). Studies revealed that individuals presented a fourfold pattern of risk attitudes (Table 5-1): risk seeking for gains and risk aversion for lo sses of low probabilities and risk aversion for gains and risk seeking fo r losses for high probabilities (Kahneman and Tversky, 1979) (Fishburn and Kochenberger, 1979, Hers hey and Said, 1987; Wehrung, 1989; and Tversky and Kahneman, 1992; Camerer, 1992). Table 5-1. Pattern of risk attitudes. Gain Loss Low Probability Risk seeking Risk aversion High Probability Risk aversion Risk seeking The next paragraphs present examples of risk measurement techniques that were developed, starting with the Pratt-Arrow th eory, and moving to its generalizations to multiattribute situations. Pratt-Arrow Model The first measurement method of risk av ersion was developed by Pratt (1964) and Arrow (1965, 1971) who developed a theory of risk based on the cu rvature prope rties of the utility function as a function of wealth around a given presen t wealth state, to interpret economic behavior in risky situations, within the von Neumann and Morgenstern framework of maximizing expect ed utility. They showed that, assuming that U is twice differentiable and increasing on X the degree of concavity of a utility function U, measured by the curving index absolu te risk aversion, or risk premium

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173 x U x U' / ', and the relative risk aversion x U x xU' / ' determines how risk attitudes varies with indivi duals and wealth levels. U is risk averse if 0 ' x U, risk seeking if 0 ' x U, and risk neutral if 0 ' x U. Defining the certainty equivalent of p p c, as the sure amount in X at which the individual is indifferent between this amount and p i.e., p p c~ and p x E p c u, U is risk averse if p x E p c,, risk seeking if p x E p c,, and risk neutral if p x E p c, Comparing two different utility functions, the more risk av erse possesses steeper indifference curves. The Pratt-Arrow theory of risk attitudes was generalized by several authors in a univariate and multivariate contexts (Dun can, 1977; Jia and Dyer, 1996; Karni, 1979; Keeney, 1973; Kihlstrom and Mirman, 1981; Levy and Levy, 1991; Li and Ziemba, 1989; Machina and Neilson, 1987; Paroush, 1975; Ross, 1981; Rubinstein, 1973; Stiglitz, 1969). Some example theories are provided below. Risk Aversion Matrix The absolute risk aversion matrix is a ge neralization of the Arrow-Pratt local risk premium to compound risks and multivariate situation (Duncan, 1977); it is defined as i ij ijU U r R, such that iU and ijU are the first and second derivatives of the utility function U on the multivariate set of outcomes. Like the authors in the univariate case (Arrow, 1971; Pratt, 1964), Duncan used Taylor series approximation to derive his risk measure. Karni (1979) introduced a positive definite matrix measure of absolute multivariate risk aversion, similar to Dun can using the concept of risk premiums. Formally, the author defined a risk premium function

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174 n i n i ij ijp y Z p y11 1, 2 1 , such that nz z Z,...,1 is a random vector of small risks such that the sum of their probability is one, p y, is a strictly concave indirect utility function of the n dimensional commodity vector y and its 1 n dimensional vector p with 1 and ij its first and second derivatives, and ij the variance. The matrix measure of local risk aversion, defined as 1 ij ijr R, is proportional to the risk premium per unit of variance on its diagona ls and to the excess of risk premiums per unit covariance on its off-diagonal elements. An other example of the matrix method is as a generalization to Rubinste in (1973) risk premium was suggested by Li and Ziemba (1989). Risk-Value Theory Risk-value theory leads to a decision ma king by explicitly trading-off between risk and value; it may be based on both expect ed and non expected utility theories of preferences (Jia and Dyer, 1997). Several models of this form have been suggested (Bell, 1988; Bell, 1995; Jia and Dyer 1996; Jia and Dyer, 1997; Sa rin and Weber, 1993). An example is provided. Jia and Dyer (1996) proposed a standard meas ure of risk that can be used implicitly or explicitly in the expected utility model defining the standard risk-value model. Formally, the standard measure of risk is defined as X X V E X R ', where .V is the value function, X is a probability distribution or a random outcome, X is its mean, and X X X the standard risk or the normalization of X to eliminate the effect of preferences; the measure of risk, pref erence function ,and trade-off factor can be

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175 treated independently. For a linear plus exponential utility model, cxbe ax x V the standard risk measure is 1 X X ce bE X R; the linear functiona l has no effect on the risk measure. This risk measure represen ts a globally risk averse risk-value function. Moments Risk-Value Model In cases when the probability distributions are not available, mean and variance are used in the risk-value trade-off model (Jia and Dyer, 1997; Ma rkowitz, 1991; Sharpe, 1991), leading to a risk adjusted value func tion. A mean-variance-skewness model is represented as 3 2' ,X X cE X X kE X X X U E (Jia and Dyer, 1997). Another example is an exponential risk adjusted utility function, where X is a multiattribute vector and r is the risk attitude, 22j j jr X r rXe e X U (Roberts and Urban, 1988). In a third example, the authors used a real-valued risk measure such that 21X X E X X R where 1 0 (Pollatsek and Tversky, 1970). De Mesquita’s Risk Model De Mesquita (1980, 1985, 1997) describes the measure of risk propensity as an indicator of the size of the tradeoff made by a coalition between pursuing political versus policy satisfaction, where political satisfaction is the desire to be seen as an essential member of the winning coalition even if at the expense of pursui ng the policy it really wants; on the other hand, policy satisfaction refers to supporting a policy outcome close to the preferred choice. While political satisfaction reflects a fear of vulnerability and a need of security, hence a tendency to be risk averse, policy satisfaction reflects a tendency to be risk acceptant. Different ly, the measure of risk provides basis for estimating the value attached to the status quo.

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176 This theory assumes that players decl are policy positions that represent a compromise between what they really want and what they be lieve is feasible. Defining n i ijEU1 as i’s perceived utility that players i j derive from challenging player i, n i ijEU1 min and n i ijEU1 max as the outcomes that would minimize and maximize player i’s vulnerability, the risk propensity of a player is defined by how proximate i’s actual policy is to these extremes, such that n i ij n i ij n i ij n i ij n i ij iEU EU EU EU EU R1 min 1 max 1 min 1 max 12 This measure is then transformed to the measure 3 1 3 1i i iR R r such that 2 5 0 ir, that increases with risk aversion. Model Development As mentioned earlier, this works undert akes the development of an applied modeling tool for common pool resources conf lict resolution, between multiple coalitions over multiple issues and desired outcomes, by combining different concepts, namely, utility, position, issue linkage, ratio of gain, salience, risk propensity, conflict level, and political uncertainty. The model describes a decision making procedure developed in a utility function optimization / game theoretic framework with a Ndimensional Euclidian players and issues spaces and a D dimensional Euclidian outcome space. A description of the model components is presented in Table 5-2. Verbally, single and multiple players coal itions’ preferences for issue outcomes are represented by a single peak ed continuous positive bounded utility function. As

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177 developed by Morrow (1986), the general form of this function reflects the differences in the (1) ideal points, (2) salience of issues, (3) combination of issues, and (4) attitudes toward risks, such that a coalition’s util ity for an outcome is the salience weighted distance between this outcome and the ideal poi nt altered by the risk attitude of that coalition. As mentioned earlier, while the ideal point affects the utility value, i.e., the preference for an outcome drops off as the distance between its ideal point and the outcome increases, the salience, separability and risk measures affect the shape, orientation, and curvature of the function, respectively. From a game theoretic angle, the mode l does not assume a cooperative or noncooperative framework; in other words, the solu tion may be self enforc ing or contractual. In addition, the game may be played as a one-shot static game, a repeated / multistage game, which consists of severa l stages of the static game over a time horizon; in both, however, coalitions move or reveal there id eal position, salience, and risk attitudes simultaneously. In a repeated game, if a co alition is dissatisfied with the expected outcome, there are essentially four course s of action by which the coalition might improve its prospects: (1) alter its own level of effort (i.e. change its salience), (2) shift its revealed position (i.e. its ideal point) (3 ) influence other coali tions to change their salience, and (4) influence othe r coalitions to change their id eal point, with respect to one or more issues, through maneuver or coercion. The model is repeated until the conflict is resolved (de Mesquita et al., 1997). Multistages allow the game to evolve into an adaptive or evolutionary situation. Revealed preferences and attitudes do not have to be actual ones, a situation referred to as bluff or bully games (Dacey and Carlson, 1996); for example (de Mesquita, 1974), a

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178 party that asks for more has a higher chance of obtaining more, until a certain limit, at which other parties loose incentives for c ooperation. In such cases, unless other coalitions know of the possibi lity of bluffing by other coalit ions, the game is one with incomplete information, such that players do not share the same information; if coalitions possess probabilistic information about other coalitions’, the game may be treated as a Bayesian game. In addition, these preferences may change for each coalition at each stage of the game. Due to cognitive limitations, coalitions are assumed to be partially or boundedly rational in adapting and learning about other coalitions and predicting iterated choices and future outcomes. As a result, the m odel may be solved in the framework of behavioral game theoretic models, namely, limited thinking or learning models. In addition, coalitions may exhibit social beha viors such as reciprocity, where those who cooperate typically expect ot hers to cooperate, and vengeance, where a coalition is willing to sacrifice its outcom e to punish others who were unfair; in such cases, the model may be solved using social preferences behavioral game theoretic models such as the inequality-aversion, me-min-us, and reciprocity theories.

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179Table 5-2. Definiti on of model components Model Component Formal Definition Description Eucledian Space n N2 1 d D2 1 Ndimensional realm of players and issues D dimensional realm of outcomes Player n i I 2 1 ˆ & n h H 2 1 ˆ Ndimensional vector of parties involved in conflict over issues in Jˆ and K ˆ Issue n j J 2 1 ˆ & n k K 2 1 ˆ Ndimensional vector of matters of dispute between the different players in I ˆ Salience i S S S S S S S S S Snn i n i n i n i i i n i i i jk i ˆ2 1 2 22 21 1 12 11 NxN dimensional positive square matrix where the diagonal elements represent salience of player i for each issue, j and off-diagonal elements measure interdependence between issues (equal to zero, when issues are separable in terms of effects); it ranges from 0 (low salience) – 1 (high salience).

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180Table 5-2 Continued Model Component Formal Definition Description Risk Propensity i P P P Pn i i i j i ˆ2 1 i P P Pj i n j j i i max 21 Ndimensional vector of risk propensity of player i to issue j iP is a scalar ranging between 0.5 (risk aversion) and 2 (risk acceptance) depending on player i’s risk perception. Note that iP is a scaled measure. Variable i x x x x xin ij i i ij ˆ2 1 Ndimensional vector of variable coordinates on issues axis for player i Ideal point i x x x x xn i j i i i j i ˆ2 1 Ndimensional vector of player’s i ideal coordinates for each issue j Distance i x x x x x x x x X X X X Xn i in j i ij i i i i n i j i i i j i ˆ2 2 1 1 2 1 Ndimensional vector of absolute distances between variable coordinates and player’s i ideal coordinates for each issue j Utility n i iU U U U U 2 1ˆ Ndimensional vector of utilities of players to a certain outcome. Note that the utility of i at its ideal coordinates vector, j ix ˆ is at its maximum.

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181Utility Function It is common practice to st udy coalition formation in characteristic function form of the games, where the value of each coalition is represented by a characteristic function; the characteristic function represents, for each coalition, the quality of the optimal solution or of the bounded rational value. A coalition’s characteristic function depends on members that are not part of it due to pos itive and negative externalities. Negative interactions include shared resources or conflicting goals, while positive externalities include partially overlapping goals (G amson, 1961a; Larson and Sandholm, 2000; Sandhlom and Lesser, 1997). A player’s utility for a multidimensional outcome is a function of the distance between that outcome and the player’s ideal point, weighted by its salience, jk iS ˆ, and altered by its risk attitude, j iP ˆ. Formally, a player’s utility, iU is stated in equation 1. Note that it is a negative f unction with a maximum of zero. 2iP T i jk i i iX S X U (1) The optimization problem is formulated as a multiobjective utility maximization statement; The optimal outcome of a player i is obtained by solving: i 2iP T i jk i i iX S X U Max (2) to Subject Political, Economic, Physical, Soci al, and Environmental constraints Let T i jk i i iX S X V ; then iP i iU V2 similarly, 2iP i iV U For mathematical simplicity, we will use iV (Equation 4) instead of iU wherever possible.

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182T i jk i i iX S X V (3) Extending Equation 3, we obtain, n i j i i i nn i n i n i n i i i n i i i n i j i i i iX X X X S S S S S S S S S X X X X V 2 1 2 1 2 22 21 1 12 11 2 1 (4) Further extensions results in the following summation, mm i m i m i i m i i m i m i m i i i i i i m i m i i i i i i iS X S X S X X S X S X S X X S X S X S X X V 2 2 1 1 2 22 2 21 1 2 1 21 2 11 1 1 (5) Equation 5 can be compacted in a random terms series summation form (Equation 6). n k n j jk i j i k i iS X X V1 (6) Hence, 2 1iP n k n j jk i j i k i iS X X U (7) To solve the problem using Linear Progr amming (LP) we need to linearize the objective function obtained in Equation 7. Fo r that purpose, Taylor series approximation is defined and applied.

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183 Let ) ˆ (Y f be a function of the vector n ny y y y Y1 2 1ˆ of n variables. Taylor series approximation, in its matrix form, applied to the function ) ˆ (Y f at the vector n na a a a A1 2 1ˆ is defined in Equation 8. 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ 2 ˆ T A AA Y Y f A Y Y f A Y A f Y f (8) Note that AY fˆˆ is the Jacobian of ) ˆ (Y f (Equation 9) at the vector A ˆ and AY fˆ 2ˆ is the Hessian ) ˆ (Y f at the vector A ˆ, which is not needed in the first order approximation. ny f y f y f Y f2 1ˆ (9) To obtain the first order Taylor approximation of iU, we derive the first order derivatives of the function iV (Equation 10) and deduce those of iU from it (Equation 12) using Equation 11. The Jacobian of iU follow in Equation 13. n l ql i lq i l i q i iS S X X V1 (10) q i i P i i q i iX V V P X Ui 1 22 (11) n l ql i l i n l lq i l i P i i q i iS X S X V P X Ui1 1 1 22 (12) n l l i l i l i n l l i l i l i P i i n i i i i i i q i iS S X S S X V P X U X U X U X Ui1 1 1 1 1 1 1 2 2 12 ˆ (13)

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184 Using Equations 8, and substituting with Equation 13, the first order Taylor series approximation of the function iU at the vector j i j ixˆ ˆ n i n i i i i ix x x 2 2 1 1ˆ is obtained in Equations 14 and 15. n k n j j i j i ij kj i jk i j i x P i i ix x S S V P Uj i i11 ˆ @ 1 22 (14) n k n j j i j i ij kj i jk i j i P n k n j jk i k i j i i ix x S S S P Ui11 1 2 112 (15) Relative Gain The ratio-differences of payoffs, or relative gains, are as important as the absolute differences, or absolute gains, in affec ting the likelihood of the implementation of a solution. In the case where the optimal so lution of a problem results in an unequal distribution of gains, competing parties are not likely to cooperate. This is included in the previously presented issue linkage model as a constraint (Equation 16). i k i kU U (16) and represent the range of allowed ra tio difference between the player utilities to ensure equity. Coalition Formation Let 12 , 123 1 , 12 , 2 1 n n n C be the set of all subsets of N with size n tt n c1 of possible number of coaliti ons. A coalition is denoted by

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185 Maoz defines maxD as a measure of conflict of inte rest which represents the actual disagreement in rankings for a set of play ers; its general formula in a group of m members on the basis of their ranking over n members is: 12 12 2 max n nm D (17) where Dmax is the maximum possible sum of s quared rank differences on the basis of all the 2 / 1 m m pairwise comparisons in this group. The actual conf lict of interest in a group of m members given n ranks is: 1 122 2 1 1 11 2 n nm r r Conn h m i m i q h q h i (18) where Con stands for group conflict, h ir is the rank assigned by player h to player i ’s policy, k qr is the rank assigned by player h to player q. The cohesion of a given coalition is simply the complement of the conflict of interest within the coalition: Con Coh1 (19) What a player derives from a coalition is a function of what the coalition can do given its resources and its cohe sion. The cohesion of a coalition is a function of the preferences of its members. A highly cohesi ve coalition is composed of members with very similar or identical prefer ences. On the other hand, a heterogeneous coalition is one composed of members with different policies. In order to form, each member must take some policy concessions. In this case, the co alition cannot be seen as a simple sum of its resources. Rather, that sum weighted by its cohesion Coh.

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186Political Uncertainty Let W, O, and B symbolize the willingness, opportunity, and occurrence of an event materialization. By definition, W and O are necessary conditions for B ; formally, the causal relation is O W B Political behavior is also uncertain, as the occurrence of of W and O, and thus B is uncertain. Formally, there exists probability measures such that ) Pr( B b ) Pr( W w ) Pr( O o where b, the probability of political behavior, is the dependent variable, and equals the product of the probabilities of willingness and opportunity (Equation 20), where o w z 2) Pr( ) Pr( ) Pr( ) Pr( z wo b O W O W B (20) Note that both w and o alone overestimate the value of b. Moreover, political behavior occurs with probability smaller than the least probable of either willingness or opportunity. The magnitude of this gap, a ge neral property of real-w orld politics can be measured by the difference, z b where 0 defines hypoprobability, 0 defines hyperprobability, and 0 as b z Informally, hypoprobability is the additional degree of political difficulty that is involved in combining willingness and opportunity to produce behavior; that extra effort that goes beyond the distinct difficulties of separately producing willingness and opportunity. Finally, assuming that the probabili ty of a coalition formation, as a political event, is directly proportiona l to the measure of the c ohesion between its members, b is estimated using Equation 21, where is the coefficient of proportionality, or scaling constant. Coh b (21)

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187 The probabilities of formation of co alitions that includes a player i should, by definition, sum up to 1. Hence, for a player i for all i i In addition, the formation of a certain coalition necessarily occurs with the formation of the other coalitions to include the rest of the players, leading to different coalition combination scenarios. The probability of a scenario is equal to the product of the independent event of its coalitions’ probabilities and is denoted as sb where scenario all for sb b. The sum of all scenarios probabilities should be 1. Modified Utility Function The utility function for a player i is obtained by summing its utilities weighed with their corresponding probabilities as part of all scenarios (Equation 22). Note that a coalition of more than one player divide its utility equally between its members. s n k n j j i j i ij kj i jk i j i P n k n j jk i k i j i i scenarios all for s ix x S S S P b Ui 11 1 2 112 (22) Hypothetical Application To demonstrate our model, we use a si mple hypothetical situation. A common conflict arises in an expanding town is the allocation of resources to the various competing parties, in this case the town residents and the agri cultural and industrial sectors. A council represents each party in negotiations. For the purpose of this study, we assume that the issues of conflict are the allocation of land, water, and financial resources. Formally, three different players 3 n are in conflict over three issues. The players, denoted by numbers, the importance th ey allocate to the different issues, and

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188 their ideal point expressed as a ratio of the available resource s, are described in Tables 53 and 5-4. Although the utility function are linear in terms of the issues, its dependence on the salience and risk attitude of the player rende rs the interpretation of its behavior rather complex without graphical analysis; the utility functions of player 1, 2, and 3 are depicted in Figures 5-2 – 5-3. Starting from the firs t cell on the left, and moving to the right and then downward, the utility function is plotte d for increasing values of issue 3, ranging from 0 to 1. In each cell, the utility functi on is plotted as a function of issues 1 and 2 (horizontal and vertical ax is, respectively) for a cons tant value of issue 3. Table 5-3. Player description Issue Importance (% of resource needed) Player Plan Financial 1 Land 2 Water 3 1 City council Public gardens and facilities for families and children Average importance (0.3) High importance (1) High importance (0.5) 2 Industrial council Industrial establishments for increase in number of job opportunities No importance (0) Av-High importance (0.5) Average importance (0.5) 3 Agricultural council Land, equipment, and water subsidies for increase in production High importance (1) High importance (1) High importance (0.7) Table 5-4. Input data Player Status quo Ideal point Salience Risk Propensity 1 0 0 0 ˆ1jx 5 0 0 1 3 0 ˆ1jx 1 0 0 0 1 0 0 0 5 0 ˆ1 jkS 71 1 5 1 5 0 5 1 ˆ1 1 P Pj 2 0 0 0 ˆ2jx 5 0 5 0 0 ˆ1jx 5 0 0 0 0 1 0 0 0 0 ˆ1 jkS 00 2 5 1 5 0 0 2 ˆ2 2 P Pj 3 0 0 0 ˆ3jx 7 0 0 1 0 1 ˆ1jx 1 0 0 0 1 0 0 0 1 ˆ1 jkS 68 0 5 0 5 0 5 0 ˆ3 3 P Pj

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189 Let’s start with the domestic council utility function, Figur e 5-2. For any value of issue 3 (any cell), for a value of issue 1, the ut ility increases with an increase in the value of issue 2. This straightforward behavior is explained by the standpoi nt of player 1 w.r.t issue 2, where it is assigned a maximum importance (salience of 1), maximum risk adversity (risk propensity of 0.5), and an idea l point of 100 percent of the total available resource. The behavior of the utility functi on with respect to issue 1 is more complex, however. For a value of issue 3 (any cell), fo r a value of issue 2, the utility increases until the ideal point of the player is reached, and then decrease s afterwards. Note that the player gives this issue a medium importance (s alience of 0.5) with a more risk acceptant attitude (risk propensity of 1. 5). Finally, moving through the cel ls from left to right, and then downward, for any values of issues 1 and 2, the utility function increases to a maximum corresponding to the pl ayer’s ideal point on issue 3 and then decreases. The salience the player assigns to issue 3 is the same of that he assigns to issue 2 (salience of 1) while the risk propensity is the same as that assigned for issue 1 (risk propensity of 1.5). The utility function is different for the indus trial council, Figure 5-3. For any value of issue 3 (any cell), the utility is unaffected by a change in issue 1. This behavior is explained by the standpoint of player 1 w. r.t issue 1, where it is assigned no importance (salience of 0), minimum risk adversity (risk propensity of 2), and an ideal point of 0 percent of the total available resource. For a value of issue 3 (any cell), for a value of issue 1, the utility increases until the ideal po int of the player is reached, 50 percent, and then decreases afterwards. Note that the player gives this issu e a medium importance (salience of 1) with a risk averse attitude (risk propens ity of 0.5). Finally, moving

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190 through the cells from left to ri ght, and then downward, for any values of issues 1 and 2, the utility function increases to a maximum corresponding to the player’s ideal point on issue 3 and then decreases. The salience and risk propensity the play er assigns to issue 3 are medium (salience of 0.5 and risk propensity of 1.5). The utility function of the ag riculture council is presente d in Figure 5-4. For any value of issue 3 (any cell), for a value of issu e 1, the utility increases with an increase in the value of issue 2. This straightforwar d behavior is explained by the standpoint of player 1 w.r.t issue 2, where it is assi gned a maximum importance (salience of 1), maximum risk adversity (risk propensity of 0. 5), and an ideal point of 100 percent of the total available resource. The behavior of the utility function with respect to issue 1 is the same. Finally, moving through the cells from left to right, and then downward, for any values of issues 1 and 2, the utility func tion increases to a maximum corresponding to the player’s ideal point (70 percen t) on issue 3 and then decreases. The salience and risk propensity the player assigns to issue 3 are the same of that he assigns to issues 1 and 2 (salience of 1 and risk propensity of 0.5). Note that in all Figures 5-2, 5-3, and 5-4, the maximum utility of zero is reached for the ideal point – 10 percent. This is due to the fact that the Taylor’s series approximation was applied at a va lue of 10 percent.

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191 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5-2. Change of the utility function for coalition 1 w. r. t. issue 1 (horizontal axis) and issue 2 (vertical axis) for increasing values of x3 (0 – 1, top to bottom, left to right).

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192 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5-3. Change of the utility function for coalition 2 w. r. t. issue 1 (horizontal axis) and issue 2 (vertical axis) for increasing values of x3 (0 – 1, top to bottom, left to right).

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193 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5-4. Change of the utility function for coalition 3 w. r. t. issue 1 (horizontal axis) and issue 2 (vertical axis) for increasing values of x3 (0 – 1, top to bottom, left to right).

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194 The rankings of players to each other ar e presented in Table 5-5. The least favorable player it the industrial council, pl ayer 2, which has the least favor for and by both other councils. Table 5-5. Players ranking matrix Player Ranks 1 2 3 1 1 3 2 2 3 1 3 Player 3 2 3 1 There are seven possible coalitions 7 c (Table 5-6) and five possible coalition combinations scenarios (A – E) (Table 5-7). Table 5-6. Coalitions Coalitions Players Comments 1 1 1 member coalition 2 2 1 member coalition 3 3 1 member coalition 4 1 and 2 2 member coalition 5 1 and 3 2 member coalition 6 2 and 3 2 member coalition 7 1, 2, and 3 All players; cooperation, total coalition Table 5-7. Coalition combination scenarios Scenario Coalition(s) Combination A 1 2 3 B 1 6 C 5 2 D 4 3 E 7 The probabilities of formation of co alitions that includes a player i should, by definition, sum up to 1 (Equation 23).

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195 1 1 17 6 5 3 7 6 4 2 7 5 4 1 b b b b b b b b b b b b (23) The same applies to the sum of the scenar ios probability (Equation 24), where the probability of each scenario is the product of the probabilitie s of coalition formation as independent events. 1 17 4 3 5 2 6 1 3 2 1 b b b b b b b b b b b b b b bE D C B A (24) Substituting by the previously defined probability (Equation 21), we obtain, 1 1 1 17 7 4 4 3 3 5 5 2 2 6 6 1 1 3 3 2 2 1 1 7 7 6 6 5 5 3 3 7 7 6 6 4 4 2 2 7 7 5 5 4 4 1 1 Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh (25) As scaling constants, the s in each individual equation are set equal and denoted 1 2 3 and respectively. 1 1 1 17 4 3 5 2 6 1 3 2 1 7 6 5 3 7 6 4 2 7 5 4 12 2 2 3 3 3 3 3 2 2 2 2 1 1 1 1 Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh Coh (26) Table 5-8 summarizes the obtained results. As expected, the probability of the industrial council forming a coalition with any of the other councils is null. As a result, two possible negotiation scenario s are studied: (1) scenario A: the councils negotiate as independent one-member coalitions (3 2 1, ), and (2) scenario C: the city and agricultural councils negotiates as one coaliti on with similar objectiv es, all the resources,

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196 salience, highest, and risk attitudes, ri sk aversion, against the industrial council (5 2, ); the input data for this scenario are presented in Table 5-9. Table 5-8. Measuring the probability of formation of different coalitions Coalition Coh b 1 1 0.571 2 1 1.000 3 1 0.571 4 0 0 5 0.75 0.429 6 0 0 7 0 0 Coalition(s) Combination (Scenario) scenarioCoh scenariob 1, 2, 3 (A) 0.460 1, 6 (B) 0 5, 2 (C) 0.540 4, 3 (D) 0 7 (E) 0 Table 5-9. Input data Coalition Status quo Ideal point Salience Risk Propensity 2 0 0 0 ˆ2jx 5 0 5 0 0 ˆ1jx 5 0 0 0 0 1 0 0 0 0 ˆ1 jkS 00 2 5 1 5 0 0 2 ˆ2 2 P Pj 5 0 0 0 ˆ3jx 0 1 0 1 0 1 ˆ1jx 1 0 0 0 1 0 0 0 1 ˆ1 jkS 68 0 5 0 5 0 5 0 ˆ3 3 P Pj The first scenario results in 9 variables, corresponding to the am ount of each of the resources allocated to each of the coalitions. The second s cenario involves 6 variables. These quantities are bound by the non-negativity constraint. Naturally, for each resource, the sum of allocated quantities to all players should a dd to unity. To assess the effect of

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197 the ration of gain constraint, th e scenarios were run with and without it. The solutions to both cases are displayed in Ta bles 5-10 and 5-11 and Figures 5-5 and 5-6, respectively. In the first case, no feas ible solution was found for above 0.68 and below 1.46 for scenario 1; for scenario 2, a solution was obtained for and equaling 1. As the industrial counc il, coalition 2, 2 had the least salience and highest risk acceptance, he was not allocated any of the resources, and still had the highes t utility. If the player is not satisfied with this result, he alters its salience, risk attitude, and / or required resources. This is an example of the im portance of apparent pl ayers’ positions, where players request a higher percenta ge of resources and / or combine it with higher salience and risk aversion to ensure that they obtain their, in this case, “unrevealed” goal. The resources were allocated between coalitions 1, 1 domestic council, and 3, 3 agriculture council. As the ag riculture council had the highe st salience and risk aversion attitude for all issues, he was allocated the mo st resources for a comparable utility to the domestic council. In the case these councils form a coalition, 5 against the industrial council, 2, they are allotted higher resources as a coalition than what each is allocated by working independently, but lesser than the su m of their resources in this latter case. The industrial council is allotted some res ources, leading to equa l utilities for both coalitions. This scenario shows that it is no t in the benefit of the domestic and agriculture councils to show their coope ration. They are better off working as independent coalitions, apparently, and then combining their resources. The expected value of the resources and utilities for each coalition of individual council are shown in Table 5-10 and plotted in Figure 5-5. The red lines in th e latter figure indicate the ideal point of each player with respect to each issue, not modified by salience and risk attitudes.

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198 In the second case, the ratio of gain constrai nt is removed. In the first scenario, the agriculture council rece ives the highest amounts of resources, followed by the domestic and industrial councils. The results are cons istent with how much importance each has allotted for each of the issues and how much ri sk they are willing to take with respect to each of the issues. In the second scenario although no limit was put on the ratio of the utilities, coalitions 2 and 5 obtained equal utilities, w ith most of the resources allotted to the latter coalition, domestic and agriculture council. The expected value of the resources and utilities for each coalition of individual council are shown in Table 5-11 and plotted in Figure 5-6. Figures 5-5 and 5-6, as men tioned earlier, show the expe cted value of the issues allocation and utilities for each council, with and without ratio of utilities constraint, respectively. The lack of a ratio of utili ties constraint resulted in unequal utilities between councils, compared to wh en that constraint is imposed The absence of equity in utilities distribution, rather than resources, suggests that th e councils are unlikely to reach a decision, which stresses the importan ce of this constraint in the model.

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199 Table 5-10. Optimum resources allocation under the different scenarios, with the ratio of gain constraint Issue Coalition Financial Land Water Utility Scenario 1 ( =0.68 and =1.46) 1 0.10 0.35 0.40 -0.1752 2 0.00 0.00 0.00 -0.1200 3 0.90 0.65 0.60 -0.1721 Scenario 2 ( =1.00 and =1.00) 2 0.10 0.20 0.10 -0.0697 5 0.90 0.80 0.90 -0.0697 Expected resources allocation and utilities 1 0.29 0.38 0.43 -0.09950 2 0.05 0.11 0.05 -0.09292 3 0.66 0.51 0.52 -0.09808 Table 5-11. Optimum resources allocation under the different scenarios, without the ratio of gain constraint Issue Coalition Financial Land Water Utility Scenario 1 ( =0 and = ) 1 0.10 0.10 0.40 -0.2481 2 0.00 0.00 0.00 -0.1200 3 0.90 0.90 0.60 -0.0002 Scenario 2 ( =0 and = ) 2 0.10 0.10 0.10 -0.0900 5 0.90 0.90 0.90 -0.0900 Expected resources allocation and utilities 1 0.29 0.29 0.43 -0.13855 2 0.05 0.05 0.05 -0.10385 3 0.66 0.66 0.52 -0.02441

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200 0.0 0.2 0.4 0.6 0.8 1.0 123Ratio of Resources-0.2 -0.2 -0.1 -0.1 0.0Utility Financial Land Water Utility City Council Industrial Council Agricultural Council Figure 5-5. Change of the utility function with the relative gain constraint 0.0 0.2 0.4 0.6 0.8 1.0 123Ratio of Resources-0.2 -0.2 -0.1 -0.1 0.0Utility Financial Land Wate r Utilit y City Council Industrial Council Agricultural Council Figure 5-6. Change of the utility function without the relative gain constraint Conclusion This paper presented a resources alloca tion model involving conflict. The model was based on a utility function that accounts for position, salience, and risk attitude of conflicting parties. The utility function wa s linearized and incorporated in a game theoretic framework. Equity considerations were included by constraining the ratio of

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201 utilities. Finally, the model was validated using a theoretical application, where three councils, domestic, industry, an agriculture coun cils, are in conflic t over three issues, financial, land, and water resources. The model showed that including salien ce and risk attitudes of coalitions significantly affects the resulting coalitions’ util ities, as compared to accounting for ideal positions alone. The industry council, with th e least salience and risk aversion for all the issues, was allotted the least of the resources for a comparable utility with the other councils. In addition, the model suggested that coal itions acting together does not guarantee them better results; the domestic and agriculture c ouncil obtained less re sources as a team than their combined resources when acti ng alone. This suggests the concept of “apparent” versus “real” coali tions’ intentions. The industr ial council is better off hiding his real salience and risk att itudes and the other two council s are better off acting alone, even if they eventually deci de to combine their resources. Finally, the model showed that the absence of the ratio of utilities constraint results in unequal utilities; the absence of equity in utilities distribution, rather than resources, suggests that the councils are unlikely to re ach a decision, which stresses the importance of this constraint in the model.

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202 CHAPTER 6 CONCLUSION He who knows and knows he knows – He is wise, follow him He who knows not and knows he knows not – He is a child, teach him He who knows and knows not he knows – He is asleep, wake him He who knows not and knows not he knows not – He is a fool, shun him Arabic Proverb As convenient and powerful formal models of decision making are, the mathematical representation of a real system fails to capture all the intricacies of natural stochasticities and complex human behavior th at play a paramount role in the actual decision making process. In decision analys is or risk management, or the field of decision making under risk and uncertainty, many techniques have been developed in an effort to account for those intricacies. In the area of single party decision maki ng under uncertainty, th ere exists a large number of techniques designed to help a decision-maker choose among a set of alternatives in light of their possible conse quences. Sensitivity and scenario analysis, while allowing the observation of model respon se versus the change in an uncertain parameter, only provides some intuition of risk. Expected value, while a valuable measure of risk, it fails to highlight extrem e-event consequences. Decision trees and hierarchical approaches, while including uncerta inty via discrete probabilities, fail to generate robust and efficient solutions in situ ations of highly uncertain environments with a large number of dynamic and correlated stocha stic factors and multiple types of risk exposures. Chance-constraine d programming, while allowing for some control of uncertainty and associated ri sks through the explicit introduc tion of statistical central

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203 moments, do not account for fat-tailed distributions and penalize positive and negative deviations from the mean risks value equally. Recourse and multi-stage stochastic optimization, while allowing for the separation of the decision-making process to accommodate information, and associated uncerta inties, available at different time steps, does not provide means to control risk, and most importantly, downside risk, which is risk associated with low probability and hi gh losses, i.e. extreme events. From the perspective of water resources management and public policy, events like dam failure, floods, water contamination, or water shortage, with low probabilities cannot be ignored. In the past ten years, a new method, the Value-at-Risk, VaR was developed as a downside risk measure with a probability associated with it. VaR however, reduces the risk information to this single number that does not provide any information about the extent and distribution of the losses that exceed it, where for the same VaR we can have very different distribution shapes with different associated maximum losses; in addition, VaR is not coherent and is unstable a nd difficult to handle mathematically. An alternative measure that was developed to overcome the limitations associated with VaR is the Conditional Value-at-Risk, or CVaR which is a simple representation of risk that accounts for risk beyond VaR making it more conservative than VaR ; CVaR is stable with integral ch aracteristics, continuous, and consistent with respect to the confidence level and convex with respect to d ecision variables, allowing the construction of efficient optimizing algorithms. CVaR however, has seldom been applied outside the fields of finance and insu rance, such as military applications. The literature of water resources management is filled with applications of scenario and

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204 sensitivity analysis, expected value, recourse stochastic optimization, chance constrained optimization, and many others; however, as discussed earlier, none of these techniques allow the explicit cont rol of risk of extreme events which cannot be ignored when managing water resources. The importan ce of this work is in applying the CVaR to a water allocation problem in order to demonstrat es the utility of such a risk management technique in water resources management in general. The work also suggests and demonstrates a continuous distribution discre tization method that in corporate an n-point moment matching technique along with optimization. In the area of multiple party decision ma king under risk and uncertainty, a large body of knowledge exists to analyze and pr edict individual preferences, attitudes, and choices. A key controversy in this field is the identification of the principle factors motivating these preferences, attitudes, and c hoices. Some main factors that have been dealt with in the literature include ideal posi tion, issues salience and linkage, risk attitude, preference similarity and coalition formation, and equity consideration. Ideal position refers to the position, or amount of resource, that a party ideally as pires to obtain and at which the party’s utility for that resource is maximum. The salience of an issue to a decision maker is defined as the importance of that issue to the decision maker or his willingness to spend influence on the issue in question depending on his priorities; issues may be separable / non separable, where the outco me of one issue is irrelevant / relevant to desired outcome of another issue, and he nce the salience a coalition gives it. Risk attitude refers to the risk a party is willing to take in pursuing its preferences; a party may have different risk attitudes, i.e. be risk aver se / acceptant, for different issues. Coalition formation refers to both the probability of occurrence of each possi ble coalition and the

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205 reward division among the coalitions’ members; however, only a few coalition formation theories have been proposed to predict the likelihood of a coalitiona l structure formation in terms of explicit probabilities. Equity research has shown that individuals resist unfairness to the extent that they are willing to sacrifice their own well-being to help or punish those who are being kind or unkind; expe riments, observed that coalitions tend to refuse outcomes that result in substantially high er gains to other coalitions, even if this outcome results in an improvement in its situation. Methods accounting for absolute, difference, and ratio of utilities have been suggested. While several research incorporate one or more of these factors, the importance of this work is in the incorporation of all of these factors into one comprehensive model that allows the prediction of optimal solutions in view of all of these factors. Note that the model does not assume a specific framework of single-shot versus dynamic, complete information versus Bayesian, cooperative versus noncooperative game and ma y be solved depending on the specific situation. In addition, this work develops a method of calculating coalitions formation probabilities based on their ranking of each othe r and by assuming that the probability of a coalition formation is proportional to the cohesion measure for its members. The work is divided into si x chapters. After the presentation of plan of work in Chapter 1, Chapter 2 defined the concepts of risk, uncertainty, and risk management and compared the various general risk manageme nt approaches with emphasis on value-atrisk and conditional value-at-risk concepts, and described the different scenario generation techniques. The chapter concl uded with a review of the various risk management modeling approaches in the fiel d of water resources management. Chapter 3 presented an application of some risk management techniques to a case of water

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206 resources allocation in the Saint John’s Rive r Water Management District caution area. The developed model was a mixed linear in teger stochastic model with uncertain supplies; it was formulated under an expected value, a two-stage stochastic, and conditional value-at-risk fram eworks, and run for two normal distributions of supplies with different standard devi ations. As Chapters 2 and 3 focused on risk management techniques in situations of single party decision situations Chapters 4 and 5 focused on multiple parties or strategic decision making, specifically, game theory and its extensions. As the core of game theory models, Chapter 4 starts with a review of individual choice models of preferences or utility, their limitations and their alternatives. The discussion is continued with the examinati on of standard game theory, its main taxonomy, solution concepts, and limitations, setting the stage fo r behavioral game theory. Chapter 5, developed a decision making model involving mu ltiple parties and multiple issues in the framework of games. Attempting to simu late real world situations, the model incorporated behavioral obser vations such as equity, salience, and risk attitudes; the model also incorporated a method to estimate th e probabilities of coalitions formation, if they form, based on the members cohesion. Note that the model may be treated as cooperative or noncooperative or as a one-shot or repeated game. Finally, the model was applied to a hypothetical situ ation involving the allocation of three issues, including water, to three parties with competing and conflicting objectives.

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207 APPENDIX SJRWMD COSTS DEPRECIATION Engineering News Records estimate a C onstruction Cost Index, CCI, on a monthly basis, which represents the unde rlying trends of construction costs in the USA. It is determined by several factors such as labor, materials, a nd others. Table A-1 lists historical yearly averages of CCI for the years 1908 – 2005. Figure A-1 is a plot of these values to obtain a best fit of the year – CCI relationship. Using the Equation of the best fit, projections of CCI were calculated fo r the years 2000 – 2025 (Table A-2). CCI was also estimated from the equation for the years 2000 – 2005 for consistency. But what is the significance of CCI and how is it used? Actually, CCI is used as a measure of change of costs between differe nt years (ENR, 2005; Mi chaels, 1996). This change, CCI for consecutive time periods, years, is estimated using Equation 2. The change in CCI can also be calculated for nonconsecutive years using Equation 3, such as t t '. 1001 1 t t t tCCI CCI CCI CCI 2 100' t t t tCCI CCI CCI CCI 3 To estimate the value of costs at time t tC the cost at time t, tC is multiplied by tCCI with t t ', Equation 4.

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208 t t tC CCI C 4 Table A-3 are the discounted cost estimates for the different SJRWMD suggested projects presented in Table 3-5. y = 1.530E-07x6 1.804E-03x5 + 8.861E+00x4 2.321E+04x3 + 3.419E+07x2 2.686E+10x + 8.790E+12 0 1000 2000 3000 4000 5000 6000 7000 8000 19101920193019401950196019701980199020002010 YearCCI Figure A-1. CCI EN R historical data

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209Table A-1. CCI ENR (1908-2005) Year CCI Year CCI Year CCI Year CCI 1908 97 1933 170 1958 759 1983 4066 1909 91 1934 198 1959 797 1984 4146 1910 93 1935 196 1960 824 1985 4195 1911 93 1936 206 1961 847 1986 4295 1912 91 1937 235 1962 872 1987 4406 1913 100 1938 236 1963 901 1988 4519 1914 89 1939 236 1964 936 1989 4615 1915 93 1940 242 1965 971 1990 4732 1916 130 1941 258 1966 1019 1991 4835 1917 181 1942 276 1967 1074 1992 4985 1918 189 1943 290 1968 1155 1993 5210 1919 198 1944 299 1969 1269 1994 5408 1920 251 1945 308 1970 1381 1995 5471 1921 202 1946 346 1971 1581 1996 5620 1922 174 1947 413 1972 1753 1997 5826 1923 214 1948 461 1973 1895 1998 5920 1924 215 1949 477 1974 2020 1999 6059 1925 207 1950 510 1975 2212 2000 6221 1926 208 1951 543 1976 2401 2001 6343 1927 206 1952 569 1977 2576 2002 6538 1928 207 1953 600 1978 2776 2003 6694 1929 207 1954 628 1979 3003 2004 7115 1930 203 1955 660 1980 3237 2005 7298 1931 181 1956 692 1981 3535 1932 157 1957 724 1982 3825

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210Table A-2. CCI ENR projecti on and relative change(2000-2030) Year CCI Percent Relative Change (CCI ) 2000 6348 2001 6489 2.2131 2002 6626 2.1085 2003 6759 2.0090 2004 6888 1.9150 2005 7014 1.8269 2006 7136 1.7444 2007 7256 1.6695 2008 7372 1.5997 2009 7485 1.5398 2010 7596 1.4869 2011 7706 1.4424 2012 7815 1.4078 2013 7923 1.3827 2014 8031 1.3689 2015 8141 1.3653 2016 8253 1.3750 2017 8368 1.3978 2018 8488 1.4329 2019 8614 1.4831 2020 8747 1.5480 2021 8890 1.6279 2022 9043 1.7239 2023 9209 1.8358 2024 9390 1.9640 2025 9588 2.1081

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211Table A-3. Discounted cost s of the SJRWMD proposed pr ojects for the caution area Year 2003 2004 2005 2006 Option CC O&MC UPC CC O&MC UPC CC O&MC UPC CC O&MC UPC 111 189 7.56 3.03 192.619 7.705 3.088 196.135 7.845 3.144 199.560 7.982 3.199 121 91 3.81 3.00 92.743 3.883 3.057 94.435 3.954 3.113 96.084 4.023 3.168 131 457 18.71 2.93 465.751 19.068 2.986 474.252 19.416 3.041 482.533 19.755 3.094 132 238 11.29 2.74 242.558 11.506 2.792 246.984 11.716 2.843 251.297 11.921 2.893 133 217 7.56 3.27 221.155 7.705 3.333 225.192 7.845 3.393 229.124 7.982 3.453 134 84 3.67 2.94 85.609 3.740 2.996 87.171 3.809 3.051 88.693 3.875 3.104 141 81 3.80 2.80 82.551 3.873 2.854 84.058 3.943 2.906 85.526 4.012 2.956 142 372 18.80 2.63 379.124 19.160 2.680 386.043 19.510 2.729 392.784 19.850 2.777 143 714 37.20 2.55 727.673 37.912 2.599 740.953 38.604 2.646 753.892 39.278 2.692 151 210 7.56 3.22 214.021 7.705 3.282 217.927 7.845 3.342 221.733 7.982 3.400 152 105 3.80 3.25 107.011 3.873 3.312 108.964 3.943 3.373 110.866 4.012 3.432 153 447 18.80 2.91 455.560 19.160 2.966 463.874 19.510 3.020 471.974 19.850 3.073 154 871 37.20 2.84 887.679 37.912 2.894 903.880 38.604 2.947 919.664 39.278 2.999 161 386 12.40 3.41 393.392 12.637 3.475 400.571 12.868 3.539 407.566 13.093 3.601 171 55 2.20 1.66 56.053 2.242 1.692 57.076 2.283 1.723 58.073 2.323 1.753 172 134 6.00 1.68 136.566 6.115 1.712 139.058 6.226 1.743 141.487 6.335 1.774 211 255 5.45 2.94 259.883 5.554 2.996 264.626 5.656 3.051 269.247 5.754 3.104 311 90 5.00 3.33 91.723 5.096 3.394 93.397 5.189 3.456 95.028 5.279 3.516 312 180 9.40 3.23 183.447 9.580 3.292 186.795 9.755 3.352 190.057 9.925 3.410 313 274 13.60 3.20 279.247 13.860 3.261 284.343 14.113 3.321 289.309 14.360 3.379 321 90 4.50 3.20 91.723 4.586 3.261 93.397 4.670 3.321 95.028 4.751 3.379 322 177 8.40 3.07 180.389 8.561 3.129 183.682 8.717 3.186 186.889 8.869 3.242 323 268 12.10 3.28 273.132 12.332 3.343 278.117 12.557 3.404 282.973 12.776 3.463 331 83 3.10 5.06 84.589 3.159 5.157 86.133 3.217 5.251 87.637 3.273 5.343 332 121 5.20 3.99 123.317 5.300 4.066 125.568 5.396 4.141 127.760 5.491 4.213 333 159 7.60 3.61 162.045 7.746 3.679 165.002 7.887 3.746 167.884 8.025 3.812 GW 0 48.91 1 0.000 49.847 1.019 0.000 50.756 1.038 0.000 51.643 1.056

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212Table A-3 Continued Year 2007 2008 2009 2010 Option CC O&MC UPC CC O&MC UPC CC O&MC UPC CC O&MC UPC 111 202.891 8.116 3.253 206.137 8.245 3.305 209.311 8.372 3.356 212.423 8.497 3.406 121 97.688 4.090 3.220 99.251 4.155 3.272 100.779 4.219 3.322 102.278 4.282 3.372 131 490.589 20.085 3.145 498.437 20.406 3.196 506.112 20.721 3.245 513.637 21.029 3.293 132 255.493 12.120 2.941 259.580 12.314 2.988 263.577 12.503 3.034 267.496 12.689 3.080 133 232.949 8.116 3.510 236.676 8.245 3.566 240.320 8.372 3.621 243.893 8.497 3.675 134 90.174 3.940 3.156 91.616 4.003 3.207 93.027 4.064 3.256 94.410 4.125 3.304 141 86.953 4.079 3.006 88.344 4.145 3.054 89.705 4.208 3.101 91.039 4.271 3.147 142 399.341 20.182 2.823 405.730 20.505 2.868 411.977 20.820 2.913 418.103 21.130 2.956 143 766.478 39.934 2.737 778.739 40.573 2.781 790.730 41.198 2.824 802.488 41.810 2.866 151 225.435 8.116 3.457 229.041 8.245 3.512 232.568 8.372 3.566 236.026 8.497 3.619 152 112.717 4.079 3.489 114.521 4.145 3.545 116.284 4.208 3.599 118.013 4.271 3.653 153 479.854 20.182 3.124 487.530 20.505 3.174 495.037 20.820 3.223 502.398 21.130 3.271 154 935.017 39.934 3.049 949.975 40.573 3.098 964.602 41.198 3.145 978.945 41.810 3.192 161 414.370 13.311 3.661 420.999 13.524 3.719 427.482 13.733 3.776 433.838 13.937 3.833 171 59.042 2.362 1.782 59.987 2.399 1.811 60.911 2.436 1.838 61.816 2.473 1.866 172 143.849 6.441 1.803 146.150 6.544 1.832 148.400 6.645 1.861 150.607 6.744 1.888 211 273.742 5.851 3.156 278.121 5.944 3.207 282.404 6.036 3.256 286.603 6.125 3.304 311 96.615 5.367 3.575 98.160 5.453 3.632 99.672 5.537 3.688 101.154 5.620 3.743 312 193.230 10.091 3.467 196.321 10.252 3.523 199.344 10.410 3.577 202.308 10.565 3.630 313 294.139 14.600 3.435 298.844 14.833 3.490 303.446 15.062 3.544 307.957 15.285 3.597 321 96.615 4.831 3.435 98.160 4.908 3.490 99.672 4.984 3.544 101.154 5.058 3.597 322 190.009 9.017 3.296 193.049 9.162 3.348 196.021 9.303 3.400 198.936 9.441 3.450 323 287.698 12.989 3.521 292.300 13.197 3.577 296.801 13.400 3.632 301.214 13.600 3.686 331 89.100 3.328 5.432 90.526 3.381 5.519 91.920 3.433 5.604 93.286 3.484 5.687 332 129.893 5.582 4.283 131.971 5.671 4.352 134.003 5.759 4.419 135.996 5.844 4.484 333 170.686 8.159 3.875 173.417 8.289 3.937 176.087 8.417 3.998 178.705 8.542 4.057 GW 0.000 52.505 1.073 0.000 53.345 1.091 0.000 54.166 1.107 0.000 54.972 1.124

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213Table A-3 Continued Year 2011 2012 2013 2014 Option CC O&MC UPC CC O&MC UPC CC O&MC UPC CC O&MC UPC 111 215.487 8.619 3.455 218.521 8.741 3.503 221.542 8.862 3.552 224.575 8.983 3.600 121 103.753 4.344 3.420 105.214 4.405 3.469 106.669 4.466 3.517 108.129 4.527 3.565 131 521.046 21.332 3.341 528.381 21.632 3.388 535.687 21.932 3.434 543.020 22.232 3.482 132 271.354 12.872 3.124 275.174 13.053 3.168 278.979 13.234 3.212 282.798 13.415 3.256 133 247.411 8.619 3.728 250.894 8.741 3.781 254.363 8.862 3.833 257.845 8.983 3.886 134 95.772 4.184 3.352 97.120 4.243 3.399 98.463 4.302 3.446 99.811 4.361 3.493 141 92.352 4.333 3.192 93.652 4.394 3.237 94.947 4.454 3.282 96.246 4.515 3.327 142 424.133 21.435 2.999 430.104 21.736 3.041 436.052 22.037 3.083 442.021 22.339 3.125 143 814.063 42.413 2.907 825.523 43.010 2.948 836.938 43.605 2.989 848.395 44.202 3.030 151 239.430 8.619 3.671 242.801 8.741 3.723 246.158 8.862 3.774 249.528 8.983 3.826 152 119.715 4.333 3.705 121.400 4.394 3.758 123.079 4.454 3.810 124.764 4.515 3.862 153 509.644 21.435 3.318 516.819 21.736 3.365 523.965 22.037 3.411 531.138 22.339 3.458 154 993.065 42.413 3.238 1007.04543.010 3.284 1020.970 43.605 3.329 1034.94644.202 3.375 161 440.095 14.138 3.888 446.291 14.337 3.943 452.462 14.535 3.997 458.656 14.734 4.052 171 62.708 2.508 1.893 63.591 2.544 1.919 64.470 2.579 1.946 65.353 2.614 1.972 172 152.779 6.841 1.915 154.930 6.937 1.942 157.072 7.033 1.969 159.223 7.129 1.996 211 290.737 6.214 3.352 294.830 6.301 3.399 298.906 6.388 3.446 302.998 6.476 3.493 311 102.613 5.701 3.797 104.057 5.781 3.850 105.496 5.861 3.903 106.940 5.941 3.957 312 205.226 10.717 3.683 208.115 10.868 3.735 210.993 11.019 3.786 213.881 11.169 3.838 313 312.399 15.506 3.648 316.797 15.724 3.700 321.178 15.942 3.751 325.574 16.160 3.802 321 102.613 5.131 3.648 104.057 5.203 3.700 105.496 5.275 3.751 106.940 5.347 3.802 322 201.805 9.577 3.500 204.646 9.712 3.550 207.476 9.846 3.599 210.316 9.981 3.648 323 305.559 13.796 3.740 309.860 13.990 3.792 314.145 14.183 3.845 318.445 14.378 3.897 331 94.632 3.534 5.769 95.964 3.584 5.850 97.291 3.634 5.931 98.623 3.684 6.012 332 137.957 5.929 4.549 139.900 6.012 4.613 141.834 6.095 4.677 143.776 6.179 4.741 333 181.283 8.665 4.116 183.835 8.787 4.174 186.377 8.909 4.232 188.928 9.031 4.290 GW 0.000 55.764 1.140 0.000 56.549 1.156 0.000 57.331 1.172 0.000 58.116 1.188

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214Table A-3 Continued Year 2015 2016 2017 2018 Option CC O&MC UPC CC O&MC UPC CC O&MC UPC CC O&MC UPC 111 227.641 9.106 3.649 230.771 9.231 3.700 233.997 9.360 3.751 237.350 9.494 3.805 121 109.605 4.589 3.613 111.112 4.652 3.663 112.665 4.717 3.714 114.280 4.785 3.767 131 550.434 22.535 3.529 558.003 22.845 3.578 565.802 23.164 3.628 573.910 23.496 3.680 132 286.659 13.598 3.300 290.601 13.785 3.346 294.663 13.978 3.392 298.885 14.178 3.441 133 261.366 9.106 3.939 264.960 9.231 3.993 268.663 9.360 4.049 272.513 9.494 4.107 134 101.174 4.420 3.541 102.565 4.481 3.590 103.999 4.544 3.640 105.489 4.609 3.692 141 97.560 4.577 3.372 98.902 4.640 3.419 100.284 4.705 3.467 101.721 4.772 3.516 142 448.056 22.644 3.168 454.217 22.955 3.211 460.565 23.276 3.256 467.165 23.609 3.303 143 859.978 44.806 3.071 871.803 45.422 3.114 883.988 46.057 3.157 896.655 46.716 3.202 151 252.935 9.106 3.878 256.413 9.231 3.932 259.997 9.360 3.987 263.722 9.494 4.044 152 126.467 4.577 3.914 128.206 4.640 3.968 129.998 4.705 4.024 131.861 4.772 4.081 153 538.389 22.644 3.505 545.792 22.955 3.553 553.421 23.276 3.603 561.351 23.609 3.654 154 1049.076 44.806 3.421 1063.50245.422 3.468 1078.367 46.057 3.516 1093.81946.716 3.567 161 464.918 14.935 4.107 471.311 15.141 4.164 477.898 15.352 4.222 484.746 15.572 4.282 171 66.245 2.650 1.999 67.156 2.686 2.027 68.094 2.724 2.055 69.070 2.763 2.085 172 161.396 7.227 2.023 163.616 7.326 2.051 165.903 7.428 2.080 168.280 7.535 2.110 211 307.135 6.564 3.541 311.358 6.655 3.590 315.710 6.748 3.640 320.234 6.844 3.692 311 108.401 6.022 4.011 109.891 6.105 4.066 111.427 6.190 4.123 113.024 6.279 4.182 312 216.801 11.322 3.890 219.782 11.478 3.944 222.854 11.638 3.999 226.048 11.805 4.056 313 330.019 16.381 3.854 334.557 16.606 3.907 339.234 16.838 3.962 344.095 17.079 4.019 321 108.401 5.420 3.854 109.891 5.495 3.907 111.427 5.571 3.962 113.024 5.651 4.019 322 213.188 10.117 3.698 216.119 10.257 3.749 219.140 10.400 3.801 222.280 10.549 3.855 323 322.793 14.574 3.951 327.231 14.774 4.005 331.805 14.981 4.061 336.560 15.195 4.119 331 99.969 3.734 6.095 101.344 3.785 6.178 102.761 3.838 6.265 104.233 3.893 6.354 332 145.739 6.263 4.806 147.742 6.349 4.872 149.808 6.438 4.940 151.954 6.530 5.011 333 191.508 9.154 4.348 194.141 9.280 4.408 196.855 9.409 4.469 199.675 9.544 4.534 GW 0.000 58.910 1.204 0.000 59.720 1.221 0.000 60.554 1.238 0.000 61.422 1.256

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215Table A-3 Continued Year 2019 2020 2021 2022 Option CC O&MC UPC CC O&MC UPC CC O&MC UPC CC O&MC UPC 111 240.870 9.635 3.862 244.599 9.784 3.921 248.581 9.943 3.985 252.866 10.115 4.054 121 115.974 4.856 3.823 117.770 4.931 3.883 119.687 5.011 3.946 121.750 5.097 4.014 131 582.421 23.845 3.734 591.437 24.214 3.792 601.065 24.608 3.854 611.427 25.032 3.920 132 303.318 14.388 3.492 308.013 14.611 3.546 313.027 14.849 3.604 318.424 15.105 3.666 133 276.554 9.635 4.167 280.836 9.784 4.232 285.407 9.943 4.301 290.328 10.115 4.375 134 107.053 4.677 3.747 108.711 4.750 3.805 110.480 4.827 3.867 112.385 4.910 3.933 141 103.230 4.843 3.568 104.828 4.918 3.624 106.535 4.998 3.683 108.371 5.084 3.746 142 474.093 23.960 3.352 481.432 24.330 3.404 489.270 24.727 3.459 497.705 25.153 3.519 143 909.953 47.409 3.250 924.040 48.143 3.300 939.082 48.927 3.354 955.272 49.770 3.412 151 267.633 9.635 4.104 271.776 9.784 4.167 276.201 9.943 4.235 280.962 10.115 4.308 152 133.817 4.843 4.142 135.888 4.918 4.206 138.100 4.998 4.275 140.481 5.084 4.348 153 569.677 23.960 3.709 578.495 24.330 3.766 587.913 24.727 3.827 598.048 25.153 3.893 154 1110.041 47.409 3.619 1127.22548.143 3.675 1145.575 48.927 3.735 1165.32449.770 3.800 161 491.936 15.803 4.346 499.551 16.048 4.413 507.683 16.309 4.485 516.435 16.590 4.562 171 70.094 2.804 2.116 71.180 2.847 2.148 72.338 2.894 2.183 73.585 2.943 2.221 172 170.776 7.647 2.141 173.419 7.765 2.174 176.242 7.891 2.210 179.281 8.027 2.248 211 324.983 6.946 3.747 330.014 7.053 3.805 335.387 7.168 3.867 341.168 7.292 3.933 311 114.700 6.372 4.244 116.476 6.471 4.310 118.372 6.576 4.380 120.412 6.690 4.455 312 229.400 11.980 4.116 232.951 12.165 4.180 236.743 12.363 4.248 240.825 12.576 4.321 313 349.198 17.332 4.078 354.603 17.601 4.141 360.376 17.887 4.209 366.589 18.196 4.281 321 114.700 5.735 4.078 116.476 5.824 4.141 118.372 5.919 4.209 120.412 6.021 4.281 322 225.577 10.705 3.913 229.069 10.871 3.973 232.798 11.048 4.038 236.811 11.238 4.107 323 341.551 15.421 4.180 346.838 15.659 4.245 352.485 15.914 4.314 358.561 16.189 4.388 331 105.779 3.951 6.449 107.416 4.012 6.549 109.165 4.077 6.655 111.047 4.148 6.770 332 154.208 6.627 5.085 156.595 6.730 5.164 159.144 6.839 5.248 161.888 6.957 5.338 333 202.637 9.686 4.601 205.774 9.836 4.672 209.123 9.996 4.748 212.729 10.168 4.830 GW 0.000 62.333 1.274 0.000 63.298 1.294 0.000 64.328 1.315 0.000 65.437 1.338

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216Table A-3 Continued Year 2023 2024 2025 Option CC O&MC UPC CC O&MC UPC CC O&MC UPC 111 257.508 10.300 4.128 262.566 10.503 4.209 268.101 10.724 4.298 121 123.985 5.191 4.087 126.420 5.293 4.168 129.086 5.405 4.256 131 622.652 25.492 3.992 634.881 25.993 4.070 648.265 26.541 4.156 132 324.270 15.382 3.733 330.638 15.684 3.807 337.608 16.015 3.887 133 295.658 10.300 4.455 301.464 10.503 4.543 307.819 10.724 4.639 134 114.448 5.000 4.006 116.696 5.098 4.084 119.156 5.206 4.170 141 110.361 5.177 3.815 112.528 5.279 3.890 114.900 5.390 3.972 142 506.841 25.615 3.583 516.796 26.118 3.654 527.690 26.668 3.731 143 972.809 50.684 3.474 991.914 51.680 3.543 1012.825 52.769 3.617 151 286.120 10.300 4.387 291.740 10.503 4.473 297.890 10.724 4.568 152 143.060 5.177 4.428 145.870 5.279 4.515 148.945 5.390 4.610 153 609.027 25.615 3.965 620.988 26.118 4.043 634.080 26.668 4.128 154 1186.718 50.684 3.869 1210.02551.680 3.945 1235.533 52.769 4.029 161 525.916 16.895 4.646 536.245 17.227 4.737 547.550 17.590 4.837 171 74.936 2.997 2.262 76.408 3.056 2.306 78.019 3.121 2.355 172 182.572 8.175 2.289 186.158 8.335 2.334 190.082 8.511 2.383 211 347.432 7.425 4.006 354.255 7.571 4.084 361.723 7.731 4.170 311 122.623 6.812 4.537 125.031 6.946 4.626 127.667 7.093 4.724 312 245.246 12.807 4.401 250.062 13.059 4.487 255.334 13.334 4.582 313 373.319 18.530 4.360 380.651 18.894 4.446 388.675 19.292 4.539 321 122.623 6.131 4.360 125.031 6.252 4.446 127.667 6.383 4.539 322 241.158 11.445 4.183 245.895 11.670 4.265 251.079 11.916 4.355 323 365.144 16.486 4.469 372.315 16.810 4.557 380.164 17.164 4.653 331 113.086 4.224 6.894 115.307 4.307 7.030 117.737 4.397 7.178 332 164.860 7.085 5.436 168.098 7.224 5.543 171.641 7.376 5.660 333 216.634 10.355 4.919 220.889 10.558 5.015 225.545 10.781 5.121 GW 0.000 66.639 1.362 0.000 67.948 1.389 0.000 69.380 1.419

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217 LIST OF REFERENCES Abdellaoui, M. (2002). "A Genuine RankDependent Generalization of the Von Neumann-Morgenstern Expected Utility Theorem." Econometrica 70(2), 717736. Abdollahian, M., and Alsharabati, C. (2003). "Modeling the Strategic Effects of Risk and Perceptions in Linkage Politics." Rationality and Society 15(1), 113-135. Abramowitz, M. (1965). Handbook of Mathematical Functi ons, with Formulas, Graphs, and Mathematical Tables Dover Publications, New York, NY, USA. Abrishamshi, A., Marino, M. A., and Af shar, A. (1991). "Reservoir Planning for Irrigation District." Journal of Water Resour ces Planning and Management 117, 74-85. Adams, G., Rausser, G., and Simon, L. ( 1996). "Modelling Mulitlate ral Negotiations: An Application to California Water Policy." Journal of Economic Behavior and Organization 30, 97-111. Aguado, E., Sitar, N., and Remson, I. (1977). "Sensitivity Analysis in Aquifer Studies." Water Resources Research 13, 733-737. Aknine, S., Pinson, S., and Shakun, M. F. (2004). "A Multi-Agent Coalition Formation Method Based on Preference Models." Group Decision and Negotiation 13(6), 513-538. Akter, T., and Simonovic, S. P. (2004). "Modelling Uncertainties in Short-Term Reservoir Operation Using Fuzzy Sets and a Genetic Algorithm." Hydrological Sciences Journal-Journal Des Sciences Hydrologiques 49(6), 1081-1097. Albrecht, P. (2003). "Risk Measures." SFB 504, dp 03-01, Sonderforschungsbereich, National Research Center on Concepts of Rationality, Decision Making and Economic Modeling at Mannheim, Universi ty of Manheim, Schloss, Germany. Alker, H. R. (1973). Mathematical Approaches to Politics Jossey-Bass, San Francisco, CA, USA.

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266 BIOGRAPHICAL SKETCH Ghina M. Yamout was born in Beirut, Lebanon, the spring of 1978, where she attended International College, an Ameri can accredited school, and graduated with honorable mention for her school achievements Directly afterwards Ghina joined the Chemistry Department at the American Univ ersity of Beirut, AUB, in October of 1996 and earned her B.Sc. in chemistry in June of 1999. After graduati on, she joined the environmental technology M.Sc. program unde r the supervision of Professor Mutassem ElFadel in the Department of Civil Engin eering, AUB. As she was introduced to the subfields of environmental engineering, Ghina found special interest in the field of water resources. She defended her thesis August 2002. In the last stages of the M.Sc. program, Ghina was accepted in the Ph.D. program at the University of Florida, Gainesville, FL, under the tutelage of Professor Kirk Hatfield She flew to the United States and started the program January of 2002. Ghina completed her doctorate in thr ee and a half years; her publications are under preparation. She defended her dissertation August 22, 2005, and graduated officially with a Ph.D. in civ il engineering, water resources and hydrology, December 2005. Ghina is currently preparing to move to West Palm Beach, Florida, where she will be part of Parsons Water and Infrastructure.


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APPLICATION OF SINGLE PARTY AND MULTIPLE PARTY DECISION MAKING
UNDER RISK AND UNCERTAINTY TO WATER RESOURCES ALLOCATION
PROBLEMS

















By

GHINA M. YAMOUT


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Ghina Yamout

































This document is dedicated to my family; my mother, father, sister, brothers, and
husband, I thank you.















ACKNOWLEDGMENTS

At last! The "there" is "here"! I started school at the age of three, finished at the age

of (), let's just say by the year 2005! Twenty-four years of schooling (do your math)! As

people set high standards for me, I set even higher ones for myself. You start something,

you finish it, completely and perfectly; you start school, you finish it: kindergarten / PhD!

It might not make sense to many; for me, there was no other way! Goals, trivial ones and

less trivial ones, all but small steps towards an ultimate aspiration, that, once seeming so

far away, now is unbelievably close. I look back at this journey of self-fulfillment, and I

realize that everything happens for a reason.

I thank God for what seemed to be "adverse" circumstances at the time and for

what still does. God grant me the blessing of being a good servant of His.

This long longed for "end" is but a beginning; it is called commencement for a

reason! At the onset of my "new" life, I would like to express my utmost gratitude to

many, whose guidance and comfort had a hand in getting me where I am today.

Firstly, I dedicate this work to my mother and father; their self-sacrifice was my

drive. My achievement is theirs. Throughout my life, they made me recognize that when

there is a will, there is always a way. It is with this will that I stand here today.

I would like to express my deepest gratitude to my advisor, Professor Kirk Hatfield,

whose character and energy added considerably to my graduate experience. I thank him

for his unyielding understanding, patience, and faith in me when I was at my best and my

worst. It has been and will always be a privilege. I doubt that I will ever be able to









convey my appreciation fully, but I owe him my life long gratitude, for providing me

with the opportunity to be part of an exceptional university, the University of Florida, in

this exceptional country. An advice for new doctoral candidates: listen to the advisor!

When he says stop taking courses, stop! There is always a "one more" course that needs

to be taken!

I would like to thank the other members of my committee: Professor Edwin

Romeijn, for his unyielding patience and indispensable tutoring which added

considerably to the value of this work; and Dr. Clayton Clark, Professor Warren

Viessman, and Professor Scot Smith for their valuable comments.

I owe my most loving thanks to the man whose support, encouragement, and

persistence were behind the completion of this work, my husband, Husam Jumaa. A

woman is so lucky to meet her better half as young as I was, as his touch can bring back

the starlight and glow of years ago, for me, the first day I met this amazing man, when I

was still twenty-two; with him, I have and will grow older and build memories, but I will

always be 22. His high expectations from me and total devotion to me only better me, to

become, one day, as amazing as he is.

When people think that their prayers are not being answered, they should look

around them, they might not be seeing clearly. Thank God everyday for his blessings,

because she is a blessing; no lesser word can describe her. Zeina Najjar's unconditional

love and guidance, so innate to her, at my worst and my best, are my sunshine in the

happiest and the darkest moments. I will not let words define what she means to me and

will use Martin Luther King's words as he said "Occasionally in life there are those

moments of unutterable fulfillment which cannot be completely explained by those









symbols called words. Their meanings can only be articulated by the inaudible language

of the heart." I would also like to lovingly recognize Amal and Wafic Dabbous, my

youngest sister and brother, whose genuineness and affection had the deepest effect on

my heart and mind.

I would also like to express my most affectionate gratitude to my sister, Aya, and

brothers, Abdel-Ghani and Mohammad. I hope I have and will always be there for them;

I hope they forgive me if at any time I haven't. The responsibility of being their older

sister and the care I have for them were my biggest motivations. I thank my dearest sister,

Aya, for being the older sister at every step of the way. I thank her for opening to me a

world of possibilities, for loving me at my best and my worst, for knowing when I am

smiling even in the dark, for being my teacher, my attorney, my stylist, even my shrink.

I would also like to extend my loving appreciation to my mother, father, and

brother in laws, Abla, Wafic, and Mohammad Luay Jumaa, for their watching eyes,

indispensable prayers, and much needed advice which helped alleviate the heaviest

obstacles. It is amazing how much comfort the mere realization of the presence of a

caring and supporting hand brings to you.

I would like to extend my warmest gratitude to three very dear people to my heart:

my uncle, Mohammad El-Wali, who answered my calls in the latest, or shall I say

earliest, hours; my gentle aunt, Hoda Yamout Kandil, whose rare warm and gracious

nature is a model very hard to follow; and my dear departed uncle, Talal Yamout, who

saluted me with the word "Dr." before I even thought of becoming one.

I would like on this occasion to extend my profound gratitude to another very dear

person to my heart, Mrs. Leila Knio, whose admirable graciousness will be a model for









me throughout my life. I thank her for giving me the privilege of being part of my life;

rare are the people who mark one's life with unconditional guardianship and love.

I would also like to express my warmest thanks to all my friends from the Water

Resources and Hydrology group, Nebiyu Tiruneh, Sudheer Satti, Anirudha Guha, Ali

Sedighi, Mark Neuman, Nanette Conrey, Qing Suny, and Sherish Bhat, with whom I

shared many laughs, debates, exchange of knowledge, and venting of frustration, making

my stay at UF unforgettable. I pray we all stay in touch.

I would also like to extend my gratitude to Ms. Sonja Lee, Doretha Ray, and Carol

Hipsley, who answered, over and over, every question and concern I had; their

welcoming faces and ready assistance had an immense part in easing my transition as I

came to the United States and my experience through my time at the University of

Florida. Thanks also go out to Mr. Anthony Murphy for all of his computer and technical

assistance throughout my graduate program. Finally, this work would not have been

submitted on time without the tremendous assistance from the editorial office at UF.

I also recognize that this research would not have been possible without the

financial assistance of the Civil Engineering Alumni Fellowship at the University of

Florida.

I must also acknowledge the Saint Johns River Water Management District for

providing me with the data used in part of this study. In particular, I would like to take

the opportunity to extend my gratitude to Mr. Ron Wycoff, private consultant for the

district, for generously giving me of his time and answering all my concerns.

For all your guidance, I wish to express my sincerest appreciation.

If I forgot somebody please forgive me, I thank you as well.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TA BLE S ...... .... .... ........ .... .... ...... ...................... ..... xii

LIST OF FIGURES ......... ....................... .......... ....... ............ xiv

ABSTRACT ........ .......................... .. ...... .......... .......... xvii

CHAPTER

1 IN T R O D U C T IO N ............................................................................. .............. ...

2 DECISION MAKING UNDER UNCERTAINTY: A COMPARATIVE REVIEW
OF METHODS AND APPLICATIONS TO WATER RESOURCES
M A N A G E M E N T .............................................................................. .......... .. .. ...5

On the Origin of Risk .................. ................................................... .6
D definition of R isk .................................................................. ........................... . 8
D definition of R isk M anagem ent ....................................................... .... ........... 13
O ur D definition .................................................................................................. ........16
Risk M anagem ent Techniques......................................................... ............... 17
M them atical N otations................................................ ............ ............... 19
Non-Probability-Based RM Techniques .................................. ............... 20
Sensitivity analysis ............ .......... ........................... 20
D decision m making criteria ....................................................................... 20
Analytic hierarchy process or decision matrix...........................................21
U utility and gam e theory ...................................................... ..... .......... 21
M ultiobjective optim ization ...................................................................... 22
Probability-Based RM Techniques................................... ....................... 22
Scenario analysis .......................................... ...... .. .. .... ...... ......... .... 22
M om ents and quantiles........................................... .......................... 23
D e c isio n tre e s ......................................................................................... 2 3
Stochastic optim ization ........................................ .......................... 24
B ayesian analysis ..................................... ............... ..... ..... 25
F u z z y se ts ............................................................................................... 2 5
Inform ation gap ........................ .. ....................... .... .. ........... 26
D ow inside risk m etrics ............................................................. .. ............. 26
The Different RM M ethods: A Discussion............................................................ 27









Value-at-Risk and Conditional Value-at-Risk.....................................30
Scenario Tree .......................................... .... ................. .......34
D iscretization...................... ................................ .... .. ............ .. 37
Risk in the Water Resources Management Literature .........................................41
C o n clu sio n ...................... .. .. ......... .. .. ................................................. 4 2

3 COMPARISON OF RISK MANAGEMENT TECHNIQUESFOR A WATER
ALLOCATION PROBLEM WITH UNCERTAIN SUPPLIES A CASE STUDY:
THE SAINT JOHNS RIVER WATER MANAGEMENT DISTRICT .....................46

M odel Form ulation ................................................... .. .......... .............. ... 47
Objective Function .................. ............................ ... .... .. ........ .... 48
D decision V ariables........... ........................................................... .... .... .... ... 49
Problem Data .................................... ............................... ........ 49
C on strains ........................................................ .......... ................ 49
Deterministic Expected Value M odel ...................................... ............... 50
S cen ario M o d el .................................................................................. 5 1
Two-Stage Stochastic Model with Recourse...........................................52
CVaR, Objective Function M odel................................... ....................... 53
CVaR, Constraint Model ............................ .......... ... ............... 54
Scenario G generation .......... .............................................................. ......... ....... 55
C ase S tu dy A rea ..................................................... ................ 59
W after Demand ................................. ........................... ............ 59
W after Su p p ly .................................................... ................ 6 5
W after Cost ................................. ... ................................ .........67
Scenario G generation .......... .............................................................. ......... ....... 68
R results and D iscu ssion .............................. ........................ .. ...... .... ...... ...... 72
5% Standard D aviation ............................................... ............................. 73
10% Standard D aviation ........................................................... ............... 85
A n a ly sis .................. ........................ ...... ................. ................ 9 6
Conclusions and recom m endations ........................................ ........................ 98

4 UTILITY, GAME, AND WATER: A REVIEW ............................................. 101

Theory of Preference ....................... .............. ... .......... ................. 103
Preference Comparison Relationship ..................................... ............... 104
Expected U utility Theory ............................................................................. 105
B ernoulli's utility theory ................................................. ............... 105
Linear expected utility theory ........... .......... ................... .............. 107
Subjective linear expected utility theory................................ ................. 111
M ultiattribute expected utility ................................................................. 112
Descriptive Limitations of LEUT and SEUT.............. .... ............... 113
Violation of independence ..... .................... ...............114
V violation of transitivity .................................. ................. ..................... 116
Probability judgm ent ............................................. ............................... 120
N on-A rchim edean preferences........................................ ............... 121









A alternatives to Expected U utility Theory...........................................................121
L inear generalizations ........................................ .......................... 122
N on-linear generalizations ........................................ ...... ............... 122
Strategic D decision M making ............................................... ............................ 131
Game Theory ................................ ......... ... ..................... ......... 131
M them atical form ulations ............................................. ............... 132
Classification of gam es .......... ......................................... .................135
Solutions concepts ........................................ ......... .. .. ....... .... .......... 137
Extensions to Standard Game Theory ............. ........................................145
M etagam e analysis ............ ...... .................... ........ .. ............. 146
Hypergame theory .................. ................ .................... 147
Analysis of options........................ ...... .... ............ ........ .... 147
C conflict analy sis............ ............................................ ...... ..... .......148
Drama theory ..................... ......... ..........148
Graph model for conflict resolution ............................................148
Theory of m oves ........... .. ....... ............................ .. .. ...... .... .. 149
Alternatives to Standard Game Theory .................................. ............... 149
Lim ited thinking m odels ........................................ ........................ 150
L earning m odels............ .... ................................ ........ ........ ..... ....... 50
Social preferences m odels ................. ....... .... ......... ............... ............... 151
Game Theory in Water Resources Management ...............................................152
Conclusion ................ ......... .................... ........ .... ..... ........ 155

5 A MULTIAGENT MULTIATTRIBUTE WATER ALLOCATION GAME
M O D E L ................................. ...................................................................1 5 7

P ow er and Preferences........................... ........................................ ............... 158
S alien ce T h eories.......... ................................................................ ......... ....... 160
Issue Linkage Theories ........................... ...................................... ............... 162
E quity T heories........... .. ........................... .. .................... .................. .... ... ... ..... .. 162
C coalition Form ation Theories........................................................ ............... 163
Minimum Resource Theory .......................... ....................163
B alan ce T h eory ........... ...... .......................................................... .. .... .. .... .. 164
M inim um Pow er Theory ............................................................................164
B bargaining Theory .......... ............ ...................... ..... .. .. ........ .... 165
Equal Surplus Theory ........................................... 165
Policy-Distance M inim ization Theory ................................... .................165
O utcom e G rouping Theory........................................... .......... ............... 166
Option Preferences Theory .................................................... ............... ... 167
Ordinal Deduction Selection System Theory .............................167
G raph M odel T h eory ...................... ......... .................................................... 16 8
Triads Theory .................... ................................ 168
Probability of Coalition Form ation........................................... .......................... 168
Size-Probability Model Theory ..... ................. ...............169
Johansen-C Probability Model Theory.... ......... ...............................169
Central Union Theory ............ ........... ........................ 170
W willingness and Opportunity Theory ..................................... ............... 170


x









Cohesion Theory ...................... ............. ................................171
Stochastic Communication Structures ............... ...... ........................171
R isk A ttitu d es ...............................................................17 1
Pratt-A rrow M odel .............................................. ............ .. .............. 172
R isk A version M atrix ............................................... ............................. 173
R isk-V alue Theory .............................................. ............ .. .............. 174
M om ents Risk-V alue M odel ........................................ ........................ 175
D e M esquita's Risk M odel ..... ............. ..................................... ...... .. 175
Model Development ................ ................... ................. 176
U utility F u n action .............................................................18 1
R elative G ain ..........................................184.............................
C coalition Form ation.......... .............................................. ..................... .. 84
Political U uncertainty ............................. ......... .. .... ........................ 186
M odified U utility Function....................................................... ............... 187
H hypothetical A application ................................................ .................................... 187
Conclusion ........................................................... ................. 200

6 C O N C L U SIO N ......... .................................................................... ......... .. ..... .. 202

APPENDIX SJRWMD COSTS DEPRECIATION.................... ..................................207

L IST O F R E F E R E N C E S ...................................................................... .....................2 17

B IO G R A PH IC A L SK E T C H ........................................ ............................................266
















LIST OF TABLES

Table p

1-1 Possible decision making environment combinations ............................................2

2-1 The use of DMuU techniques in water resources management.............................44

3-1 Standard normal discrete distribution approximation for N=10 and up to the 2nd,
4th, 6th, and 8th moments constraints ................ .... .............. 60

3-2 M om ents con strains ........................................................................ .................. 60

3-3 Least square regression analysis results ............. ........................................... 61

3-4 SJWRMD caution area water demand projections ...............................................65

3-5 Supply sources, capacities, and costs .............................. ...............69

3-6 Scenarios of supply capacities at 5% and 10% standard deviation........................71

4-1 Choice m odel classification ............................................................................102

4-2 Applications of GT in water resources management grouped into area of
application ........................ ...................... ..... 156

5-1 Pattern of risk attitudes ........................... .................. .................. ............... 172

5-2 Definition of model components............................... ............... 179

5-3 Player description ......... .... ............. ..... ... .... .... .. .... .. ............ 188

5-4 Input data .................................................................................................. 188

5-5 P lay ers ranking m atrix ............................................ ......................................... 194

5 -6 C o alitio n s ...............................................................19 4

5-7 Coalition combination scenarios ....... .. ......................... ....... .. ............... 194

5-8 Measuring the probability of formation of different coalitions............................196

5-9 Input data ............... ....... ............................. ...........................196









5-10 Optimum resources allocation under the different scenarios, with the ratio of
gain constraint .................................... .............................. ......... 199

5-11 Optimum resources allocation under the different scenarios, without the ratio of
gain constraint .................................... .............................. ......... 199

A -1 C C I EN R (1908-2005) ................................................ ............................... 209

A-2 CCI ENR projection and relative change(2000-2030).....................................210

A-3 Discounted costs of the SJRWMD proposed projects for the caution area............211















LIST OF FIGURES


Figure pge

1-1 D issertation organizational diagram ................................. ...................................... 3

2-1 Chapter 2 organizational diagram ........................................ ......................... 6

2-2 G generic decision tree ............................................................... .. .............. 24

2-3 A visualization of VaR and CVaR concepts .......... ............................................32

2-4 M measure of CVaR, scenario generation ....................................... ............... 36

2-5 Scenario tree for a two-stage stochastic problem with recourse ...........................36

3-1 Chapter 3 organizational diagram ........................................ ....................... 47

3-2 Discretized standard normal distribution (moments in parentheses, i.e., (qth), are
the unm watched m om ents) .............................................................. .....................62

3-3 Discretized standard normal distribution (moments in parentheses, i.e., (qth), are
the unm watched m om ents) .............................................................. .....................63

3-4 M om ents least-square regression analysis plots................................... ................64

3-5 Priority water resource caution areas in the SJRWMD, Florida, USA....................64

3-6 Approximate locations of potential alternative water supply projects ...................66

3-7 Efficient frontier for a'= 50, 75, 80, 85, 90, 95, and 99 percent, 5% STD. ............75

3-8 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 50%, 5% STD........78

3-9 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 75%, 5% STD.........79

3-10 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 80%, 5% STD ........80

3-11 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 85%, 5% STD ........81

3-12 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 90%, 5% STD ........82









3-13 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 95%, 5% STD ........83

3-14 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 99%, 5% STD ........84

3-15 Efficient frontier for a'= 50, 75, 80, 85, 90, 95, and 99 percent, 10% STD. .........86

3-16 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 50%, 10% STD ......89

3-17 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 75%, 10% STD.......90

3-18 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 80%, 10% STD.......91

3-19 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 85%, 10% STD.......92

3-20 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 90%, 10% STD.......93

3-21 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 95%, 10% STD.......94

3-22 Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 99%, 10% STD.......95

3-23 Efficient frontiers for a'= 50, 75, 80, 85, 90, 95, and 99 percent, (A) 5% STD
and (B) 10% STD. ...................................... .. ........... ........ ..... 97

3-24 Comparison of CVaR,, CVaR+, CVaR,, and VaRy values calculated using
m odel 3, (A) 5% STD and (B) 10% STD ................................... ..................99

3-25 Comparison of CVaR, values calculated using model, 5% and 10% STD............99

4-1 Chapter 4 organizational diagram ............................................. ............... 103

4-2 Expected utility indifference curves................ ... ..... .................. 109

4-3 Fanning-out effect ......... ..... ......... ......... ......... ........ ... 115

5-1 Chapter 5 organizational diagram ............................................. ............... 158

5-2 Change of the utility function for coalition 1 w. r. t. issue 1 (horizontal axis) and
issue 2 (vertical axis) for increasing values of x3 (0 1, top to bottom, left to
rig h t). ............................................................................. 19 1

5-3 Change of the utility function for coalition 2 w. r. t. issue 1 (horizontal axis) and
issue 2 (vertical axis) for increasing values of x3 (0 1, top to bottom, left to
right). ...............................................................................192









5-4 Change of the utility function for coalition 3 w. r. t. issue 1 (horizontal axis) and
issue 2 (vertical axis) for increasing values of x3 (0 1, top to bottom, left to
right). ...............................................................................193

5-5 Change of the utility function with the relative gain constraint.............................200

5-6 Change of the utility function without the relative gain constraint........................200

A -1 C C I EN R historical data........................................................................... .... ... 208















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

APPLICATIONS OF SINGLE PARTY AND MULTIPLE PARTY DECISION
MAKING UNDER RISK AND UNCERTAINTY TO WATER RESOURCES
ALLOCATION PROBLEMS

By

Ghina Yamout

December 2005

Chair: Kirk Hatfield
Cochair: Edwin Romeijn
Major Department: Civil and Coastal Engineering

Decision theory refers to the analysis, formalization, and prediction, through

mathematical models, of optimal and real decision-making; it involves the process of

selection of perceived solutions, actions, and outcomes to a given problem from a set of

possible alternatives. Depending on the extent of possible quantification, presence of

uncertainty, number of decision makers, and number of objectives, decision theory is

classified as quantitative or qualitative, deterministic or stochastic, single or multiple

party, and single or multiobjective. The management of water resources systems involves

unavoidable natural and social conditions of risk and uncertainty and multiple competing

or conflicting parties and objectives, which introduce the risks of high economic and

social costs due to wrong decisions, necessitating the formulation of models that

adequately represent a given situation by incorporating all factors affecting it.









Hence, the adequate modeling of such systems should incorporate risk and

uncertainty; decision theory under risk and uncertainty is called decision analysis or risk

management. Risk management approaches may be broadly categorized as non-

probability and probability based techniques. Non-probability based techniques include

sensitivity analysis, decision criteria, analytical hierarchy, and game theory; probability

based techniques include scenario analysis, moments, decision trees, expected value,

stochastic optimization, Bayesian and fuzzy analysis, and downside risk measures such as

Value-at-Risk and Conditional Value-at-Risk metrics. Many of these methods, though

very useful, suffer from critical shortcomings: sensitivity and scenario analysis only

provide some intuition of risk, expected value fails to highlight extreme-event

consequences, decision trees and hierarchical approaches fail to generate robust and

efficient solutions in highly uncertain environments, central moments do not account for

fat-tailed distributions and penalize positive and negative deviations from the mean

equally, recourse does not provide means to control risk, and Value-at-Risk does not

provide information about the extent and distribution of the losses that exceed it and is

not coherent. Game theory, used in situations of multiple party decision making, suffers

from several systematic violations, such as the common consequence, preference

reversal, and framing effects. To account for some of these shortcomings, several

extensions and alternatives have been suggested such as the conditional-Value-at-Risk

and behavioral game theory.


xviii















CHAPTER 1
INTRODUCTION

Decision theory is an interdisciplinary area of study, related to and of interest to

practitioners in many fields such as mathematics, statistics, economics, philosophy,

management, sociology, political science, and psychology. Its main concern is the

analysis, formalization, and prediction, through conceptual, physical, and mathematical

models, of optimal and real decision-making, defined as the process of selection of a

perceived solution, action, and corresponding outcome, to a given problem, from a set of

possible alternatives, in a given situation. A situation is usually described by the extent

of possible quantification, the presence of uncertainty, the number of decision makers,

and the number of issues or objectives (Hipel et al., 1993; Radford et al., 1994); these

four factors result in sixteen combinations, depicted in Table 1-1. Water resources

systems management involves unavoidable natural and social conditions of risk and

uncertainty and multiple competing or conflicting parties and objectives, which introduce

the risks of high economic and social costs due to wrong decisions, necessitating the

formulation of models that adequately represent a given situation by incorporating all the

factors affecting it (Haimes, 2004). Hence, this research is focused on the formal

representation of multiple objective situations involving risk and uncertainty, for single

party and multiple parties decision-making (the bolded sections of Table 1-1). The

diagram in Figure 1-1 previews the plan of this dissertation. Chapter 2 reviews the

concepts of uncertainty, risk, and probability.












Table 1-1. Possible decision making environment combinations
Factors
Combination Uncertainty Quantification Objectives Decision makers
Absent Present Qualitative Quantitative Single Multiple Single Multiple
1 /
2 /
3 /
4 /
5
6 V V
7
8 /
9 /
10 ,
11/ /
12
13 / /
14 /
15 /
16










Chapter 1 Plan of dissertation



Chapter 2 Decision making under uncertainty: methods comparative
review and applications to water resources management


Chapter 3: Comparison of single decision-maker risk management
techniques using a water allocation case study


Chapter 4: Utility theory, game theory, and water resources
management applications


Chapter 5: Multiagents multattributes water allocation game model:
development and application


Chapter 6 Conclusions and future research recommendations




Figure 1-1. Dissertation organizational diagram

The chapter also compares different probability and non-probability based

analytical techniques used in risk management, focusing on the conditional value-at-risk

method, CVaR,. The chapter also presents the different methods used for scenario

generation representation of uncertainty. The chapter concludes with an extensive review

of the application of risk management techniques to water resources problems.

Chapter 3 applies and compares different single party risk management techniques,

presented in Chapter 2, to a water resources management problem, where risk is

quantified as cost. These methods are the expected value, scenario model, two-stage

stochastic programming with recourse, and CVaR,. They were built into a mixed integer

fixed cost linear programming framework. Uncertainty was introduced via water









supplies and results were compared for two discrete distributions with equal means, and

different standard deviations, developed using one of the scenario generation methods

presented in Chapter 2. The models were applied to a case study, using the Saint Johns

River Water Management District (SJRWMD), Priority Water Resource Caution Areas

(PWRCA), in East-Central Florida (ECF), as a study area.

Chapter 4 presents a review of the development of utility theory and game theory,

as theories of individual and strategic decision making under risk and uncertainty. The

chapter starts with a summary of the formal conception of expected utility theory,

followed by its critique as a human choice predictive tool in decision making situations,

and an overview of its alternatives, and continues with the examination of standard game

theory, its main taxonomy, and solution concepts, setting the stage for behavioral game

theory. The chapter concludes with an extensive review of the application of game

theory to water resources decision making problems.

Chapter 5 presents the main concepts of research in the allocation of multiple

resources between multiple competing or conflicting parties, such as preferences, risk,

equity, salience, and power. Subsequently, the chapter proposes an applied modelling

tool for common pool resources conflict resolution that combines essential concepts in an

n parties game theoretic framework. Specifically, these concepts are utility, ideal

position, issue linkage, equity, salience, risk propensity, conflict level, and political

uncertainty. The chapter concludes with the illustration of this model using a

hypothetical conflict situation over water, land, and financial resources among three

different parties.
















CHAPTER 2
DECISION MAKING UNDER UNCERTAINTY: A COMPARATIVE REVIEW OF
METHODS AND APPLICATIONS TO WATER RESOURCES MANAGEMENT

Decision Theory, DT, is a subset of Operations Research, OR, and Systems

Analysis (SA);1 it is a body of knowledge and related analytical techniques, of different

degrees of formality, designed to help a decision-maker choose among a set of

alternatives in light of their possible consequences. Decision Analysis (DA) is a subset

of DT that refers to the discipline of Decision-Making under Uncertainty (DMuU)

(Haimes, 2004; Winston, 1994). The process of Decision-Making under Uncertainty is

also equivalent to another process, namely, Risk Management, RM (Haimes, 2004).

Decision Analysis, Decision-Making under Uncertainty, and Risk Management are used

interchangeably in the rest of the text.

Although DMuU and RM refer to the same processes, their designations employ

terms, namely uncertainty and risk, which, historically, have generated a great deal of

argument. In the next sections we clarify the concepts of uncertainty, risk, and

probability in the framework of RM. Subsequently, we present the general structure of

the latter. These overviews are not aimed at presenting the field specific definitions and

applications of risk and its management; rather they are meant to provide a summary of

the general definitions. Following, we describe and compare different probability and

1 Originally, OR and SA referred to the use of quantitative techniques as a scientific approach to decision-
making and the analysis of complex interdependent elements of a system in a holistic interdisciplinary
manner, respectively, to aid in decision-making. Currently, the analytical tools and evaluative techniques
that both sciences utilize overlap; as a result, these terms are sometimes used interchangeably (Winston,
1994).









non-probability based analytical techniques used in RM, focusing on the conditional

value-at-risk method. As a major issue in DMuU is the representation of uncertainty, we

also present the different methods used for scenario generation. We conclude this chapter

with an extensive review of the application of RM techniques to water resources

problems. The diagram in Figure 2-1 exhibits the plan of this chapter.


Risk Origin and Definition



Risk Management Definition and Structure



Risk Management Comparison of probability and non-probability
Techniques based Tools


Scenario Generation Methods



Conditional Value- Formal definition
at-Risk


Application of RM Review
to Water Resources



Figure 2-1. Chapter 2 organizational diagram

On the Origin of Risk

The first recorded practice of risk analysis, qualitative, dates to as back as early

Mesopotamia, about 3200 B.C. in the Tigris-Euphrates valley, where the Ashipu served

as consultants for difficult decisions in ancient Babylonia; the Ashipu created ledgers of

alternatives, their corresponding outcomes, and favorability (Oppenheim, 1977).









Evidence of games of chance have been found in archeological ancient Egyptian,

Sumerian, and Assyrian sites, where talis, the predecessor of dice, was used. Marcus

Aurelius was regularly accompanied by his master of games (Covello and Mumpower,

1986).

Quantitative risk analysis can be traced to early religious ideas concerning the

probability of an afterlife. Plato addressed it in Phaedo in the fourth century B.C.

Arnobius the Elder, a major church figure in the fourth century A.D. in North Africa,

proposed a two-by-two matrix to analyze the alternatives of God's existence/inexistence

versus accepting Christianity/being a pagan (Covello and Mumpower, 1986). In 1518, in

response to a cash flow problem, the Catholic Church sanctioned usury as long as there

was risk on the part of the lender (this definition was rescinded in 1586 to be resanctioned

later in 1830) (Grier, 1981).

The formal mathematical theories of probability, however, did not appear until the

time of Pascal, who introduced probability theory in 1657. Following the steps of Pascal,

the late seventeenth century witnessed a surge of related intellectual activity by authors

such as Arbuthnot, who argued that probabilities of an event's causes can be calculated;

Graunt, in 1662, and Halley, in 1693, who published life expectancy tables; and

Hutchinson, who studied the trade-offs between probability and utility in risky situations.

In the early eighteenth century, 1738, Cramer and Bernoulli proposed solutions for the

Saint Petersburg paradox.2 In 1792, LaPlace analyzed the probability of death as a


2 In probability theory and decision theory the St. Petersburg paradox is a paradox that exhibits a random
variable whose value is probably very small, and yet has an infinite expected value. This poses a situation
where decision theory may superficially appear to recommend a course of action that no rational person
would be willing to take. That appearance evaporates when utilities are taken into account. The paradox is
named from Daniel Bemoulli's original solution, published in 1738 in the "Commentaries of the Imperial
Academy of Science of Saint Petersburg."









function of smallpox vaccination (Covello and Mumpower, 1986). In the nineteenth

century, Von Bortkiewicz examined ten year records of Prussian soldiers' mortality to

study the event of death by kicks from horses (Campbell, 1980). Before the century was

out, with the need to quantify risk, authors, mainly in the fields of economics, finance,

and accounting, had begun to explore the relationship between uncertainty, probability,

and risk. The next section shows how that task was revealed to be not as simple and

straightforward a matter as it appears (McGoun, 1995).

Definition of Risk

Historically, the concept of risk has been far from easy to define. Its

comprehension and quantification challenged and confused professionals such as

philosophers, psychologists, economists, social scientists, physical scientists, natural

scientists, mathematicians, and engineers (Haimes, 2004). The association between

uncertainty, probability, and risk was a matter of great debate in the late nineteenth and

early twentieth centuries (McGoun, 1995).

Haynes (1895) argued that for many risks, historical relative frequency statistics are

unreliable or inexistent and that risk exists even when statistics are not. Ross (1896)

distinguished between variation, as the unquantifiable descriptive of possible outcomes

with no statistics, and uncertainty as the consequence of this variation; he argued that the

latter is equivalent to risk. Willett (1901) differentiated between probability or chance,

uncertainty, and risk: uncertainty is the degree of rational ambivalence between two

alternatives and also is the deviation of a probability from its normal value; risk, related

to both uncertainty and probability, is the quantification of uncertainty in the form of

mean absolute deviation. Willett also defined risk and uncertainty as the objective and

subjective aspects of apparent variability, where the former is the effect of the









psychological effect of the latter. Fisher (1906) described probability as a measure of

ignorance; as such it is subjective and, in many cases, undefined; it is necessary

information to assess risk, but it is not a measure of risk (McGoun, 1995).

Lavington (1912) and Pigou (1920) were the first to explicitly define risk, as

measurable, as the dispersion of relative frequency distribution. Knight (1921)

underlined risk as "measurable uncertainty" where objective probability exists and

uncertainty, or "unmeasurable uncertainty" as the cases where no quantitative distribution

or only subjective probability exists. Lavington (1925) defined risk as the probability of

a loss and uncertainty as the confidence in that probability. Florence (1929) elicited three

values associated with uncertainty and risk: that of probability itself, the meaning of that

value, and its quality or precision as objective statistics or subjective confidence. Fisher

(Fisher, 1930) rejected the probabilistic measure of risk, which is the synonym of

uncertainty or lack of knowledge.

By 1930, probabilistic measurement of risk had been rejected (McGoun, 1995). It

was with Fisher (1930) and Hicks (1931), in economic research, that the probabilistic

measure of risk returned. Makower and Marschak (Makower and Marschak, 1938;

Marschak, 1938) continued the movement toward an objective probabilistic measure of

risk. In 1944, Domar and Musgrave (1944) acknowledged the skepticism toward the

probabilistic measurement of risk; however, they recognized that, with the need of the

quantification of values and their associated risks and with the absence of a satisfactory

alternative approach to the subject of risk, this method should be adopted.

Lawrence (1976) defined risk as a measure of the probability and severity of

adverse effects. He also distinguished between risk and safety; whereby measuring risk









is an empirical, quantitative, and scientific activity, judging safety is judging the

acceptability of risk, a normative, qualitative, political activity. His definition was

adopted by Haimes (2004) who stated that risk is a complex composition of two

components: (1) "real" potential adverse effects and consequences, and (2) "imagined"

mathematical intangible construct of probability.

The Principles, Standards, and Procedures (PSP) published in 1980 by the U.S.

Water Resources Council (USWRC, 1980) made a clear distinction between risk,

uncertainty, imprecision, and variability as follows. In a situation of risk, the potential

outcomes can be described using a reasonable well-known probability distribution. In

situations of uncertainty, the potential outcomes cannot be described in terms of

objectively known probability distributions or subjective probabilities. In situations of

imprecision, the potential outcomes cannot be described in terms of objectively known

probability distributions, but can be estimated by subjective probabilities. Finally,

variability is the result of inherent fluctuations or differences in the quantity concerned.

The PSP identified two major sources of risk and uncertainty: (1) measurement errors of

the variable complex natural, social, and economic situations and (2) unpredictability of

future events that are subject to random influences.

Kaplan and Garrick (1981) defined risk as a set R = {< S,,L,, X, where S,, L,

and X, denote the risk scenario i, its likelihood, and its damage vector, respectively.

Subsequently, Kaplan (1991; 1993) added a subscript c to indicate that the set of

scenarios should be complete. He also added the idea of "success" or "as planned"

scenario S,; the risk of a scenario S, is then visualized as the deviation from So. This

idea matured into the theory of scenario structuring (TSS) (Kaplan et al., 2001; Kaplan et









al., 1999). Haimes (2004) refined this risk definition as Rp = {< S,, L,, X, where Rp

is an approximation to R based on the partition P of the underlying risk space, stating

that the scenarios S, are finite in number, disjoint, and complete.

Lave (1986) defined hazard as referring to some undesirable event that might

occur; the probability of occurrence of a hazard is how frequently this hazard would be

expected to occur. He defined risk as the expected loss or the sum of all products of each

possible hazard and its probability of occurrence.

Morgan and Henrion (1990) defined probability as a formal quantification of

uncertainty. They distinguished between the classical, objective, or frequentistic and the

Bayesian, subjective, or personalistic views of probability; the former defines the

probability of an event occurring in a particular trial as the frequency with which it

occurs in a long sequence of similar trials, while the latter defines a probability of an

event as the degree of belief that a person has that it will occur given all the relevant

information known by that person at that time, so it is a function not only of the event,

but also of the state of information. Note that regardless of the view, probability

assignments must be consistent with the axioms of probability.

In his book, Technical Risk Management, Micheals (1996) distinguished between

the terms hazard, peril, and risk. A hazard is a condition or action, following a decision,

which may result in perilous conditions. Peril is the undesirable event resulting from a

hazard. The peril probability is the probability of occurrence of that peril. The number

of possible hazard peril combinations is an indication of the complexity of the system.

In a system with four hazard factors such as product, process, intrinsic, and extrinsic

hazard factors, and four impacts such as on quality, functionality, affordability, and









4
profitability, there exists 4 x Y n/r! (n r)!, in this case sixty, hazard peril
r=l

combinations. Peril recovery is the corrective action cost and time to recover from that

peril.

In that framework, Michaels (1996) defined risk as the uncertainty surrounding the

loss from a given peril and risk cost and time as the product of corrective action costs and

times, respectively, for recovery from peril and probability of the peril; they are

probabilistic measures of risk to the cost and time, respectively, of corrective actions.

Risk exposure is the sum of risk costs and risk times for a given system. Risk avoidance

or reduction is the action taken to reduce the risk exposure. Risk recovery is the

corrective action taken to offset perturbations caused by a materialized peril. A risk

determinant factor is the quantified attribute, or risk determinant, that serves as a measure

of factors contributing to risk exposure. Risk metrics are a system of risk determinant

factors quantifying risk exposure.

Micheals also categorized risk as (1) objective or subjective, which the former can

be described by statistics and the latter as a reflection of attitudes and states of mind and

subject to perceptions; (2) speculative or pure, where in the former there is uncertainty

about both hazards and perils, while in the latter, there is only uncertainty as to whether a

hazard leading to a peril will occur and not as to whether the resulting peril produces loss.

Holton (2004a), in the context of financial and economic analysis, stated in his

Contingency Evaluation website that risk has true meaning only when it refers to the

possibility of incurring loss, mainly financial, directly or indirectly. In his book (Holton,

2003) and publications (Holton, 1997; Holton, 2004b), he defined uncertainty as

ignorance, a personal experience, and risk as exposure to uncertainty. As such, risk has









two components: uncertainty and exposure to it; if both are not present concurrently,

there is no risk. Holton then argued that since ignorance is a personal experience, risk is

necessarily subjective. Probability is a metric of uncertainty; at best, it quantifies

perceived uncertainty.

There are many more efforts to define risk; a further review of definitions,

however, is superfluous for the purpose of this work. In the next section, we present

some frameworks of risk management.

Definition of Risk Management

There are almost as many definitions for Risk Management as there are

management disciplines. Everybody makes decisions: scholars and analysts in the fields

of economics, finance, and accounting, psychology and social sciences, biology,

toxicology, and medicine, mathematics and computer science, and engineering, all have

been addressing the field of risk management. What all disciplines have in common

though, is the definition of RM as a feedback process consisting of several steps. The

following definitions are based on the corresponding ones, if available, presented in the

previous section.

Kaplan and Garrick (1981) defined risk assessment as the process of identifying

what can go wrong, the likelihood of it going wrong, and the consequences. Haimes

(1991) used this definition to define risk management as a process that builds on risk

assessment and finds the available management alternatives and solutions, their costs,

benefits, and risk trade-offs, and the impacts of the management decisions on future

options. It requires the synthesis of the empirical and normative, the quantitative and

qualitative, and the objective and subjective.









Lave (1986) defined the risk management process as a cyclical succession of (1)

risk identification, or the identification of hazard and its associated risks, (2) risk

assessment, or the identification of the quantitative magnitudes of the hazards, (3)

management options, or the identification of goals, (4) decision analysis, or the

identification of alternatives, and (5) implementation and monitoring.

Michaels (1996) defined RM as the executive function of controlling hazards and

their consequential perils that causes some kind of loss. Its aim is to reduce risk exposure

rather than recovery; hence it stresses risk avoidance as first line of defense and risk

recovery as a backup. He divided RM to three concurrent processes: (1) risk

identification, (2) risk quantification, and (3) risk control. The first step includes

determining the scope of investigation and, establishing the baseline model, and

identifying the hazards and perils. The second consists of deriving the risk hierarchy,

selecting the risk metrics and formulation, establishing a risk model, calculating risk

exposure, and estimating contingency reserve. The third includes establishing risk

organization and funding it, propagating best practices, implementing audits, initiating

motivational programs, and rewarding performance.

Haimes (2004) defined risk assessment and management as two overlapping

processes; he used two perspectives, qualitative normative and quantitative empirical.

In his qualitative normative perspective, Haimes defined risk assessment as the set

of five logical, systemic, and well-defined activities of (1) risk identification, (2) risk

modeling, quantification, and measurement, (3) risk evaluation, (4) risk acceptance and

avoidance, and (5) risk management.









Haimes distinguished risk identification, the first step of RM, as the process of

identifying the sources and nature of risk and the uncertainty associated with it; this stage

attempts to uncover and describe all risk-based events that might occur, be it natural

hydrologicc, meteorological, and/or environmental,) or man-made (demographic,

economic, technological, institutional, and/or political) causes. The second step, risk

modeling, quantification, and measurement, involves assessing the occurrence likelihood

of adverse events through objective or subjective probabilities and modeling the causal

relationship among the different sources of risk, or adverse events, and their impacts. In

other words, it involves the quantification of the input and output relationships of the

random and decision variables and their relationship to the state variables, objective

functions, and constraints. In the third step of RM, risk evaluation, various policy options

are formulated, developed, and optimized. Risks, benefits, and costs tradeoffs are

generated and evaluated. The fourth step, risk acceptance and avoidance, is the decision-

making step where the level of acceptability of risk is determined by evaluating the

considerations that fall beyond the modeling and quantification process; it answers the

question of "how safe is safe enough?". The fifth and final step, risk management, is the

execution step where the chosen policy option is implemented.

In his quantitative empirical perspective, Haimes defined risk assessment as the set

of three major, though overlapping, activities: (1) information measurement, (2) model

quantification and analysis, and (3) decision-making. Information measurement includes

data collection and processing. Model quantification and analysis includes the

quantification of risk and other objectives, the generation of Pareto-optimal solutions and

their trade-offs, and the conduct of impact and sensitivity analysis. Finally, decision-









making involves the interaction between analysts and decision-makers and the subjective

judgment for the selection of preferred policies.

Haimes also defined total risk management as the process that harmonizes risk

management with the overall system management; it addresses hardware, software,

human, and organizational failures involving all aspects of the system's life cycle,

planning, design, construction, operation, and management. Finally, he defined risk-

based decision-making refers to a decision-making process that accounts for uncertainties

through some process in the formulation of policy options.

Our Definition

For the purpose of this dissertation, we do not dwell on the philosophical concerns

associated with uncertainty, probability, and risk, mainly, the concepts of their existence

and the extent of their objectivity.

We start with Holton's (1997; 2003; 2004a; 2004b) distinction between the terms

metric, measure, and measurement. A metric is the designation of a tool. An operation

or algorithm that supports a metric is called a measure. The value obtained from

applying a measure is a measurement. Hence, a measure is used to obtain a measurement

of a metric; there may be many measures to one metric.

We associate uncertainty with ignorance. We define stochasticity as a special type

of uncertainty associated with randomness. For practicality, we classify uncertainty into

two main groups: natural and man-made; natural uncertainty is associated with a natural

system's components, while man-made uncertainty is associated with a man generated

system's components. Uncertainty is quantified using probability.

We adopt the classical objective theory of statistics; we assume that probability

exists and can be quantified. The methods of its generation, through historical









information or mathematical algorithms, and their accuracy and precision are outside the

scope of this work. Probability is a metric of uncertainty.

We associate risk with loss; any type of loss, resulting from a decision-making

policy or action. Loss, or risk, is hence represented using risk metrics. The choice of a

risk metric and measure depends on the problem at hand and the decision-maker's

objectives and priorities.

The description of risk is done through measurement of the risk metric and its

associated statistics.

We define risk management, RM, as the processes of risk identification, risk

estimation, risk evaluation, and risk monitoring. Risk identification consists of

uncovering and describing natural, man-made risk-based events that might occur. Risk

estimation refers to the quantification of these events, their probability of occurrence, and

their causal relationship. In the third step of RM, risk evaluation, various policy options

are formulated, developed, and optimized. Risks, benefits, and costs tradeoffs are

generated and evaluated. This definition is based on Haimes (Haimes, 2004). Finally,

risk monitoring is the continuous process of the first three steps. RM is a feedback

process.

Risk Management Techniques

"To manage risk, one must measure it" is an adage that Haimes (2004) uses in his

book Risk Modeling, Assessment, and Management. Public interest in the field of RM

has expanded significantly during the last two decades as an effective and comprehensive

procedure that complements/supplements the management of almost all aspects of our

lives. As federal and state legislators and regulatory agencies have been addressing the

importance of the assessment and management of risk, industrial and government









agencies in many management disciplines such as financial management, health care,

human safety, manufacturing, the environment, and physical infrastructure (e.g. water

resources, transportation, and power generation) all started incorporating risk analysis in

their decision-making process. In parallel, the scholastic community witnessed an

unprecedented release of articles covering the development of theory, methodology, and

practical applications (Haimes, 2004). What are the methods, how are they distinguished,

and how are they categorized?

This section is devoted to the presentation of the main classes of RM techniques.

Note that it is not meant to be an exhaustive reference of every available tool, sub-tool,

and application developed in every discipline, but a comprehensive categorized overview

of RM tools.

As we undertake the task of classification, we are faced with the multiplicity of

possible ways with which this task may be approached, depending on our interests and

objectives. For the purpose of this dissertation, we base our classification, on a first

level, on the concept of probability. Therefore, we classify risk-based decision-making

methodologies into probability-based and non-probability-based techniques. Note that

the presented classes of methods are not mutually exclusive; in other words, in many

instances, their concepts may overlap and multiple methods may be incorporated into one

model, as always, depending on the problem at hand and the decision-maker's objectives.

We start this section by defining the general mathematical notations we use.

Following, we describe non-probabilistic and probabilistic risk management techniques.

The non-probabilistic methods presented are sensitivity analysis, decision-making

criteria, decision matrix, multiobjective optimization, and game theory. The probabilistic









techniques presented are scenario analysis, moments and quantiles, decision trees,

stochastic optimization, downside risk metrics, and utility theory.

Mathematical Notations

An uncertain parameter Z may be continuous or discrete. A continuous parameter

can assume all values in a specified interval. A discrete parameter assumes different

values with different associated probabilities. Either way, the uncertainty of Z is

described via its statistics; such statistics are its mean, variance or standard deviation,

and, at best, its probability density function, pdf, and cumulative distribution function,

cdf The pdf and cdf are represented by the functions f(Z) and F(Z), which are

formalized differently for continuous and discrete parameters. In the following sections,

we consider only cases of discrete parameters; we assume that continuous ones may be

discretized.

The occurrence of each value, Z of Z, with a probability p(s ), is represented

by a scenario, s where j = 1,2,...,J denotes the scenario's name or number. Each

scenario s, occurs with a probability p(s ) of Z We denote by a, the decision or

action alternatives adopted by the decision-maker, where i = 1,2,..., is the alternative's

name or number.

We also define the pair (a ,s ) as the outcome from the combination of the

decision a, and the scenario s,. The risk resulting from a pair (a,s, ) is r.

Finally, the parameter value Z the corresponding scenario, s,, and risk or loss r

have the same p(s ), f(Z), and F(Z).









Non-Probability-Based RM Techniques

Although the management of risk, generally, connotes the quantification of risk

through reliance on probability and statistics (Haimes, 2004), several risk-based decision-

making methodologies do not require the knowledge of probabilities. These methods

include sensitivity analysis, decision-making criteria, decision matrix, multiobjective

optimization, and game theory; they are described in the following sections.

Sensitivity analysis

Sensitivity or what-if analysis is the process of varying model input parameters

over a reasonable range (range of uncertainty in values of model parameters) and

observing the relative change in model response. The purpose of this type of analysis is

to demonstrate the sensitivity of the model simulations to uncertainty in values of model

input data. The sensitivity of one model parameter relative to other parameters is also

demonstrated. Sensitivity analysis is also beneficial in determining the direction of future

data collection activities. Data for which the model is relatively sensitive would require

future characterization, as opposed to data for which the model is relatively insensitive

(Morgan and Henrion, 1990; Winston, 1994).

Decision making criteria

Decision making criteria are methods for handling risk and uncertainty without

adhering to probability (Haimes, 2004; Winston, 1994) in an optimization formulation.

The three most common criteria are the pessimistic rule, the optimistic rule, and the

Hurwitz rule.

Pessimistic rule. Also called the maximin or minimax criterion because it consists

of maximizing the minimum gain or minimizing the maximum loss; the rationale is that,









by applying this rule, a decision-maker will at least realize the minimum gain or avoid the


maximum loss. Its formulation is min maxr .
1<1
Optimistic rule. Also called the maximax or minimin criterion because it consists

of maximizing the maximum gain or minimize the minimum risk. Its formulation is


min min r .
1<1
Hurwitz rule. This rule is a compromise between the two extreme criteria through

an a index, where 0 < a <1 that specifies the degree of the decision-maker's optimism:

the smaller the a the greater the optimism; it is the linear combination of the minimax


and minimin criteria formulated as min r, (a) = min a max r, + (1- a)min r
1K<1 111 <]
Analytic hierarchy process or decision matrix

The Analytic Hierarchy Process, AHP, also called decision matrix method for the

evident reason that it is based on a ranking matrix of decision and corresponding

attributes, such as risk. It is a semi-quantitative decision making tool for situations where

the attributes are not amenable to explicit quantification. The attributes are also assigned

weighing factors. The decision option with the highest weighted sum of attributes is

considered the best solution. Changing the weights of the assigned attributes is

performed for sensitivity analysis (Haimes, 2004; Winston, 1994).

Utility and game theory

Utility is used to represent individual preferences, therefore predict their choice

behavior. The theory of games uses utility to model strategic interactions between

competing and conflicting decision-makers (Heap, 2004; Myerson, 1991). Both theories,

their limitations, and alternatives are discussed in details in Chapter 4.









Multiobjective optimization

Multiobjective optimization, MO, also known as Multi-Criteria Decision Making,

MCDM, refers to optimization problems with several, possibly conflicting or competing,

objectives. Objectives are weighted according to their priority. There are several

methods for solving MO problems. The most commonly used method is Goal

Programming, where the objectives are given goals that are ranked by weighting factors

and the problem is reduced to a single objective function of the weighted minimization of

the deviations from the assigned goals. Another method is the weighted sum approach,

where the objectives are assigned weighing factors and combined into one objective

forming a single optimization problem. A third method is the hierarchical optimization

method, where the objectives are ranked in a descending order of importance and each

objective is then optimized individually subject to a constraint that does not allow the

optimum for the new function to exceed a prescribed fraction of a minimum of the

previous function. Other methods are the trade-off, constraint, or F method, the global

criterion method, the distance function method, and min-max optimization (Haimes,

2004; Winston, 1994).

Probability-Based RM Techniques

This section describes the following probability-based RM technique: scenario

analysis, moments and quantiles, decision trees, stochastic optimization, downside risk

metrics, and utility theory.

Scenario analysis

Scenario analysis combines sensitivity analysis and the expected value metric,

where the uncertain parameter is assigned different values, corresponding to different









scenarios, with associated probabilities, and the expected value of the scenarios results is

calculated (Haimes, 2004; Winston, 1994).

Moments and quantiles

The expected value operator, E, mean, or first central moment, is a central metric

that for a discrete parameter multiplies the parameter's values, such as loss, from

different scenarios, by their corresponding probability of occurrence and then sums these

products. For example, in an optimization framework, the expected value of loss for all


policy options is minimized; formally, for a discrete problem: min Y ps )r
j=1

The variance, "2, or second central moment, and standard deviation, C, are

measures of dispersion of the values of a parameter around its mean. For example, the


variance of a discrete loss is o2 = E(r ))p() E= E(r2- Er)
J-1

A quantile is the generic term for any fraction that divides the values of a parameter

arranged in order of magnitude into two specific parts. For example, the 90th percentile

of the loss is the value for which the value of F(Z) is 90% or 0.9; in other words, 90% of

the losses lay below the value of the 90th percentile.

Decision trees

Decision Trees are one of the most used tools in risk-based decision-making; it

relies both on a graphical descriptive and an analytical probability-based representations

(Haimes, 2004; Winston, 1994). The basic components of a decision tree are the decision

nodes, designated by squares, chance nodes, designated by circles, and consequences,

designated by rectangles (Figure 2-2). Branches emanating from a decision node

represent the various decisions or actions, a,, to be investigated. Branches emanating









from a chance node represent the various scenarios, s,, with their associated

probabilities, p(s ); at their end are the consequences, /u,, associated with the

scenario/action pair (a,,s (Figure 2-2).


s' ] I(a, 1, )



s12 (al s 2 )



sZ p J1 (a3, s)

s2 IU32 (a3 s2)

Figure 2-2. Generic decision tree (adopted from Haimes, 2004)

Stochastic optimization

The theory and applications of Stochastic Programming, SP, appeared in the 1950s,

when authors such as Beale, Dantzig, Charnes, and Cooper (Beale, 1955; Charnes and

Cooper, 1959; Dantzig, 1955) realized the need to incorporate uncertainty in Linear

Programming, LP. Stochastic optimization, SO, refers to optimization in the presence of

uncertain parameters, with the uncertainty quantified statistically by continuous or

discrete probability distributions. Depending on the way the uncertainty is expressed and

modeled, SP models can be categorized as recourse problems, SPR, or chance-

constrained problems, CCP; these methods are briefly explained below (Birge and

Louveaux, 1997; Di Domenica et al., 2003).

Recourse optimization. Recourse optimization, RO, is also referred to as multi-

stage optimization, MSO, or dynamic optimization, DO. Recourse is the ability to take









corrective actions after an uncertain event has taken place. An example is two-stage

recourse problem; in simple terms, the problem involves choosing a variable to control

what happens in the present time and taking some recourse corrective action after an

uncertain event occurs in the future.

Chance-constrained optimization. Chance-constrained optimization problems,

CCP, or probabilistically constrained optimization problems, PCP, are optimization

problems that involve statistical terms in their objectives and/or constraints.

Bayesian analysis

Bayesian analysis involves uncertainty caused by incomplete understanding or

knowledge. One technique is Bayesian network, also known as belief networks, causal

networks, Bayesian nets, qualitative Markov networks, influence diagrams, or constraint

networks. Bayesian networks use a graphical structure to represent the complex causal

chain linking decisions and consequences via a sequence of conditional relationships;

variables are represented by a round node and the dependence between two variables is

represented by an arrow. Dependence is represented by a conditional probability

distribution for the node at the end of the arrow, based on Bayes formula. The graphical

network constitutes a description of the probabilistic relationships among the system's

variables (Batchelor and Cain, 1999; Borsuk et al., 2004; Bromley et al., 2005).

Fuzzy sets

Fuzzy set theory was suggested (Zadeh, 1965) to deal with decision situations

involving risk and uncertainty without using probabilities. It deals with situations

characterized by imprecise information described by membership functions (Hatfield and

Hipel, 1999).









Information gap

Information gap models, also know as convex models, define uncertainty to be an

information gap between what is known and what is needed to be known for the decision

making process; its aim is to quantify and reduce this information gap (Ben-Haim, 1997;

Hatfield and Hipel, 1999).

Downside risk metrics

Under this category are the second, first, and zero order lower partial moments,

LPM, value-at-risk, and conditional value-at-risk metrics. In general, these metrics are

referred to as measures in the literature (Albrecht, 2003). Note that an LPM of order n is

computed at some fixed quantile q and defined as the nth moment below q. Developed

by Bawa (Bawa, 1975) and studied by Fishburn (Fishburn, 1977), LPM measure risk by a

probability weighted mean of deviations below a specified target level q; the higher the

n, the higher the risk aversion.

Second order LPM or semi-variance. The second order LPM, or SLPM, is also

referred to as semi-variance, SV; it describes the downside risk computed as the average


of the squared deviations below a target loss. Formally, it is SLPM = q r )2 p ).
j=1

First order LPM. The first order LPM, or FLPM, describes the downside risk

computed as the average of the deviations below a target loss. Formally, it is defined as


FLPM = (q- r)p(s ). It refers to risk neutral behavior.
J-1









Zero order LPM. The zero order LPM, or ZLPM, describes the downside risk

computed as the average of the probabilities below a target loss; it coincides with the

q
cumulative probability of q. Formally, it is defined as ZLPM = p(s )= F(q).
J-1

Value-at-Risk. Value-at-Risk is denoted by VaR In simple terms, VaR, is a

quantile. If the cumulative probability of q is denoted F(q) = p, then VaR, is the

inverse of the ZLPM, such as VaRX = -F- (p) = -q, and is defined as the maximum

potential loss with a confidence level a = p. A detailed and more correct discussion of

VaR, is presented later in this chapter.

Conditional Value-at-Risk. Conditional Value-at-Risk is denoted by CVaR,; it is

equivalent to expected shortfall, ES. Generally, it measures the expected value of losses

exceeding VaR At a confidence level a, CVaR, = E(rr < VaRy). A detailed and

more correct discussion of CVaR, is presented later in this chapter.

The Different RM Methods: A Discussion

In the past, risk was investigated using a variety of ad hoc tools, such as sensitivity

and scenario analysis. Although these techniques allow the observation of model

response versus the change in an uncertain parameter using a deterministic model, they

only provide some intuition of risk. Risk was also commonly quantified through the

mathematical expected value concept. Although an expected value provides a valuable

measure of risk, it fails to highlight extreme-event consequences, which are adverse

events of high consequences and low probabilities in advantage of events of low

consequence and high probabilities, regardless of the former' potential catastrophic and









irreversible impacts. From the perspective of public policy, events like dam failure,

floods, water contamination, or water shortage, with low probabilities cannot be ignored.

This is exactly what the use of expected value would ultimately generate. The pre-

commensuration of these low probability high damage events with high probability low

damage events into one expectation function by the analyst markedly distorts the relative

importance of these events as they are viewed, assessed, and evaluated by the decision-

makers (Haimes, 2004).

Other methods developed to surpass the drawbacks of these quasi-deterministic

techniques relied on decision trees and hierarchical approaches where uncertainty is

introduced via discrete probabilities of uncertain parameters. These methods fail to

generate robust and efficient solutions in situations of highly uncertain environments with

a large number of dynamic and correlated stochastic factors and multiple types of risk

exposures.

The explicit introduction of statistical central moments, such as variance and

standard deviation, which have well established calculation methods, into stochastic

simulation and/or optimization approaches, such as chance-constrained programming,

allowed for some control of uncertainty and associated risks. Central moments, however,

do not account for fat-tailed distributions and penalize positive and negative deviations

from the mean risks value equally (Cheng et al., 2003). The introduction of the recourse

and multi-stage concepts into stochastic optimization allows for the separation of the

decision-making process to accommodate information, and associated uncertainties,

available at different time steps. Recourse, however a very useful concept in practical









decision-making applications, does not provide means to control risk, and most

importantly, downside risk, which is risk associated with low probability and high losses.

In the past ten years, a new method became popular in industry regulations, the

Value-at-risk, VaRy, which was introduced by the leading bank, JP Morgan. Unlike the

past methods, VaR, provides a downside risk measure with a probability associated with

it. VaR,, however, has several shortcomings. The reduction of the risk information to

this single number may lead to misleading interpretations of results. VaRy does not

provide any information about the extent and distribution of the losses that exceed it,

where for the same VaRy, we can have very different distribution shapes with different

associated maximum losses; Hence, it is incapable of distinguishing between situations

where losses that are worse may be deemed only a little bit worse, and those where they

could well be overwhelming. In addition, the recent research on the axiomatic

characterization of risk metrics revealed that VaR, is not coherent.

The coherence concept was first introduced by Artzner, Delbaen, Eber, and Heath

(Artzner et al., 1999), who defined a coherent measure of risk as one that satisfies the

following four axioms:

1. Subadditivity: p(X + Y) < p(X)+ p(Y)

2. positive homogeneity: if A > 0, p(AX) = Ap(X)

3. Monotonicity: if X < Y, p(X)> p(Y)

4. Translation invariance: p(X + W)= p(X)- W

Where p is a risk metric, X,Y, and W are different risk functions.









A major drawback of VaR, is that it does not satisfy axiom 1, i.e., it is not (with a

few exceptions) subadditive. In practical terms, this signifies that the total risk associated

with a certain project may be larger than the sum of individual risks resulting from

different sources (Cheng et al., 2003). In addition, VaRy is not convex; hence, may have

many local extremes, which makes it unstable and difficult to handle mathematically.

An alternative measure that was developed to overcome the limitations associated

with VaR, is the Conditional Value-at-Risk, or CVaR, (Rockafellar and Uryasev, 2000;

Rockafellar and Uryasev, 2002). Rockafellar and Uryasev defined expectation-bounded

risk measures as satisfying axioms 1, 2, 4, and axiom 5, which is:

5. p(X) > E(- X) if X non constant, and p(X)= E(- X) if X constant.

If axiom 3, monotonicity, is also satisfied, then the risk measure is coherent and

expectation-bounded.

CVaR, is both coherent and expectation-bounded. It is a simple representation of

risk that accounts for risk beyond VaRy, making it more conservative than VaR,.

CVaR, is also stable as it has integral characteristics. It is continuous and consistent

with respect to the confidence level a CVaR, is also a sub-additive convex function

with respect to decision variables, allowing the construction of efficient optimizing

algorithms; it can be optimized using linear programming techniques, which makes it

efficient.

Value-at-Risk and Conditional Value-at-Risk

To set the ground for formal definition of CVaR,, we start with the formal

definition of VaR, VaR is a quantile; it has three components: a time period, a









confidence level, and a loss amount; it answers the question: with a given confidence

level, a, what is our maximum loss, VaR,, over a specified time period, T? Note that

there is always a probability, 1-a, that the actual loss will be larger. What does this

mean? In reference to Figure 2-3, which presents the probability and cumulative

distribution functions for two loss functions (solid and broken lines), we can read that

with a confidence (60 percent), we expect that our worst loss, over time T, will not

exceed VaR, (5.5 loss units, for both loss functions); there is a probability 1-a (40

percent), that this measure may be exceeded in the right tail of the distribution (> 5.5 loss

units) (shaded area). Increasing the confidence level will result in an increase in VaR,,

by moving into this right tail; for a confidence level a of 95 percent, the VaR,

corresponds to 8 and 9.5 units for the broken and solid lines loss functions, respectively.

In reference to Figure 2-3, CVaR, quantifies the losses in the right tail of the

distribution, the shaded area. Most importantly for applications, CVaRY can be

expressed by a minimization formula that can be incorporated into problems of

optimization that are designed to minimize risk or shape it within bounds, such as the

minimization of CVaR, subject to a constraint on loss, the minimization of loss subject

to a constraint on the CVaR,, and the maximization of a utility function that balances

CVaR, against loss.

But how do these concepts translate mathematically? And how is this risk

measure, CVaR,, calculated and controlled?





















02
100













0 2 4 6 8 10 12
Loss


Figure 2-3. A visualization of VaR and CVaR concepts

Note that the following mathematical formulation is limited to losses with discrete

distribution functions; those with continuous distribution functions can be discretized.

For the mathematical formulation using losses with continuous functions refer to the

literature (Rockafellar and Uryasev, 2000; Rockafellar and Uryasev, 2002).

Let L(x, ) be a loss function depending on a decision vector x and a stochastic

vector Let Y(x, <) be its cumulative distribution function. Assume that the behavior

of the stochastic parameter can be represented by a discrete probability distribution

function, from which a scenario model can be constructed. Index the scenarios

s = 1,..., S corresponding to the stochastic parameter ,s, with corresponding

probabilities ps, such that the losses are listed in an increasing order

L(x, 1 ) ...< L(x, s ). In this setting, we can define the following terms:

* VaR, is the value of L(x, <) corresponding to the confidence level a Formally,
VaR, = min [L(x, )|Y(x, ) a] = L(x, r ).









* CVaR+, or upper CVaR,, is the expected value of L(x, <) strictly exceeding
VaR,; it is also called Mean Excess Loss and Expected Shortfall. Formally, for
equally probable scenarios,
CVaR+ = E[L(x, () L(x, ) > VaR, ] = 1 (x, ,).
S Sa s=s+l

* CVaR or lower CVaR,, is the expected value of L(x, <) weakly exceeding
VaR,, i.e., values of L(x, ) which are equal to or exceed VaR,; it is also called
Tail VaR, Formally, for equally probable scenarios,

CVaR- = E[L(x, )L(x, )>VaRj= ] ZL(x, 4).
S s, + 1s=Sa

* Y ,a is the probability that L(x, ) does not exceed VaR Formally,
Y, = max [Y(x, JL(x, )< VaR where a < Ya <1.

* A is a weighing factor. Formally, A = (Y, -C a(1- a), where a < Y V <1
and 0 A <21.

* Finally, CVaRc is the A weighted average of VaRy and CVaR+. Formally,
CVaR, = A VaR, + (1- A) CVaR .

Having defined all the terms, we can distinguish four cases in the calculation of

CVaR,. These cases are demonstrated in Figures 2-4(a) through 2-4(d). The examples

provided correspond to cases where we have 6 scenarios, s = 1,...,6, and 4 scenarios,

s = 1,...,4, with equal probabilities p1= p2 = = 4 = p5 6 = 1/6 and

P = P 2 3 = 4 = 1/4, respectively:

6. a corresponds to the cumulative probability of one of the scenarios sc
Y, = a, A = 0, VaR, = L(x, ), and VaR, < CVaR, < CVaR, = CVaR.

7. a does not correspond to the cumulative probability of one of the scenarios:
Y(VaR, )> a, A > 0, VaR, = L(x, ), and VaR, < CVaR, < CVaR, < CVaR .









8. a does not correspond to the cumulative probability of one of the scenarios and is
greater than that of the last scenario: W(VaR) = 1 > a, A =1 > 0,
VaR, = L(x, ,v, ), and VaR, = CVaR, = CVaR, with CVaR undefined.

9. a corresponds to the cumulative probability of the last scenario s,: Ya = 1 = a,
A is undefined, VaR, = L(x, ), and VaR, = CVaR~ = CVaR, with
CVaR undefined.

A major issue in decision-making under uncertainty is the representation of the

underlying uncertainty. Usually, we are faced with either continuous distribution or a

large amount of data, making the problems too complex or too large to solve regardless

of the algorithm or computing capacity. Hence, we need to pass from the continuous

distribution, or data, to a discrete distribution with a small enough number of realizations,

or scenarios, for the stochastic program to be solvable, and a large enough number of

scenarios to represent the underlying continuous distribution or data as close as possible

(Dupacova et al., 2000b; Dupacova et al., 2003).

Scenario Tree

The process of creating this discrete distribution is called scenario generation; it

results in a scenario tree (Di Domenica et al., 2003; Dupacova et al., 2000b; Hoyland et

al., 2003; Hoyland and Wallace, 2001). Formally, The uncertainty in the model is

represented by the parameter with a probability density function, pdf, f(<) and a

corresponding cumulative distribution function, cdf, F(s). The true pdf, f(s), of is

approximated by a discrete probability function, or mass distribution function, mdf,

denoted P(O), concentrated on a finite number of scenarios s = 1,..., S corresponding to

the stochastic parameter s,, with corresponding probabilities p, = P(O, ),such that










S
p, = 1. The scenario tree for a two-stage stochastic problem with recourse is
s=l

illustrated in Figure 2-5.


Probability

1
6
-0-

fI
Loss


a= 2/3 =4/6, ,,


CVaR

6 6 6 6


f3 /14 5i f6

VaR CVaR- CVaR+


S4/6 = a, A = 0,CVaR4/6 = CVaR+ = 1/2


.................. ................. .. (a )

f5,+1/2f6


CVaR


Probability

1
6


f,
Loss


VaR CVaR-


a =7/12, =8/12 >a, = 1/5,CVaR,


Probability

4
4


5 + /f6 =CVaR... ................. (b)


7/12 =1/5 VaR12 + 4/5CVaR = 1/5 f4 +2/5f, + 2/5 f


CVaR

41
SjIT


a = 7/8, yT.


8/8 = > a, A = 1, CVaRI,


'4
VaR


= VaRV,


........................ ................. .. (c )

f4, CVaR is undefined











Probability CVaR
1 1 1 1 1
4 4 4 4 8


fl f, f3 f4'
Loss
VaR ...................... ... ............ (d)

a = 8/8, T, = 8/8 = 1 > a, is undefined CVaR = VoR = f4, CVaR/ is undefined



Figure 2-4. Measure of CVaR, scenario generation

The process of scenario generation is done through the discretization of a

continuous process, the aggregation of a discrete process, or internal sampling. In the

following paragraphs we provide an overview of the first process, i.e., the methods used

for discretization or quantization of continuous distributions.


-0 Os=l

0Os=2

0 0Os=3



0 s=S

First Stage ,, Second Stage

I I I I I I
t=l t=2 t=k t=k+l t=k+2 t=T


Figure 2-5. Scenario tree for a two-stage stochastic problem with recourse

Aggregation or reduction processes consist of deleting scenarios or data from an

already existing large collection. An overview of the reduction methods is outside the

scope of this work; for examples, refer to the literature (Chen et al., 1997; Consigh and

Dempster, 1996; Dupacova, 1996; Dupacova et al., 2000a; Dupacova et al., 2003; Wang,









1995). Internal sampling methods sample the scenarios during the solution procedure,

without using a pre-generated scenario tree; some of these methods are stochastic and L-

shaped decompositions and stochastic quasigradient methods. The reader is referred to

the literature (Casey and Sen, 2002; Dantzig and Infanger, 1992; Dempster and

Thompson, 1999; Ermoliev and Gaivoronski, 1992; Higle and Sen, 1991; Infanger, 1992;

Infanger, 1994).

Discretization

Discretization or quantification of a continuous distribution function is the process

of constructing a discrete distribution function from this continuous distribution. Several

methods have been developed; they can be classified into three main groups: Monte

Carlo simulations, bracket methods, moment-matching methods, and optimization

methods (Dagpunar, 1988; Hoyland et al., 2003; Pfeifer et al., 1991).

One of the most widely used techniques to generate discrete values from a

continuous distribution are Monte Carlo simulations, which draw randomly from the

distribution of a parameter. Monte Carlo simulations require a sequence of random

numbers, usually provided by random number generators, RNG. RNG may be broadly

classified as mixed linear congruential, multiplicative linear congruential, and general

linear congruential, such as Fibonacci, Tauseworthe, shuffled, and portable generators.

For univariate distributions, some general methods for generating random numbers are

inversion, composition, stochastic model, envelope rejection, band rejection, ratio of

uniforms, Forsythe, alias rejection, and polynomial sampling methods. The applicability

and degree of suitability of each method varies with the type of distribution (Beaumont,

1986; Dagpunar, 1988; Pfeifer et al., 1991).









The simplest and most trivial discrete approximations are the bracket methods.

There are two traditionally used bracket methods: the bracket-median and bracket-mean

methods. In the bracket-median approximation the cdf scale is divided into a number of

equal intervals, or brackets, and the median of each is assigned the probability of its

interval. The error in calculating the moments can be substantial if only a few intervals

are used; the most commonly used is the five-point equiprobability bracket median

approximation. The bracket-mean method is similar to the bracket-median method

except that the intervals are represented by their means rather than their median (Clemen,

1991; Hoyland and Wallace, 2001; Miller and Rice, 1983; Smith, 1993; Tagushi, 1978;

Zaino and D'Errico, 1989a).

Another type of approximations is the moment-matching approximations.

Generally, an n-point moment-matching discrete distribution approximation, nPDDA,

consists of n values and their corresponding probabilities of occurrence chosen to

approximate the pdf of a continuous parameter (Di Domenica et al., 2003; Kaut and

Wallace, 2003).

Usually, the n values are specified fractiles from the cdf of the uncertain parameter

with specified probabilities to work well in estimating moments of the pdf. The standard

type of nPDDA is the three-point approximations, 3PDDA, since at least three points are

needed to represent the underlying pdf well while the number of scenario tree paths

increases exponentially with n. In 3PDDA, the pdf is substituted by a three-point mdf.

3PDDA provide a convenient and simple way to approximate a pdf; in addition, it can be

constructed to match the first three moments of a pdf exactly (Keefer, 1994). Several

3PDDA have been developed based on different methods such as the Pearson-Tukey, P-T









(Pearson and Tukey, 1965), the Extended Pearson-Tukey, E-PT (Keefer and Bodily,

1983), the Brown-Kahr-Peterson, B-K-P (Brown et al., 1974), the Swanson-Megill, S-M

(Megill, 1977), the Extended Swanson-Megill, ES-M (Keefer and Bodily, 1983), the

Miller-Rice One-Step, M-RO (Miller and Rice, 1983), the McNamee-Celona Shortcut,

M-CS (McNamee and Celona, 1987), the Zaino-D'Errico Tagushi,Z-DT (D'Errico and

Zaino, 1988), and the Zaino-D'Errico Improved, Z-DI (Zaino and D'Errico, 1989b)

approximations.

Miller and Rice (Miller and Rice, 1983) introduced the use of the Gaussian

quadrature technique for approximating n-point mdf. The result is an n-point discrete

distribution that matches the first 2n-1 moments of the underlying continuous

distribution. The values and probabilities for many distributions are obtained as solutions

to polynomials; they are tabulated for different distributions in many references

(Abramowitz, 1965; Beyer, 1978; Stroud and Secrest, 1966).

Smith (Smith, 1993) developed another n-point the moment-matching

approximation for approximating mdf using the Gaussian quadrature technique; like

Miller and Rice, the result is an n-point discrete distribution that matches the first 2n-1

moments of the underlying continuous distribution. A characteristic of this method is

that it incorporates extreme values of the latter.

Hoyland, Kaut, and Wallace (2003) developed a new moment-matching

approximation by applying a least-square model to minimize the distance between the

generated and target first four moments and correlations for multivariate problems.

Another general method to construct a discrete distribution is optimal

discretization. Hoyland and Wallace (2001) suggested a nonconvex optimization









problem by minimizing a measure of distance between the moments of the constructed

distribution and the ones of the underlying distribution. The model is rerun heuristically

from different starting points until a local minimum is obtained. Other optimization

techniques were developed by Pflug (2001).

The accuracy of a discrete approximation of a probability distribution is measured

by the extent to which the moments of the approximation match those of the original

distribution. Several authors undertook the task of comparing the performance of the

previously presented methods for different underlying distributions.

Miller and Rice compared their Gaussian quadrature based method to the brackets

methods for uniform, normal, beta, and exponential distributions. Their method resulted

in smaller approximation errors on the mean, variance, skew, and kurtosis; in addition,

that error decreased as the number of points in the discrete approximation increased The

bracket methods resulted in the underestimation of almost all moments.

Keefer and Bodily compared several two-point, three-point, five-point, and bracket

approximations in estimating the mean, variance, and the cdf, for beta and log-normal

distributions. They showed that different methods result in different approximation

errors depending on the performance measures and types of distributions. The best

performance was observed for the EP-T method followed by the ES-M method; these

methods, however, perform poorly in cases of extremely skewed or peaked distributions.

Zaino and D'Errico performed a Monte Carlo Simulation to compare different

bracket and three-point approximations. They showed that all methods perform

comparably well in estimating the mean and the variance and that the Z-DI was superior

when estimating higher order moments.









Smith compared the bracket-media, bracket-mean, EP-T, and his moment-matching

methods for normal, log-normal, beta, and gamma distributions. He concluded that the

bracket-mean method accurately approximates the mean but generally underestimates all

the other moments (Miller and Rice, 1983) and that the EP-T approximation accurately

estimates the mean and the variance.

Keefer (1994) compared the performance of six three-point approximations in

estimating the mean, variance, and certainty equivalent, at different risk aversion, for beta

and log-normal distributions. He demonstrated that while 3PDDA methods can exactly

match the first three moments of the pdf, they do not approximate the certainty equivalent

accurately, except for the EP-T and Z-DI approximations. Keefer concluded that the

choice between the different approximations should depend on the trade-off between the

approximation accuracy and the reliability required. For example, these methods'

accuracy in representing extremely skewed or peaked pdfs can be very poor.

The authors argued that most of the errors associated with these discrete

approximations can be reduced by taking more points; however, the tree branches grow

in proportion to the number of points, n, raised to the power of the number of uncertain

parameters being discretized (Hoyland and Wallace, 2001; Smith, 1993). It is also

argued that the errors generated from the use of these approximations are acceptable

based on the premise that very little information is available about the actual distribution

anyways.

Risk in the Water Resources Management Literature

In the water resources management literature, a large number of research

emphasized the need to account for the uncertainties and dynamism. Authors used

methods such as two stage programming with recourse, chance constrained









programming, dynamic and goal programming, fuzzy analysis, genetic algorithm, neural

networks, Bayesian methods, probabilistic analysis, and scenario and sensitivity analysis,

to name a few, to manage reservoir operation, groundwater pumping, groundwater

contamination, water quality, conjunctive supply, irrigation, etc. (Table 2-1).

Most of these studies considered expected values of the issue of interest; only a few

accounted for the risks of low probability events. Rouhani (1985) minimized the mean

square error of differences between measured and predicted groundwater head values in

the design of monitoring networks. Asefa, Kemblowski, Urroz, McKee, and Khalil

(2004) used support vector machines to minimize the bound on generalized risk of the

difference. Feyen and Gorelick (2004) inspected the effect of uncertainty in spatially

variable hydraulic conductivity on optimal groundwater production scheme via a multiple

realization approach. Wang, Yuan, and Zhang (2003) applied reliability and risk analysis

for reservoir operation and flood analysis using Lagrange multipliers. Sasikumar and

Mujumdar (2000) used fuzzy sets of low water quality to manage a river system. Ziari,

McCarl, and Stockle (1995) introduced a variance term in a two-stage stochastic model to

manage irrigation scheduling and crop mix. Others used scenario or sensitivity analysis

to model hydrologic uncertainties (refer to Table 2-1).

These methods provide estimates of risk but no means of controlling or managing

this risk, other than through trial and error (Watkins and McKinney, 1997). In the next

chapter, we propose the application of CVaR, to water resources management problems.

Conclusion

The process of Risk Management may be viewed through many lenses depending

on the discipline and objectives. In this chapter we introduced the fundamentals of risk









and its management. We reviewed the different definitions or risk and risk management

and provided a definition that was adopted in this work. We then categorized and briefly

described the various general risk management approaches with emphasis on value-at-

risk and conditional value-at-risk concepts. The approaches were compared and the

advantages and drawbacks were outlined. As representing uncertainty is as important as

modeling it, we also described and compared different scenario generation techniques.

Finally, we presented a review of the various risk management modeling approaches in

the field of water resources management. In the following chapter, we present an

application of the recommended approaches to a case of water resources management.









Table 2-1. The use of DMuU techniques in water resources management


Application
Reservoir
operation


Groundwater
management


Optimization technique
- Two-stage (Ferrero et al., 1998; Huang and Loucks, 2000; Loucks, 1968;
Wang and Adams, 1986)
- Chance-constrained (Abrishamshi et al., 1991; Askew, 1974; Azaiez et al.,
2005; Ouarda and Labadie, 2001; ReVelle et al., 1969)
- Dynamic (Ben Alaya et al., 2003; Burt, 1964; Chaves et al., 2003; El Awar
et al., 1998; Estalrich and Buras, 1991; Karamouz and Mousavi, 2003;
Kelman et al., 1990; Mousavi et al., 2004b; Nandalal and Sakthivadivel,
2002; Philbrick and Kitanidis, 1999; Stedinger and Loucks, 1984; Stedinger
etal., 1984; Trezos and Yeh, 1987; Wang et al., 2003)
- Fuzzy sets (Chang et al., 1997; Chang et al., 1996; Chaves et al., 2004;
Hasebe and Nagayama, 2002; Maqsood et al., 2005; Maqsood et al., 2004;
Mousavi et al., 2004a)
- Bayesian networks (Wood, 1978)
- Multistage stochastic programming (Pereira and Pinto, 1985; Pereira and
Pinto, 1991; Watkins et al., 2000)
- Optimal control (Georhakakos and Marks, 1987; Hooper et al., 1991)
- Neural networks and genetic algorithms (Akter and Simonovic, 2004; Hasebe
and Nagayama, 2002; Ponnambalam et al., 2003)
- Two-stage stochastic (Wagner et al., 1992)
- Chance constrained programming (Chan, 1994; Eheart and Valocchi,
1993; Hantush and Marino, 1989; Morgan etal., 1993; Tung, 1986;
Wagner, 1999; Wagner and Gorelick, 1989)
- Dynamic(Andricevic, 1990; Andricevic and Kitanidis, 1990; McCormick
and Powell, 2003; Provencher and Burt, 1994; Whiffen and Shoemaker,
1993)
- Optimal control (Georgakakos and Vlasta, 1991; Whiffen and Shoemaker,
1993)
- Neural networks and genetic algorithms (Hilton and Culver, 2005;
Ranjithan et al., 1993)
- Scenario and sensitivity analysis (Aguado et al., 1977; Burt, 1967; Feyen
and Gorelick, 2004; Flores etal., 1975; Gorelick, 1982; Gorelick, 1987;
Hamed et al., 1995; Kaunas and Haimes, 1985; Maddock, 1974; Mao and
Ren, 2004)
- Bayesian networks (Batchelor and Cain, 1999)
- Fuzzy sets (Bogardi et al., 1983; Dou et al., 1999)
- Other (Asefa et al., 2004; Bell and Binning, 2004; Rouhani, 1985;
Tiedeman and Gorelick, 1993; Wagner and Gorelick, 1987)









Table 2-1. Continued


Application
Water quality


Floodplain



Eutrophication

Water transfer
Estuary
Conjunctive
management







Lake/wetland

Irrigation


Planning





Hydrologic
Cycle
Runoff


Optimization technique
- Chance constrained (Fujiwara et al., 1988; Huang, 1998)
- Goal programming (Al-Zahrani and Ahmad, 2004)
- Genetic algorithm (He et al., 2004)
- Bayesian networks (Ricci et al., 2003; Varis, 1998; Varis and Kuikka,
1999)
- Neural networks (Boger, 1992)
- Fuzzy sets (Baffaut and Chameau, 1990; Chang et al., 2001; Chaves et al.,
2004; Jowitt, 1984; Julien, 1994; Koo and Shin, 1986; Lee and Chang,
2005; Lee and Wen, 1997; Liou et al., 2003; Liou and Lo, 2005; Ning and
Chang, 2004; Sasikumar and Mujumdar, 2000)
- Scenario and sensitivity analysis (Chu et al., 2004; Kawachi and Maeda,
2004a; Kawachi and Maeda, 2004b; Mao and Ren, 2004; Vemula et al.,
2004)
- Two stage (Lund, 2002)
- Fuzzy sets (Esogbue et al., 1992)
- Neural networks (Sahoo et al., 2005)
- Bayesian networks (Despic and Simonovic, 2000)
- Simple recourse (Somlyody and Wets, 1988)
- Bayesian networks (Borsuk et al., 2004)
- Two stage (Lund and Israel, 1995)
- Optimal control (Zhao and Mays, 1995)
- Chance constrained (Nieswand and Granstrom, 1971)
- Evolutionary annealing (Rozos et al., 2004)
- Sensitivity analysis (Escudero, 2000; Jenkins and Lund, 2000)
- Mean/Variance (Maddock, 1974)
- Two stage (Cai and Rosegrant, 2004; Watkins and McKinney, 1998; Ziari
et al., 1995)
- Dynamic (Bergez et al., 2004)
- Two stage (Lund and Israel, 1995)
- Bayesian networks (Varis and Kuikka, 1999)
- Fuzzy sets (Hobbs, 1997)
- Chance constrained (Dupacova et al., 1991; Loucks, 1976)
- Bayesian networks (Batchelor and Cain, 1999)
- Fuzzy sets (Suresh and Mujumdar, 2004)
- Scenario and sensitivity analysis (Pallottino et al., 2005)
- Goal programming (Sutardi et al., 1995)
- Bayesian networks (Bromley et al., 2005)
- Fuzzy sets (Alley et al., 1979; Babovic et al., 2002; Bender and Simonovic,
2000; Chen and Fu, 2005; Faye et al., 2005; Slowinski, 1986; Sutardi et al.,
1995; Virjee and Gaskin, 2005; Yi and Zhang, 1989)
- Information gap (Hipel and Ben-Haim, 1999)


- Fuzzy set (Cheng et al., 2002)















CHAPTER 3
COMPARISON OF RISK MANAGEMENT TECHNIQUESFOR A WATER
ALLOCATION PROBLEM WITH UNCERTAIN SUPPLIES
A CASE STUDY: THE SAINT JOHNS RIVER WATER MANAGEMENT DISTRICT

In this chapter, we applied and compared different risk management techniques to a

water resources management problem, where risk is quantified as cost. These methods

are the expected value, scenario model, two-stage stochastic programming with recourse,

and CVaR,. They were built into a mixed integer fixed cost linear programming

framework. Five models were developed: (1) a deterministic expected value model, (2) a

scenario analysis model, (3) a two-stage stochastic model with recourse, (4) a CVaR,

objective function model, and (5) a CVaR, constraint model. Uncertainty was

introduced via water supplies. Assuming continuous normal distribution for the

allowable withdrawals, two discrete distributions with equal expected values, or means,

and different standard deviations were developed based on the method developed by

Miller and Rice (1983). The two different central dispersion parameters were assumed

for the additional assessment of extreme events effect on the results. The result is 9

models formulations. To compare the performance of the different formulations, the

models were applied to a case study, using the Saint Johns River Water Management

District, SJRWMD, Priority Water Resource Caution Areas, PWRCA, in East-Central

Florida, ECF as a study area.

The diagram in Figure 3-1 exhibits the plan of this chapter.










Model Formulation



Scenario Definition



Study Area



Results and Analysis



Conclusions


Definition of decision variables, state variables,
constraints, and different formulations


Description of the uncertainty representation
process


Definition of the water allocation case study



Presentation and comparison of results from
different models


Summary of main findings


Figure 3-1. Chapter 3 organizational diagram.


Model Formulation

Five models were developed: (1) a deterministic expected value model; (2) a

stochastic single stage scenario model; (3) a two-stage stochastic model with recourse,

which is the base of, (4) a CVaR, objective function model and (5) a CVaR, constraints

model. Uncertainty was introduced via water supplies. Two discrete distributions, with

equal expected values, or means, of supplies, were developed by assuming a standard

normal underlying continuous distributions; for additional assessment of extreme events

effect on the results, two different central dispersion parameter, variance, were assumed

for each distribution, leading to the notations (a) and (b). The result is 10 (2 x 5) models

formulations denoted l(a), l(b), 2(a), 2(b), 3(a), 3(b), 4(a), 4(b), 5(a), 5(b). Note that

since the expected value of supplies is the same for both assumed distributions, models









l(a) and l(b) are equivalent, and hence denoted by 1, reducing the total number of

formulations to 9. This section presents the different model formulations, their decision

variable and problem data, objective function, and constraints.

Objective Function

The problem at hand is a fixed cost problem. Given several options (i,j,k), each of

which corresponds to one ofm supply sources i (i=, ...,m), one ofn locations

(j=1, ...,n), and one of P capacity levels k (k = 1,..., ), the decision makers have to decide

in which of the options to invest in which time period t, in a way to incur minimum cost

over the planning horizon (t=l,...,T), while satisfying the situation's constraints. Once

an investment is made, the corresponding option will be available for the remainder of

the planning horizon. The objective is to minimize the sum of two main terms: (1) the

total fixed costs, FC, corresponding to the initial investments, or capital costs, CC,

incurred to make the chosen options available from the chosen times and the

corresponding yearly operation and maintenance costs, OMC, and (2) the total variable

costs, ContC, corresponding to the continuous operational costs of withdrawing water

from each option after it is made available and the total penalty costs, PC, penalizing

unsatisfied demand, or alternative source cost, AC from using an alternative source:


FC = CC + OMC = I T CJk XJk, + Ok X Jkt
=1 =l k=l t=1 r1=

Sn T T m n m n T T
VC = ContC + PC = Ckt kt + D xkt kt P kt + ptD
,=1 j=1 k=l t=l t=l i=1 j=1 k=1 i=1 j=1 k=l t=l t=l


Note that the last term of VC is a constant and is irrelevant in the optimization.









Decision Variables

* Xjkt denotes the decision variable for the investment option (i,j,k) at time t: it is a
binary variable which assumes the value of 0 if investment option (ij,k) will not be
made available at time t and its corresponding investment will not be made, or 1 if
option (i,j,k) will be made available and its corresponding investment will be made.

* xkt denotes how much water will be withdrawn from option (i,j,k) in period t.

Problem Data

* Skt is the total capacity of option (i,j,k) in period t.

* Wt is the total capacity of source i in period t.

* Cikt is the fixed cost of making option (i,j,k) available at time t; it is a one-time
investment cost. For example, you could have C ak = at-1C where Ck is the
nominal fixed cost of making option (i,j,k) available and a e (0,1) is a discount
factor representing the time value of money.

* Oijkt is the O&M cost incurred every period t starting the time of making option
(i,j,k) available; it is a yearly cost.

* c,kt is the unit cost of supplying water from option (i,j,k) in period t. For example,
you could have ckt = at 'lc, where c,k is the nominal unit cost of withdrawing
water from option (i,j,k) and a e (0,1) is a discount factor representing the time
value of money.

* pt or is the unit cost of not supplying water in period t. For example, you could
have p, = a'- p, where is the nominal penalty/alternative cost and a e (0,1) is a
discount factor representing the time value of money. The penalty/alternative cost
could either be the unit cost for acquiring water from an alternative supply, or it
could be a penalty cost to indicate that shortages are undesirable.

Constraints

The problem is subject to different sets of constraints depending on the

formulation; below is a description of all the used constraints:


1. For all ij,k,t: xykt
option (i,j,k) in period t only if option (ij,k) was made available before or at time t.










m n 1
2. For all t: D, V x ,k > 0, which states that the total water made available in
i=1 ]=1 k=l
period t from all options should not exceed the total water demand in period t.
m n I
Note that any shortage will be penalized; or Y xkt > Dt
i=1 j=1 k=l

n
3. For all i, t: I xklk <_ W,, which states that for each source i, the maximum
j=1 k=l
allowable withdrawal in period t should not be exceeded.

I T
4. For all i,j: X k Xkt 1, which ensures that for each source i and location, only
k=1 t=1
one capacity k can be chosen; also, investment option (i,j,k) will be built at most
once.

5. For all ij,k,t: Xkt e {0,1}, which are the binary constraints on the investment
choice variables.

6. For all i,j,k,t: xkt > 0, which are the non-negativity constraints on the quantities of
water withdrawn.

Summarizing, the entire optimization model now reads:

Deterministic Expected Value Model

The deterministic expected value model, model 1, treats the uncertainty of supplies

by averaging them into one number, the expected value of allowed withdrawals, resulting

in the following formulation:











min rCUk X +O jk (x X A )xkt
=l -1 1k= t=1 r= T-l j 1 k=l t=l
subject to
t
Skt < Sjkt Xykr for all i = 1,...,m; j = 1,...,n; k = 1,...,; t = 1,...,T
r=1
m n
D- ZZZ kt >0 for all t = ,...,T
i1 j=1 k=l
n 1
I xIjkt ]=1 k=1
I T
Xkt <1 for ali =1,..., m; j = 1,...,n
k=1 t=1
X kt 1 for alli = ,...,m; j = ,...,n;k = ,...,; t = 1,.., T

xjkt > 0 for all i = 1,...,m; j = 1,...,n; k = 1,..., 1; t = 1,.., T
Scenario Model

The scenario model, model 2, is equivalent to a single-stage deterministic model.

Unlike the expected value method, which considers only the expected value of supplies,

the uncertain supplies are considered by different independent scenarios; the result is the

minimization of the expected value of the objective function over the scenarios set.

In T t m n I T
min E' SCk Xkkt + Ok, Xk, + P t)kt
=Il =-1 k=- t=1 =1 ) = l =1 k=l t=l
m n I T t m n I T
Or, E' min C (kXJk + ,Jkt Xy', + Ckt )k
=-l =-1 k=- t=1 =l j=1 k-= t-=
subject to









t
kt Skt X kt for alli 1,...,m; j ,...,n; k 1,...,;

t = ,...,T; s =l,...,S
m n I
Dt ZZZXkt > 0 for all t = 1,...,T; s = 1,...,S
z=1 j=1 k=1
n 1 l
ZX t < kt for all i = 1,...,m; t = 1,...,T; s = 1,...,S
ilk l
]=1 k=1
I T
ZZ Xkt <1 for all i = 1,...,m; j = 1,...,n
k=1 t=1
X kt 0, 1 for all i= ,...,m; j = ,...,n; k = ,...,; t = 1,...,T
xAkt > 0 for all i = 1,...,m; j = 1,...,n; k = 1,...,1;
t = 1,...,T; s =l,...,S

Two-Stage Stochastic Model with Recourse

The two-stage stochastic model, model 3, like the scenario model, represents

uncertainties by a scenarios set. Unlike the scenario model, however, in this model, the

scenarios are linked by a set of variables, referred to as the first-stage variables; the first

stage variables are the same for all the scenarios, and hence are scenario independent.

The expected value operator is applied only to the terms involving the rest of the

variables, referred to as the second stage variables; the second stage variables are

scenario dependent and hence they are different for each scenario. The result is a fixed

cost problem with recourse.

In other words, the model allows the decision-maker to make two sets of decision:

(1) a first stage decision, an uncertainty independent decision, consisting, in this case, of

the fixed costs of making supplies available, and (2) a second stage decision, an

uncertainty dependent decision consisting, in this case also, of the variable costs of

allocating the resources from the supplies made available by the first decision. This










process allows the decision maker to postpone his allocation decisions until information

about the uncertainties is revealed. The model is formulated below.

m n I T t m n I T ('
min C .ykt +OUkt X,,k + E[S k- P kJk
,=l j-1 k-1 '71 1 = =l k- 1 t- 1
subject to
t
x kt < Sjkt Xk for alli 1,...,m; j = ...,n; k = ...,;
r=1
t = l,...,T; s = l,...,S
m n
D ZZZXkt 0 for all t = 1,...,T; s = 1,..., S
i=1 j=1 k=1
n l


I T
ZZXkt <1 for all = ,...,m;j =,...,n
k=1 t=1
X kt e {0, 1} for all i = 1,..., m; j = 1,...,; k = 1,..., t 1,...,T

xkt > 0 for all i = 1,..., m;j = 1,...,n;k ,...,;

t = l,...,T; s = l,...,S


CVaR, Objective Function Model

The CVaR, objective function model, model 4, is based on the two-stage recourse


model. In this case however, the CVaR, operator is applied to the extreme events of


high costs scenarios only, corresponding to a cumulative probability > a; high costs

events are associated with high risk events of water shortage.


min CVaR { (CkX, + O I Xk + k -- pt )xk
=1 jl k- 1 t- 1 to1 -= j=1 k- 1 t- 1
subject to










t
x jkt < Sijkt f Xzjki
T=1


m n
ZZZ jkt ->
i=1 j1 k=1
n 1
xkt < ts
j lkl t it



k=1 t=1

Xzkt E (0, 1}

xk > 0
i jkt --


for all i = 1,...,m; j

t = l,...,T; s = l,...,S

for all t = 1,..., T; s


for all i = 1,...,m; t


for alli = 1,...,m; j =


for all i = 1,...,m; j

for all i = 1,...,m; j:

t = ,...,T; s = ,...,S


1,...,n;k = ,...,1;




: 1,..., S


1,...,T; s = I,...,S


1,..., n

,...,n; k = ,...,; t = ,...,T

- 1,...,n; k = 1,...,1;


CVaR, Constraint Model.


The CVaR, constraint model, model 5, is also based on the two-stage recourse


model. In this case however, the CVaR, of high risk events are restrained to a value < P


rather than minimized, while the total two-stages costs are minimized.


m in T t m n'T([
min C yktX kt +Oykt X"kt +E kt t Pykt
Il 1 k=1 t=1 r=1 t =l J=1 k=1 t=1
subject to












[. I=1 j=- k=- t=-
t
xijkt -< S"t Zj t Xzjk-
= 1



kZZZ jkt -
n I

j=1 kT=
I T


Oijkt XJkt +kt
=-1 1=1 j=1 k=1 t=1

for all i = 1,..., m; j = 1,..., n; k

t= ,..., T; s=,..., S

for all t = 1,..., T; s = 1,...,S


1,..., ;


for all i = 1,..., m; t


Xk 1 for alli = 1,..., m; j = 1,..., n
k=1 t=1
X jkt e {0, 1} for all i= 1,..., m; j = 1,..., n; k = 1,...,; t = 1,..., T

xkt >0 for all i = ,..., m; j = ,..., n; k = ,...,;
t= ,..., T; s= ,..., S

Scenario Generation

As noted in Chapter 2, a major issue in stochastic optimization is the accurate

representation of underlying uncertainties. In the case where those uncertainties are

represented by continuous distributions, this may be done by scenario generation through

discretization. The various discretization methods were discussed in Chapter 2.

Discretization allows for the approximation of the uncertain parameter, ,


underlying continuous distribution, f(s), by a discrete probability function, or mass


distribution function, mdf, denoted P(I), concentrated on a finite number of scenarios


s = 1,..., S corresponding to the stochastic parameter ,s, with corresponding

S
probabilities p, = P(O,),such that V p, = 1. In this case, e, corresponds to the
s=1

available water supply from different projects, denoted W, in the previously presented


model formulation section.


1,...,T; s = I,...,S









We chose to use the an optimization framework of the method developed by Miller

and Rice (Miller and Rice, 1983), who suggested a moment-matching approximation that

allows the construction of an n-point moment-matching discrete distribution, which

consists on n pairs of probability-value for the uncertain parameter, chosen to represent

the pdf of a continuous parameter that matches 2n-1 moments of the latter. There method

is based on the Gaussian quadrature technique of numerical integration, which

approximates functions integrals by a linear set of polynomials summation. The values,

z,, for different total number of pairs (i.e. order of polynomial approximation), n, and

many functions can be easily obtained from tables in published literature (Abramowitz,

1965; Beyer, 1978; Stroud and Secrest, 1966), as the solutions to the polynomials. For

example, for a standard normal distribution we used Table 5, page 218 in Stroud and


Secrest. The table gives values of the integral fe x f(x)dx, zlI; hence to obtain the


values, z, for a standard normal distribution, the table values were multiplied by F2 .

The values corresponding probabilities are then obtained as a solution to N linear

equations, by substituting these values into the set of equations for the moments of the

approximate discrete distribution. Following the model formulation notation, these

N
equations are of the form (q)= p g where q <1,2,..., 2N 1 is the moment order.
s=1

For example (eq) is the sum of probabilities, the mean, and the variance for q equal to

0, 1, and 2, respectively; they are equal to 1, 0, and 1, respectively, for a standard normal

distribution.









We chose to represent our uncertain parameter by a continuous normal distribution,

with equal mean and different standard deviations, to simulate different cases of

parameters scattering and extreme events. The normal distribution is applicable to a very

wide range of phenomena that occur in nature, industry, and research and is the most

widely used in statistics. Physical measurements in meteorological and hydrological

experiments as well as manufactured parts are often adequately represented using normal

distributions. In addition, the normal distribution is in many conditions a good

approximation to other distributions. It is also the asymptotic form of the sum of random

variables or parameters under a wide range of conditions, if the underlying phenomena

are additive (DeGroot, 2002; Evans et al., 2000; Walpole, 1989).

We present here the results for standard normal distributions based on 10 pairs,

corresponding to 10 scenarios (Tables 3-1 and 3-2). The values, <,, can be transformed

for specific normal distributions by as simple transformation using the specific mean, j/,

and standard deviation, a For normal distribution, this is obtained using the formula

5, = z, + /, where z, are the values obtained for a standard normal distribution.

Note that for a standard normal distribution, symmetric around the mean, 0, i.e.,

with +n/2 = -, if N is even, the sum is zero for all odd q. In addition, its symmetry

allows the reduction the number of linear equations to n/2, as ps+n/2 = Ps

In the case of 10 scenarios, N = 10, nineteen moments may be matched. Since N

in this case is even, all odd moments are equal to zero, and the linear system reduces to

10 equations with 10 unknowns, corresponding to the 0th, 2nd, 4th, 6th, 8th, 10th, 12th, 14th

16t, and 18th moments. As p, = p,, the unknown probabilities are reduced to 5 and for

all practical reasons we restrict ourselves to the first even moments equations.









Pl +P2 +P3 +P4 +P5 = 1/2
p12+p2 +p3 2+p4 1+p2 =/2
P P12 2 P 3 3 4r2 P5 52 =1/2
P 14 P24 3 34 4 +5 54 =3/2
PI +6 P22 +P33 +P444 +p 5 =15/2
Pm1 +P22 +P3V +p4 +P5 = 105/2
2 3 4 =105/2

The qth moments were calculated from the distributions' moments generating


function, M(y); My)= e 2 for o < y < +c, for a normal distribution, where the

qth moments equals the qth derivative of M(y) at zero, M(")(0); for details refer to the

literature (Beaumont, 1986; DeGroot, 2002).

This system was solved using the excel What's Best Optimization package, where

the sum of probabilities was minimized subject to the set of the first two, three, four, and

five linear equations as constraints. Additional constraints were imposed setting higher

probabilities for values closer to the mean. The resulting sets of probabilities for

matching up to the 2nd, 4th, 6th, and 8th moments, a total of four minimization problems,

are presented in Table 3-3. Note that matching the 4th moment was a source of

infeasibility in the minimization problems; hence, its corresponding constraint was not

satisfied, i.e., it was relaxed, in the four minimizations. Matching up to the 8th moment

resulted in violation of the 6th moment too. The moments from each minimization in

comparison with the continuous standard normal distribution moments are presented in

Table 3-2. The resulting histograms and smooth distributions are compared in Figures 3-

2 and 3-3.

The visual comparison of the results reveals that constraining all the moments

results in the best discrete approximation. To confirm this observation, a least-square

regression analysis was run on the results. Analysis of Table 3-3 and Figure 3-4









regression findings verified our observation. Constraining up to the 8th moments resulted

in the best fits, with the highest r2, slope closest to unity, and intercept closest to zero, and

smallest corresponding errors.

Case Study Area

Groundwater serves as the primary water source for most of the State of Florida.

Faced with continuous growth, local utilities throughout the state are working on

identifying alternative viable and economic sources of potable water.

Recognizing the need to develop new sources and plan well in advance, the Saint

Johns River Water Management District, SJRWMD, one of five water management

districts in Florida (Figure 3-5), initiated in the year 2000 several water supply plans that

(1) identified limit of fresh groundwater in Priority Water Resources Caution Areas,

PWRCA (Figure 3-5), which are areas where existing and anticipated sources of water

and conservation efforts are not adequate; (2) identified alternative water resources

options and development projects with cost data and likely project users; (3) initiated the

Alternative Water Supply Construction Cost Sharing Program in 1996 to provide

cooperative funding for the construction of alternative water supply facilities (Vergara,

2004; Wilkening, 2004; Wycoff and Parks, 2005).

Water Demand

The total water demand of the SJRWMD PWRCA is projected to linearly reach

830 MGD in the year 2025 (Table 3-4). This demand consists of public supply,

domestic, agriculture and recreational irrigation, commercial, industrial, institutional, and

power generation water needs (Wilkening, 2004; Wycoff, 2005).











Table 3-1. Standard normal discrete distribution approximation for N=10 and up to the 2nd, 4th, 6th, and 8th moments constraints
q 0th, 2nd 0th, 2nd, 4th 0th, 2nd, 4th, 6th 0th, 2nd, 4, 6, a 8th

zI, zs pdf cdf pdf cdf pdf cdf pdf cdf
-3.43616 -4.85946 0.001534 0.001534 0.02464 0.02464 0.000000 0.000000 0.000000 0.000000
-2.53273 -3.58182 0.001534 0.003067 0.02464 0.04928 0.031523 0.031523 0.016867 0.016867
-1.75668 -2.48433 0.001534 0.004601 0.02464 0.07392 0.053653 0.085176 0.068881 0.085748
-1.03661 -1.46599 0.247699 0.252301 0.02464 0.098559 0.053653 0.138829 0.068881 0.154629
-0.3429 -0.48494 0.247699 0.500000 0.401441 0.500000 0.361171 0.500000 0.345371 0.500000
0.3429 0.48494 0.247699 0.747699 0.401441 0.901441 0.361171 0.861171 0.345371 0.845371
1.03661 1.46599 0.247699 0.995399 0.02464 0.92608 0.053653 0.914824 0.068881 0.914252
1.75668 2.48433 0.001534 0.996933 0.02464 0.95072 0.053653 0.968477 0.068881 0.983133
2.53273 3.58182 0.001534 0.998466 0.02464 0.97536 0.031523 1.000000 0.016867 1.000000
3.43616 4.85946 0.001534 1.000000 0.02464 1.000000 0.000000 1.000000 0.000000 1.000000

Table 3-2. Moments constraints
Discrete Approximation Moments
q V Moment Oth, 2nd Oth, 2nd, 4th Oth, 2nd, 4th, 6th Oth, 2nd, 4, 6, a 8th
Constraints 2 Moment Constraints 2 Moment Constraints 2 Moment Constraints 2 Moment
0 0.5 0.500 =0.500 =0.500 =0.500
2 0.5 0.500 =0.500 =0.500 =0.500
4 1.5 0.656 Not= 1.197 Not= 0.936 Not= 0.871
6 7.5 2.324 18.870 =7.500 Not= 5.737
8 52.5 26.258 382.538 79.716 =52.500











Table 3-3. Least square regression analysis results
Regression Analysis Results
Moments Constraints 0th, 2nd 0th, 2nd, 4th 0th, 2nd, 4th, 6th 0th, 2nd, 4, 6, a 8th
Slope 0.501 7.464 1.538 1.008
Intercept -0.218 -12.580 -1.398 -0.575
Standard error of slope 0.0179 0.361 0.0397 0.0192
Standard error of intercept 0.425 8.558 0.943 0.454
Correlation coefficient r2 0.996 0.993 0.998 0.999
Standard error on ordinate 0.808 16.266 1.793 0.864
















05


04


03


02


01



-486 -3 58 -2 48 -1 47 -0 48 048 147 2 48 358 486
pdf(a) cdf(a)


Moments Matched: 0th, 1st, and 2nd

05


-486 -358 -248 -147 -048 048 14,
-pdf(a)


Moments Matched: 0th, st, 2nd, and (4th)

05 1


2 48 358 486
- cdf(a)


-486 -358 -248 -147 -048 048 147 248 358 486 -486 -358 -248 -147 -048 048 147 248 358 486
pdf(a) cdf(a) pdf(a) cdf(a)


Moments Matched: 0th, 1st, 2nd, (4th), and 6th Moments Matched: 0th, st, 2nd, (4th), 6th, and (8th)



Figure 3-2. Discretized standard normal distribution (moments in parentheses, i.e., (qth), are the unmatched moments)
















05


04


03


02


01


0 -A- -A---- --
-4 86 -3 58 -2 48 -1 47 -0 48 048
Spdf(a)

Moments Matched: th, 1st, and 2nd

05


04

I V


1 47 2 48 358 486
cdf (a)


01




-486 -358 -248 -1 47 -048 048
Spdf(a)

Moments Matched: 0th, 1st, 2nd, and (4th)

05


04


03


02


0 1


1 47 2 48 358 486
cdf(a)




6 6


-486 -358 -248 -1 47 -048 048 147 248 358 486 -486 -358 -248 -1 47 -048 048 147 248
E pdf(a) + cdf(a) E pdf(a) + cdl

Moments Matched: 0th, 1st, 2nd, (4th), and 6th Moments Matched: 0th, 1st, 2nd, (4th), 6th, and (8th)



Figure 3-3. Discretized standard normal distribution (moments in parentheses, i.e., (qth), are the unmatched moments)


358 486
f(a)


~


41



1











400




300




' 200




100




0


100 200 300
Continuous pdf Moments


Figure 3-4. Moments least-square regression analysis plots


Figure 3-5. Priority water resource caution areas in the SJRWMD, Florida, USA
(Vergara, 2004; Wilkening, 2004)










Table 3-4. SJWRMD caution area water demand projections (Wilkening, 2004)
Year Demand
2010 676
2011 686
2012 697
2013 707
2014 717
2015 727
2016 738
2017 748
2018 758
2019 768
2020 779
2021 789
2022 799
2023 809
2024 820
2025 830

Water Supply

The demand is currently supplied from the Floridan Aquifer groundwater. It is

projected that the aquifer's capacity for the caution areas, 670 mgd, will be reached

before the year 2010 (Wilkening, 2004; Wycoff, 2005).

With that in mind, the SJRWMD identified three main potential alternative sources

for water, with a total capacity of 335 mgd: (1) 175 mgd from the Saint Johns River basin

(SJR) at seven locations; (2) 100 mgd from the Lower Oklawaha River (LOR),3 at one

location; and (3) 60 mgd from Collocated seawater (CSW) at three locations. These rates

correspond to the maximum allowable withdrawals from the sources; they are assumed to




3 Note that although 100 mgd may be withdrawn from the Oklawaha River, a project capacity of 21.5 mgd
has been suggested since this source is at a remote location with respect to the priority caution areas of the
SJRWMD, rendering the transmission costs from this source prohibitive.











be the source of uncertainty in the models (Vergara, 2004; Wilkening, 2004; Wycoff and

Parks, 2005).

In sum, the SJRWMD identified 11 alternative water supply projects; the

approximate locations of these projects are shown in Figure 3-6. Various project

development scenarios were examined by the district to provide examples of various

water supply quantities and costs associated with each project. The projects details are

presented in Table 3-5. Each of these projects has a maximum allowed withdrawal,

which is the total water that can be withdrawn from this source while subject to

constraints such as upstream and downstream water levels, sea and riverine ecology, and

aesthetics. Cost estimates were provided for several possible average capacities at each

location; only one of these capacities may be chosen at each location. The maximum

capacities are only design capacities and not demand capacities. The design, permitting

and construction of new source facilities will likely take 5 to 10 years (Vergara, 2004;

Wycoff and Parks, 2005).



St Jal. River ar Lake G orge
nLml L ower \V L__kawaha River in Putn am Counly


-7 Intc.astal Waterway at
." -\Nw Smyrna Be ch



U J Si ,onr- P .- n Lake ..,
L \ Ind. ---.e, Loa .1r aA
-an ri -. 1.o:3-
La A pka 'k .I FP&L Ceap anIral
( [- -_ p --, i -r .,J a 6 6, Z a :,.


,- 1"-, s ... .. S jr.na r r r B
01 A T r Crk Rervolr





Figure 3-6. Approximate locations of potential alternative water supply projects
(Vergara, 2004; Wilkening, 2004)









Water Cost

Table 3-5 presents a summary of the estimated total capital, O&M, and unit costs

of water from different projects locations and capacities. The method followed for these

estimates are briefly summarized in this section. Note that the last row in Table 3-5

stands for existing groundwater supplies, hence, they are not part of the proposed

projects, have no associated fixed cost, a one significant figure unit cost of $1/1000

gallons, and an O&M cost of 0.2/100 gallons, based on year 2000 Dollar (Wycoff, 2005).

Total capital cost is the sum of construction cost, non-construction capital cost,

land cost, and land acquisition cost. The Operation and Maintenance, O&M, Cost is the

estimated annual cost of operating and maintaining the water supply project when

operated at average day capacity. The Equivalent Annual Cost is the total annual life

cycle cost of the water supply project based on facility service life and time value of

money. Equivalent annual cost, expressed in dollars per year, accounts for total capital

cost and O&M costs with facility operating at average day design capacity. Finally, the

Unit Production Cost is Equivalent annual cost divided by annual water production. The

unit production cost is expressed in terms of dollars per 1,000 gallons produced (Wycoff

and Parks, 2005).

These costs are in year 2003 Dollar. They were converted to future years Dollar

values using the Construction Cost Index (Michaels, 1996), or CCI; results are tabulated

in Appendix A.

CCI is estimated by Engineering News Records, ENR, on a monthly basis, and

represents the underlying trends of construction costs in the USA. It is determined by

several factors such as labor, materials, and others (ENR, 2005). Table A-i lists

historical yearly averages of CCI for the years 1908 2005. Figure A-i is a plot of these









values to obtain a best fit of the year CCI relationship. Using the Equation of the best

fit, projections of CCI were calculated for the years 2000 2025; CCI was also estimated

from the equation for the years 2000 2005 for consistency.

But what is the significance of CCI and how is it used? Actually, CCI is used as a

measure of change of costs between different years. This change, ACCI, for consecutive

time periods, years, is estimated using Equation 2. The change in CCI can also be

calculated for non-consecutive years using Equation 3, such as t'< t.

ACCI (CCI CCIt )
ACCI x 100 2
CCIt-1


ACCI, (c C ) x 100 3
CCI,,

To estimate the value of costs at time t, C,, the cost at time t', C, is multiplied by

ACCI,, with t'< t, Equation 4.

C, = ACCI, x C,, 4

Scenario Generation

The scenarios were defined around the uncertainty in supply. The previously

described projects are assigned deterministic expected values of water supply designated

as average withdrawals capacities. Assuming a normal distribution, two mass

distribution functions, (a) and (b), each with 10 scenarios, were defined, assuming two

different standard deviations, at 5 and 10 percent of the mean of each supply, for each

supply (Table 3-6). This was based on the Miller and Rice (1983) moment matching

method in an optimal discretization framework. Both methods were discussed earlier in

this chapter.











Table 3-5. Supply sources, capacities, and costs4
Source (Maximum Allowed Location (J) System Capacity Capital O&M Unit Production
Withdrawals) () (mgd) Cost Cost Cost
Average Maximum k ($M) ($M/yr) ($/1,000
__gallons)
Saint Johns River Saint Johns Near SR 520/528 (J= 1) 20 30 k=1 189 7.56 3.03
Basin (175 MGD) River Near SR 50 (j = 2) 10 15 k=1 91 3.81 3.00
il1
Near Lake Monroe (= 3) 50 75 k=1 457 18.71 2.93
30 45 k=2 238 11.29 2.74
20 30 k=3 217 7.56 3.27
9.6 14.4 k= 4 84 3.67 2.94
Near Lake Monroe 10 15 k=1 81 3.80 2.80
(j=4) 50 75 k=2 372 18.80 2.63
100 150 k=3 714 37.20 2.55
Near DeLand (j = 5) 20 30 k =1 210 7.56 3.22
10 15 k=2 105 3.80 3.25
50 75 k=3 447 18.80 2.91
100 150 k= 4 871 37.20 2.84
Near Lake George (j= 6) 33 49.5 k= 1 386 12.40 3.41
Taylor Near Cocoa (j = 7) 10 15 k =1 55 2.20 1.66
Creek 25 37.5 k=2 134 6.00 1.68
Reservoir
Lower Lower Putnam( = 1) 21.5 32.25 = 1 255 5.45 2.94
Ocklawaha River Ocklawaha
(100 MGD) River
i=2


4 (Vergara, 2004; Wilkening, 2004)











Table 3-5 Continued
Source (Maximum Allowed
Withdrawals) (i)


Location (J)


System Capacity
(mgd)


Capital
Cost
($M)


O&M
Cost
($M/vr)


Unit Production
Cost
($/1,000 gallons)


Collocated Indian FP&L Cape Canaveral 10 15 k = 1 90 5.00 3.33
Seawater (60 River Power Plant (J =1) 20 30 k = 2 180 9.40 3.23
MGD) Lagoon 30 45 k =3 274 13.60 3.20
i = 3 Reliant Power Plant 10 15 k = 1 90 4.50 3.20
(j = 2) 20 30 k=2 177 8.40 3.07
30 45 k =3 268 12.10 3.28
Intracoastal Near New Smyrna 5 7.5 k = 1 83 3.10 5.06
Waterway Beach (J =3) 10 15 k = 2 121 5.20 3.99
15 22.5 k =3 159 7.60 3.61


Floridan Aquifer5


670


48.91


5 Wycoff, R. (2005). "Phone Interview." Consultant, Saint Johns River Water Management District.












Table 3-6. Scenarios of supply capacities at 5% and 10% STD
Supply Source (Mean)
1 (175.0) 2(21.5) 3 (60.0) GW (670)
Scenario, s = 8.75 (a) = 17.5 (b) = 1.07 (a) = 2.15 (b) a = 3 (a) = 6 (b) =33.5 (a) a = 67 (b)
1 132.4797 89.9594 16.27608 11.05215 45.42161 30.84322 507.208 344.416
2 143.659 112.3181 17.64954 13.79908 49.25453 38.50906 550.0089 430.0178
3 153.2621 131.5243 18.82935 16.1587 52.54702 45.09404 586.7751 503.5502
4 162.1726 149.3452 19.92406 18.34812 55.60203 51.20407 620.8894 571.7787
5 170.7568 166.5136 20.97869 20.45739 58.54519 57.09039 653.7547 637.5093
6 179.2432 183.4864 22.02131 22.54261 61.45481 62.90961 686.2453 702.4907
7 187.8274 200.6548 23.07594 24.65188 64.39797 68.79593 719.1106 768.2213
8 196.7379 218.4757 24.17065 26.8413 67.45298 74.90596 753.2249 836.4498
9 206.341 237.6819 25.35046 29.20092 70.74547 81.49094 789.9911 909.9822
10 217.5203 260.0406 26.72392 31.94785 74.57839 89.15678 832.792 995.584









Results and Discussion

As explained in the previous sections, five model formulations were considered:

the expected value of supplies formulation, model 1, the scenario or expected value of

costs model, model 2, the two-stage stochastic model with recourse, model 3, the CVaR,

minimization model, model 4, and the CVaR, constraint model, model 5. As the

objective of this work was to demonstrate the tradeoffs between costs and risk, as

CVaR,, the next sections focus on the Pareto efficient frontier. The significance of this

method is to provide for a given CVaR, /cost value, the minimum cost/CVaR, that can

be obtained without exceeding that CVaR, /cost value.

Focusing on model 5, model 5 was run for three constraints corresponding to three

confidence levels a', 50, 75, 80, 85, 90, 95, and 99 percent; the models are designated as

5 a', 5-50, 5-75, and 5-95, respectively. In addition, each of the models 5 a' was run

at different values of /, designated as 1,...,b,...B in an increasing order; the values of ,/

ranged between the smallest feasible value to the value at and after which no change was

observed. Different runs are referred to by the model number, the confidence level of the

constraint, and the number of the / value used in the constraint, i.e., 5 a'-b. For the

confidence level a', the Pareto efficient frontier, for a given confidence level a, was

generated by plotting a point, corresponding to total cost on the abscissa and CVaR, on

the ordinates, for each model run, or value of f8; there exists a different frontier for each

confidence level a and a set of frontiers for each confidence level a'. To compare

solutions of models 1, 2, and 3, the costs solutions from these models were added as









points to the efficient frontiers plots; this allows to see whether these models solutions are

efficient in the Pareto sense, i.e., undominated.

Note that model 3 can be obtained from model 5 by deleting the CVaRY constraint

or setting p / oc ; hence, the solution to model 3 corresponds to one of the endpoints of

the Pareto efficient frontier. Also note that model 4 minimizes CVaRY of the total cost

and does not control costs that are below the confidence level at which CVaRY is

minimized; as a result, the two-stage total cost result of this model does not possess any

practical significance; this model, however, finds the smallest / for which a feasible

solution can be obtained, providing the other endpoint of the Pareto efficient frontier,

hence, model 4 is used to find this point and not as a model by itself.

The models were ran for two normal distributions of allowable withdrawals, W,

with equal means and different standard deviations, namely 5 and 10 percent. In this

section, we present and compare the different model results within and across both

distributions, (a) and (b).

5% Standard Deviation

The results for the distribution (a), corresponding to 5 percent standard deviations

are summarized in Figures 3-7 to 3-14. Figure 3-7 presents plots of the change in cost

with p = CVaR,,, where a' is the confidence level, a, of the constraint; these plots

correspond to the efficient frontiers at different a'. The figure demonstrates that, for all

values of a', the cost increases as / decreases; in other words, a tighter constraint

results in an increase in cost. In addition, intuitively, a higher value of a' results in a

higher range of f/. The range of / consists of a lower minimum below which the model









can no longer be feasible and an upper limit beyond which the solution is independent of

the constraint. The lower limit coincide with the minimized CVaR,, for a = a', of

model 4; the upper limit corresponds to CVaR,, for a = a', calculated from the solution

of model 3. The cost solutions obtained from model 1, model 2, and model 3, i.e.,

expected value solution, individual scenarios, and two-stage stochastic solutions, are

plotted in the graphs at values of / equal to the cost for model 1 and model 2 scenarios

and at / = CVaR,, for a corresponding to a', for model 3. Note that the efficient

frontier delineates a risk-return space corresponding to the lowest risk for a given level of

return or -cost or the best possible return or minimum cost for a given level of risk;

points below the concave frontier line correspond to inefficient or suboptimal solutions

and points above the frontier are infeasible. Figure 3-7 shows that, for all a', (1) the

expected value solution cannot be achieved for any level of risk, (2) only two of the

scenarios of model 2, corresponding to high risk values, are feasible but inefficient, and

(3) the two-stage stochastic solution is an efficient solution corresponding to the lowest

possible cost and high level of risk.

Figures 3-8 to 3-14 demonstrate the dependency of CVaR,, calculated at different

confidence levels, on (A) a, (B) f/, and (C) cost, for each model 5 a' simulated, 50, 75,

80, 85, 90, 95, and 99 percent, respectively. For each a', for different / values, the

model's CVaR, were recalculated and plotted; for example, if the model was run for 10

different constraints values f/, 10 lines were plotted, monitoring the change of the

recalculated solution CVaR, with the change in the confidence level, the constraint, and

the cost solution. The results were consistent with each other and the theory for all plots;












a higher confidence level corresponds to a higher solution CVaR,, a tighter constraint


corresponds to a lower solution CVaR,, and a higher cost corresponds to solution a


higher CVaR,,.


1.17 0.60


-I -1P


4.'


1.29 0.60

r\


U


4.,


01


-2


-X- 3


50%


40
50%


0



C-


o
O






-18
1.09


0






o


40
75%


01


-2


X3


-+- 80%


-18 -18
1.20 1.35 0.60 4.40
P P 80%
Figure 3-7. Efficient frontier for a'= 50, 75, 80, 85, 90, 95, and 99 percent, 5% STD.


01


-2


X3


75%


-18 L
1.17


0 F-


`O~I-


OK


_-1 _-I












0






o







-29
1.25


0






o







-29
1.34

0






0
o






-31
1.50


Figure 3-7 Continued


0






0







-29
1.46 0.60


0





0







-29
1.46 0.60


0 0






0







-31
1.91 0.60


01


-2



X3


85%


4.40
85%


01


-2



x3


90%


4.40
90%


01


-2



X3


-8-95%


4.40


95%












0







0







-34


01



-2



X 3



99%


0



)


o







-34
2.07


2.77 0.60


99%


Figure 3-7 Continued.


4.40


i










34




a

0


-5 10
15 20
- 25 30
--35 -40
45 50
55 60
65 -x70
-75 80
-85 90
95 99
- Cost


1.16


-5
15
-25
--35
45
55
65
-75
-85
95
X1
03


--10
20
--30
-40
50
60
--70
80
90
99
-2


-Cost


Figure 3-8. Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 50%, 5% STD.


- 3
-- 5-50-1
5-50-2
5-50-3
-5-50-4
-- 5-50-5
- 5-50-6
- 5-50-7
5-50-8
5-50-9
5-50-10


34













0
1.10


34







O
CU
0










34






CU






0
0


34







0





0
1.17
1.17


1.29


--5
15
-25
-35
45
55
65
--75
-85
95
X1
03


-10
20
--30
-40
50
60
-70
80
90
99
-2


-Cost


Figure 3-9. Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 75%, 5% STD.


-- 5-75-1

5-75-2

5-75-3

5-75-4

-- 5-75-5

5-75-6

100


--5 -- 10
15 20
25 --30
-35 -40
45 50
55 60
65 70
-75 80
-85 90
95 99
Cost


34





ro






80


34 -3
-5-80-1
3 increasing 5-80-2
5-80-3
-- 5-80-4
-*- 5-80-5
y- 5-80-6
5-80-7
> 5-80-8
5-80-9
5-80-10
5-80-11
5-80-12
--5-80-13
0 5-80-14
0 100
c (A)
34 -e-5 -10
15 20
a increasing -*25 -e-30
-I 35 -40
45 50
C 55 60
0 65 70
75 80
-85 90
95 99
0 .-- Cost
1.20 1.35
P (B)
34 -5 ---10
15 20
--25 --30
-35 -40
45 50
55 60
> 65 -70
0 \ --75 80
t\ -85 90
a increasing 95 99
I 1 -2
0 03
-17 0
-Cost (C)


Figure 3-10. Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 80%, 5% STD.










34






CU






0
0
0-






34






CU






0
1.25

34






C
>


--5 10
15 20
-25 --30
-35 -40
45 50
55 60
65 -70
--75 80
-85 90
95 99
- Cost


1.46


-5
15
-25
-35
45
55
65
-75
-85
95
X1
03


--10
20
- 30
-40
50
60
-70
80
90
99
-2


-Cost


Figure 3-11. Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 85%, 5% STD.


- 3
-- 5-85-1
5-85-2
5-85-3
- 5-85-4
- 5-85-5
- 5-85-6
- 5-85-7
5-85-8
5-85-9
5-85-10
5-85-11
5-85-12
- 5-85-13
-5-85-14










34







O
(0


1.46


34






CU



0
0-



1.34


34






CU
>


-10
20
--30
-40
50
60
-70
80
90
99
-2


-Cost


Figure 3-12. Change of CVaR, with (A) /, (B) a, and (C) cost for a'= 90%, 5% STD.


--3
-- 5-90-1
5-90-2
5-90-3
5-90-4
-- 5-90-5
5-90-6
5-90-7
5-90-8
5-90-9
100

--5 -10
15 20
-25 --30
-35 -40
45 50
55 60
65 -70
--75 80
-85 90
95 99
Cost


-5
15
-25
--35
45
55
65
--75
-85
95
X1
03