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EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES PROBLEMSA GAME THEORETIC APPROACH By ELLIOT KIN NG 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 1985 To my parents and my wife, Eileen, and children, Matthew, Michelle, Michael ACKNOWLEDGMENTS I would like to thank my chairman, Dr. James P. Heaney, for the many hours spent guiding this research. His encouragement, support, and friendship during my three years at the University of Florida have been invaluable. I would also like to thank the other members of my supervisory committee, Dr. Sanford V. Berg, Dr. Donald J. Elzinga, Dr. Wayne C. Huber, and Dr. Warren Viessman, for their time and support. In addition, I wish to thank the U.S. Air Force for giving me the opportunity to pursue the Ph.D. degree. Thanks are also due to several fellow students who have made my program enjoyable and memorable. In particu lar, I wish to thank Mr. N. Devadoss, Mr. MunFong Lee, and Mr. Robert Ryczak. I would also like to give special thanks to Mr. Robert Dickinson for keeping an extra copy of the LP80 and Mrs. Barbara Smerage for doing such an excellent job typing this manuscript. I am extremely grateful to my parents for instilling in me a desire to seek further education. Furthermore, I am especially thankful to my wife, Eileen, for typing initial drafts of this manuscript and for her love, encouragement, and sacrifices throughout my program. We will miss the iii croissants, pizzas, and hoagies that supplemented my late night studies. Finally, I wish to thank my children, Matthew, Michelle, and Michael, for their love and under standing during the countless times I have chased them out of my study. TABLE OF CONTENTS Page ACKNOWLEDGMENTS...................................... iii LIST OF TABLES....................................... vii LIST OF FIGURES...................................... ix ABSTRACT.............................................. x CHAPTER 1 INTRODUCTION............................... 1 2 LITERATURE REVIEW ....................... 4 Efficiency Analysis .......................... 4 Equity Analysis........................... 5 Conclusions .............. ................. 8 3 EFFICIENCY ANALYSIS........................ 10 Introduction... ...... .... ................. 10 Partial Enumeration Techniques............. 12 Total Enumeration Techniques................. 15 Modeling Network Problems as Digraphs....... 16 The Total Enumeration Procedure............ 21 Computational Considerations.............. 30 Summary..................... ....... ........ 38 4 EQUITY ANALYSIS............................ 39 Introduction. ....... .... ......... .... ... 39 Cost Allocation for Regional Water Networks............................. 40 Criteria for Selecting a Cost Allocation Method................... 45 Ad Hoc Methods ............................ 48 Defining Identifiable Costs as Zero... 49 Defining Identifiable Costs as Direct Costs........................ 54 Cooperative Game Theory..................... 64 Concepts of Cooperative Game Theory... 65 Unique Solution Concepts.............. 75 Empty Core Solution Concepts.......... 87 Cost Allocation in the Water Resources Field....... ................ ... .... 88 Separable Costs, Remaining Benefits Method.............................. 90 Minimum Costs, Remaining Savings Method............................. 95 Allocating Cost Using Game Theory Concepts. 99 The k Best System.................. 99 The Dummy Player....................... 108 Comparing Methods..................... 115 Summary...................................... 119 5 EFFICIENCY/EQUITY ANALYSIS .... ............... 120 Introduction............................... 120 Maximum Cost.............................. 122 Minimum Cost............................... 129 Fairness Criteria........................... 132 Summary............... ........ ... .... ..... 133 6 CONCLUSIONS AND RECOMMENDATIONS............ 135 APPENDIX EFFICIENCY/EQUITY ANALYSIS OF A THREE COUNTY REGIONAL WATER NETWORK WITH NONLINEAR COST FUNCTION..................... 142 REFERENCES......... .............. ..... .. ........... 150 BIOGRAPHICAL SKETCH................................. 159 LIST OF TABLES Table Page 31 Example of Total Enumeration Procedure for 3Node Digraph............................... 27 32 The Number of Independent Calculations to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs................. 31 33 Summary of Computational Effort for Digraphs Shown in Figure 34.......... ..... ............ 34 34 Efficiency Analysis of a ThreeUser Water Supply Network with Nonlinear Cost Function Using Lotus 123................ ............ 37 41 Projected Population Growth and Projected Average Per Capita Demand.................... 41 42 The Costs and Percent Savings for All Options............................................. 44 43 Cost Allocation Matrix....................... 50 44 Cost Allocation of Optimal Network Based on Population.................................... 52 45 Cost Allocation of Optimal Network Based on Demand ..................... ................... 53 46 Cost Allocation of Optimal Network with Use of Facilities Method......................... 56 47 Cost Allocation for the Use of Facilities Method ............... ........................ 57 48 Cost Allocation of Optimal Network with Direct Costing/Equal Apportionment of Remaining Costs Method....................... 60 49 Cost Allocation for Direct Costing/Equal Apportionment of Remaining Costs Method...... 61 vii 410 Core Geometry for ThreePerson Cost Game Example......................................... 73 411 Cost Allocation for ThreeCounty Example Using the Shapley Value...................... 80 412 Cost Allocation for ThreeCounty Example Using the Nucleolus.......................... 85 413 Empty Core Solution Methods.................. 89 414 Cost Allocation for ThreeCounty Example Using the SCRB Method......................... 94 415 Cost Allocation for ThreeCounty Example Using the MCRS Method......................... 98 416 Nominal Versus Actual Core Bounds for Optimal Network Game......................... 100 417 Cost Allocations for the Optimal Network and the Second Best Network ($).................. 103 418 Cost Allocation for Option 3 as a TwoPerson Game Using the SCRB Method................... 109 419 Comparing Cost Allocations for Option 3 as TwoPerson Game and ThreePerson Game Using the SCRB and MCRS Methods.................... 111 420 Core Bounds for Option 3 as a ThreePerson Game......................................... 112 421 Core Bounds for Option 3 as a ThreePerson Game with County 1 as a Dummy Player......... 114 422 Core Bounds for Option 3 as a TwoPerson Game........................................... 116 423 Comparison of Methods Discussed for Allocating Costs of Water Resources Projects. 117 51 Using Independent Calculations from the Total Enumeration Procedure to Find c(i), c(S), and c(N) for the ThreeCounty Regional Water Network Problem......................... 121 52 Efficiency/Equity Analysis of the Optimal Network..... .......... ............. ............ 126 viii LIST OF FIGURES Figure Page 31 Types of Cost Functions....................... 13 32 Example Digraph Representing a Regional Water Network Problem for Three Users ......... 18 33 Flow Diagram of Total Enumeration Procedure for nNode Digraph............................ 23 34 Examples of 3,4,5Node Digraphs............... 33 41 Lengths of Interconnecting Pipelines.......... 43 42 Geometry of Core Conditions for ThreePerson Cost Game Example............................... 71 43 Core for the Optimal Network Game (C(N) = $4,556,409)............................ 101 44 Core for the Second Best Network Game (C(N) = $4,556,826)............................ 102 45 Reduction inkCore as c(N) Increases from c (N) to c (N) ............................. 107 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 EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES PROBLEMSA GAME THEORETIC APPROACH By Elliot Kin Ng August, 1985 Chairman: James P. Heaney Major Department: Environmental Engineering Sciences Successful regional water resources planning involves an efficiency analysis to find the optimal system that maxi mizes benefits minus costs, and an equity analysis to appor tion project costs. Traditionally, these two problems have been treated separately. This dissertation incorporates efficiency analysis and equity analysis into a single regional water resources planning model. A reliable total enumeration procedure is used to find the optimal system for regional water network problems. This procedure is easy to understand and can be implemented using readily available computer software. Furthermore, the engineer can use realistic cost functions or perform detailed cost analysis and, also, examine good suboptimal systems. In addition, this procedure finds the optimal system for each individual and each subgroup of individuals; hence, an equity analysis can be accomplished using the theory of the core from cooperative nperson game theory. Game theory concepts are used to perform an equity analysis on the optimal system as well as good suboptimal systems. For any system, an equitable cost allocation exists if a core exists. However, if a game is not properly defined, even a cost allocation in the core may be inequitable. A rigorous procedure using core conditions and linear programming is described to determine the core bounds. An individual's lower core bound and upper core bound unambigu ously measure the individual's minimum cost and maximum cost, respectively. Traditional approaches for quantifying minimum cost and maximum cost assume that either a regional system involving the grand coalition is built or all the individuals will goitalone. However, this rigorous procedure accounts for the possibility that a relatively attractive system involving subgroups may form. Furthermore, this rigorous procedure gives a general quantitative definition of marginal cost and opportunity cost. Once the minimum cost and maximum cost for each individual are determined, a basis for equitable cost allocation is available. Finally, efficiency analysis and equity analysis are not separable problems but are related by the economics of all the opportunities available to all individuals in a project. CHAPTER 1 INTRODUCTION In situations where multiple purposes and groups can take advantage of economies of scale in production and/or distribution costs, a regional water resources system is an attractive alternative to separate systems for each purpose and each group. However, a regional system imposes complex economic, financial, legal, sociopolitical, and organiza tional problems for the water resources professionals. This dissertation examines two problems associated with regional water resources planning that are typically treated separately, yet are closely related. The first problem involves performing an efficiency analysis to determine the economically efficient or optimal regional system that maximizes benefits minus costs. Once the optimal regional system is determined, a major task still remains to allocate project costs; therefore, an equity analysis must be performed to apportion project costs in an equitable manner. This second problem is viewed from the perspective of each purpose and each group because they must each be convinced that the optimal regional system is their best alternative; otherwise, voluntary participation will be difficult. No doubt, each purpose's and each group's decision to participate in the optimal regional system depends on its allocated cost, and not necessarily on what is best for the region. The prevailing belief is that efficiency analysis and equity analysis are separate problems and, therefore, research has either focused entirely on efficiency analysis or equity analysis. Research on efficiency analysis has mainly been on the application of partial enumeration tech niques to find optimal regional systems, while research on equity analysis has continued to explore the application of concepts from cooperative game theory to allocate project costs. The purpose of this dissertation is to integrate efficiency analysis and equity analysis into a single regional water resources planning model characterized by economies of scale. The model to be presented incorporates a total enumeration procedure along with concepts from cooperative game theory for efficiency/equity analysis. The specific application is to determine the least cost regional water supply network and to determine a "fair" allocation of costs among the multiple users. Chapter 2 reviews selected works on efficiency analysis and equity analysis of water resources problems. Chapter 3 presents a reliable total enumeration procedure for effi ciency analysis of regional water supply network problems. However, unlike traditional partial enumeration techniques used for efficiency analysis that give only the optimal solution, this procedure also gives all the optimal solu tions for each user and each subgroup of users which are necessary information to perform an equity analysis using concepts from cooperative game theory. In addition, this procedure gives all the suboptimal solutions. Chapter 4 shows how the information from the total enumeration procedure is used to perform an equity analysis of not only the optimal solution, but also "good" suboptimal solutions. Chapter 5 reveals how efficiency analysis and equity analysis are related. Finally, Chapter 6 summarizes the results and conclusions. CHAPTER 2 LITERATURE REVIEW Efficiency Analysis During the past two decades, the problem of finding the economically efficient or optimal regional water system has been extensively modeled as a mathematical optimization problem. A review of selected works on efficiency analysis of regional water systems that includes Converse (1972), Graves et al. (1972), McConagha and Converse (1973), Yao (1973), Joeres et al. (1974), Bishop et al. (1975), Jarvis et al. (1978), Whitlatch and ReVelle (1976), Brill and Nakamura (1978), and Phillips et al. (1982) indicates a variety of partial enumeration techniques, e.g., nonlinear programming, for finding optimal regional systems. These optimal regional systems can be a least cost system or a system that maximizes benefits minus costs. Generally, regional water resources planning problems exhibit economies of scale in cost and, therefore, involve nonlinear concave cost functions. Consequently, to a great extent, the selection of the partial enumeration optimization technique to apply to a particular problem depends on the characteri zation of the nonlinear concave cost functions. For instance, linear programming can be applied if the nonlinear 4 5 concave cost functions are represented by linear approximations. Equity Analysis Unfortunately, successful regional planning is not merely knowing the optimal regional system but must also include an equity analysis to find an acceptable allocation of costs among the participants. Otherwise, the optimal system will be difficult to implement. Of the publications cited in the preceding paragraph, only McConagha and Converse (1973) dealt with both efficiency and equity in regional water planning. In addition to presenting a heuristic procedure for finding the least cost regional wastewater treatment facility for seven cities, they evalu ated the equity of several cost allocation procedures. Although they recognized that an equitable cost allocation should not charge any city or subgroup of cities more than the cost of an individual treatment facility, they did not include the possibility of subgroup formation in their analysis. Giglio and Wrightington (1972) introduced concepts from cooperative game theory as a way to consider the possibility of subgroup formation in allocating costs of water projects. However, their treatment of cooperative game theory was incomplete. Therefore, they concluded that the game theory approach rarely yields a unique cost allocation and proceeded to recommend the separable costs, remaining benefits (SCRB) method or methods based on measure of pollu tion. Shortly thereafter, several researchers applied popular unique solution concepts from game theory like the Shapley value and the nucleolus to allocate the costs of regional water systems. Heaney et al. (1975) applied the Shapley value to find an equitable cost allocation of common storage units for storm drainage for pollution control among competing users. Suzuki and Nakayama (1976) applied the nucleolus to assign costs for a water resources development along Japan's Sakawa and Sagami Rivers. Loehman et al. (1979) used a generalization of the Shapley value to allocate the costs of a regional wastewater system involving eight dischargers along the lower Meramec River near St. Louis, Missouri. Subsequently, Heaney (1979) established that the fair ness criteria used for allocating costs in the water resources field and the concepts used in cooperative game theory are equivalent. Moreover, Straffin and Heaney (1981) showed that a conventional method for allocating costs used by water resources engineers is identical to a unique solu tion concept used by game theorists. More recently, Young et al. (1982) compared proportionality methods, game theoretic methods, and the SCRB method for allocating cost and concluded that the game theoretic methods may be too complicated while the SCRB method may give inequitable cost allocations. Meanwhile, Heaney and Dickinson (1982) revealed why the SCRB method may fail to give equitable cost allocations and proposed a modification of the SCRB method that uses game theory concepts along with linear programming to insure an equitable cost allocation can be found if one exists. The possibilities of using concepts from cooperative game theory as a basis for allocating costs of water projects continue to develop. In fact, concepts from coop erative game theory are gaining acceptance in other fields as well. Researchers in accounting are looking toward coop erative game theory as a possible solution to the arguments by Thomas (1969, 1974) that any cost allocation scheme in accounting is arbitrary and hence not fully defensible. Recent works by Jensen (1977), Hamlen et al. (1977, 1980), Callen (1978), and Balachandran and Ramakrishnan (1981) applied concepts from cooperative game theory to evaluate the equity of existing and proposed cost allocation schemes in accounting. Meanwhile, in economics, concepts from cooperative game theory are frequently used as a basis for evaluating subsidyfree and sustainable pricing policies for decreasing cost industries, e.g., the work of Loehman and Whinston (1971, 1974), Faulhaber (1975), Sorenson et al. (1976, 1978), Zajac (1978), Panzar and Willig (1977), Faulhaber and Levinson (1981), and Sharkey (1982b). Conclusions Three conclusions can be made from reviewing the literature on efficiency analysis and equity analysis of regional water resources planning. First, there is a gap in the research to jointly examine efficiency and equity in regional water resources planning. In spite of a continual effort to find economically efficient regional water systems and equitable cost allocation procedures, no published work incorporates both efficiency analysis and equity analysis in a single regional water resources planning model using realistic cost functions. Heaney et al. (1975) and Suzuki and Nakayama (1976) used linear cost models while Loehman et al. (1979) used conventional cost curves. Secondly, the cost allocation literature in the water resources field has consistently allocated the costs of treatment and piping together even though federal guidelines suggest that piping cost be allocated separately from treatment cost to the responsible users (Loehman et al., 1979; U.S. Environmental Protection Agency, 1976). Finally, the cost allocation literature has dealt with allocating the cost of the optimal system. However, situations in practice may require that "good" suboptimal systems be considered; therefore, an acceptable cost allocation procedure should be able to allocate the costs of several systems under consideration in an equitable manner. These three conclusions formed the basis for the research undertaken in this dissertation. 9 Chapter 3 begins integrating efficiency analysis and equity analysis by searching for a computational procedure to simultaneously perform an efficiency analysis and calculate all the necessary information to perform an equity analysis using concepts from cooperative game theory. CHAPTER 3 EFFICIENCY ANALYSIS Introduction The importance of both efficiency analysis and equity analysis in planning regional water resources systems is well recognized. Over the years, researchers have applied methods ranging from simple costbenefit analysis to sophis ticated mathematical programming techniques to search for economically efficient or optimal regional water resources systems. Yet, the implementation of regional systems is difficult unless an equitable financial arrangement is found to allocate project costs among individuals (or partici pants) in a project. Until recently, a theoretically sound basis for allocating costs has eluded the water resources professional. However, there is increasing interest in using the theory of the core from cooperative nperson game theory as a basis for allocating costs, e.g., see Suzuki and Nakayama (1976), Bogardi and Szidarovsky (1976), Loehman et al. (1979), Heaney and Dickinson (1982), and Young et al. (1982). The theory of the core is based on principles of individual, subgroup, and group rationality. This means that no individual or subgroup of individuals should be allocated a cost in excess of the cost of nonparticipation, 10 while total cost must be apportioned among all individuals. The cost of nonparticipation is simply the cost that each individual and each subgroup of individuals must pay to independently acquire the same level of service by the most economically efficient means. As a result, to evaluate efficiency/equity for a regional system with n individuals, it is necessary to determine 2 1 optimal solutions. Although the close association between efficiency analysis and equity analysis is recognized, there have been few attempts to incorporate these two analyses in regional water resources planning. A typical efficiency analysis usually ends with determining the optimal solution for a problem without addressing cost allocation, and a typical equity analysis begins by assuming the 2n1 optimal solu tions are available to accomplish the cost allocation. This disjointed approach to efficiency/equity analysis is fostered by a belief that these two problems are independent (James and Lee, 1971; Loughlin, 1977). Furthermore, reliable techniques for finding the 2n1 optimal solutions to accomplish an efficiency/equity analysis of most problems encountered in actual practice are unavailable. This chapter begins by evaluating the applicability of partial and total enumeration techniques for finding the 2n1 optimal solutions for problems with different types of cost functions. Subsequently, a computational procedure is described to examine a regional water supply network problem wherein we need to find the economic optimum and a "fair" allocation of costs among the individuals in the project. In order to do the cost allocation we need to find the costs of the optimal systems for each individual and each subgroup of individuals since these costs are going to be the basis for cost allocation. Partial Enumeration Techniques The difficulty of finding the optimal solution for a particular problem depends on the nature of the cost func tions. Generally, a cost function can be classified as either linear, convex, concave, Sshape, or irregular (see Figure 31). To find the optimal solution for problems with either linear or convex cost functions is straightforward using readily available and reliable linear programming codes. Accordingly, a vast body of overlapping theoretical results is available from classical economics and operations research, e.g., convex programming, for finding the optimal solution to problems with convex cost functions. However, problems with linear and convex cost functions are unable to characterize the economies of scale in cost typically encountered in regional water resources planning. The concave cost function is generally used to represent economies of scale, and several partial enumera tion techniques are available for dealing with this cost $ a) Linear $ c) Concave $ b) Convex SSShape d) SShape e) Irregular Figure 31. Types of Cost Functions. function. One approach surveyed by Mandl (1981) is separable programming which takes advantage of readily available linear programming codes by using a piecewise linear approximation of the concave cost function. Unfortunately, this approach is rather tedious to use and guarantees only a local optimal solution. A second approach is to retain the natural concave cost function and apply a general nonlinear programming code. However, according to surveys by Waren and Lasdon (1979) and Hock and Schittkowski (1983), general nonlinear programming codes may converge to local optima and may be subject to other failures, e.g., termination of code. A final approach used by Joeres et al. (1974) and Jarvis et al. (1978) is to approximate the concave cost function with several fixedcharge cost functions and apply a mixedinteger programming code. This approach guarantees a globally optimal solution, but standard mixedinteger programming codes are expensive to use. More importantly, unresolved problems remain as to how to properly define a fixed charge problem. If the fixed charge formulation is used because it is computationally expedient, then the resulting cost estimates may distort the cost allocation procedure. Given the current status of partial enumeration techniques for finding the optimal solutions to perform efficiency/equity analysis for problems with concave cost functions, one can conclude that other methods must be used. Obviously, this conclusion applies to problems with Sshape and irregular cost functions as well. Total Enumeration Techniques Total enumeration techniques can be used to find the optimal solution for a problem regardless of the types of cost functions involved. The ability to handle irregular cost functions is especially important because this type of cost function is frequently used by stateoftheart cost estimating models like CAPDET, i.e., Computer Assisted Procedure for Design and Evaluation of Wastewater Treatment Systems (U. S. Army Corps of Engineers, 1978) and MAPS, i.e., Methodology for Areawide Planning Studies (U. S. Army Corps of Engineers, 1980). For example, in MAPS, the cost function for constructing a force main is composed of separate cost functions for pipes, excavation, appurten ances, and terrain. Furthermore, each of these cost func tions is based on sitespecific conditions. For instance, the cost function for pipe includes the cost of purchasing, hauling, and laying the pipe and depends on the material, diameter, length, and maximum pressure. No doubt, the composite sitespecific cost function for a force main may be nonlinear, nonconvex, multimodel, and discontinuous. Another advantage with a total enumeration technique is that it presents and ranks all of the alternative solu tions. Unlike partial enumeration techniques which only present the optimal solution for consideration, total enumeration techniques allow examination of suboptimal solutions which may be preferable when factors other than cost are considered. For example, proven engineering design or sociopolitical values are difficult to incorporate into an optimization model even if the problem is well defined, so the optimal solution may be so unrealistic that another solution must be selected. Depending on the size of the problem, a possible drawback with total enumeration techniques may be the compu tational effort to enumerate all possible solutions. However, for some problems, total enumeration may be the only meaningful approach. For these problems, the challenge with using a total enumeration approach is to find ways to reduce the computational effort by applying mathematical techniques or engineering considerations. After a discus sion on modeling network problems as digraphs, a total enumeration procedure that does not require extensive compu tational effort to find the least cost network for each individual and each group of individuals is presented. Modeling Network Problems as Digraphs Consider a situation wherein an existing water supply source, S, is going to serve n users with demands of Q1, Q2' S. Qn, respectively. Assume that the water source is able to supply the total demand by the n users without facility expansion except for a new regional water network. Furthermore, consider a particular system with three users that can be served directly by the source, and engineering considerations, e.g., gravity flow, have determined that it is feasible to send water from user 1 to both user 2 and user 3, and from user 2 to user 3. For this particular system, assume the total cost function for constructing a pipeline is rather simple. From Sample (1983), the total cost function for constructing a pipeline is characterized by economies of scale and can be expressed as a linear function of distance and a nonlinear function of flow; or C = aQbL (31) where C = total cost of pipeline, dollars Q = quantity of flow, mgd L = length of pipeline, feet, and a, b = parameters, 0 Given this situation, the objective of the regional water authority is to determine the least cost water network for each user and each group of users in order to perform efficiency/equity analysis. This problem can be modeled as a digraph or directed graph (see Figure 32) consisting of nodes to represent the source and users, and directed arcs to represent all Figure 32. Example Digraph Representing a Regional Water Network Problem for Three Users. 19 possible interconnecting pipelines. If water can be sent in either direction between two users, then the pipeline is represented by two oppositely directed arcs. Consequently, any regional water network problem can be modeled by a digraph. Before continuing, a few brief definitions and concepts are necessary since the nomenclature used in the network and graph theory literature is not standardized. A digraph or directed graph, D(X,A), consists of a finite set of nodes, X, and a finite set of directed arcs, A. A directed arc is denoted by (i,j) where the direction of the arc (shown by an arrow) is from node i to node j; node i is called the initial node and node j is called the terminal node. A subdigraph of D(X,A) has a set of nodes that is a subset of X but contains all the arcs whose initial and terminal nodes are both within this subset. A path from node i to node j is simply a sequence of directed arcs from node i to node j. An elementary path is a path that does not use the same node more than once. A circuit is an elementary path with the same initial and terminal node. A directed tree or an arborescence is a digraph without a circuit for which every node, except the node called the root, has one arc directed into it while the root node has no arc directed into it. A spanning directed tree of a digraph is a directed tree that includes every node in the digraph. If a cost, C(i,j) is associated with every arc (i,j) of a digraph, then the cost of a directed tree is defined as the sum of the costs of the arcs in the directed tree. Finally, a minimum spanning directed tree of a digraph is the spanning directed tree of the digraph with the least cost. For the reader desiring more information regarding networks and graphs, numerous texts are available, e.g., Christofides (1975), Minieka (1978), and Robinson and Foulds (1980). The problem of finding the least cost water network for each user and each group of users is the same as finding the minimum spanning directed tree rooted at node S for all possible subdigraphs as well as the digraph shown in Figure 32. In general, not every digraph has a spanning directed tree; however, for a realistic problem one can assume a pipeline is available to serve all individuals participating in a regional system. Thus, a spanning directed tree exists for digraphs representing realistic regional water network problems. Although algorithms are found in Gabow (1977) and Camerini et al. (1980a, 1980b) for finding the minimum spanning directed tree or the K best spanning directed trees, these algorithms assume a linear cost model in which the cost on each arc is given prior to initiating the algorithm. As a result, these algorithms are not applicable to problems with nonlinear costs on each arc. That is, the cost along each arc cannot be determined in advance 21 because the cost is a function of the quantity of flow along the arc; yet, the quantity of flow along the arc is a function of the path in which the arc belongs. The Total Enumeration Procedure The procedure for enumerating and calculating the costs of all the spanning directed trees for all possible sub digraphs as well as the digraph is based on recognizing that a large number of spanning directed trees of a digraph can be constructed from specific spanning directed trees of subdigraphs. These specific spanning directed trees are characterized by one arc emanating from the root node and are referred to as "essential spanning directed trees." In contrast, "inessential spanning directed trees" are charac terized by more than one arc emanating from the root node. The procedure sequentially calculates the costs of essential spanning directed trees for subdigraphs with increasing number of nodes, until the costs of essential spanning directed trees are calculated for all possible subdigraphs and for the digraph. Meanwhile, the cost of each inessen tial spanning directed tree for all possible subdigraphs as well as the digraph is calculated simply by summing the costs of essential spanning directed trees of subdigraphs that are associated with each arc emanating from the root node of the inessential spanning directed tree. That is, each arc emanating from the root node belongs to an 22 essential spanning directed tree of a subdigraph. By apply ing this procedure the costs of all the spanning directed trees can be systematically enumerated for all possible subdigraphs as well as the costs of all the spanning directed trees for the digraph. As a result, the least cost network for each user and each group of users is found. In the following discussion, "nnode" means the number of nodes, not including the root node, is n; e.g., an inode digraph or subdigraph consists of i+l nodes if the root node is counted. The total enumeration procedure for the nnode digraph is summarized by the flow diagram shown in Figure 33. Step 1 begins the procedure for evaluating all subdi graphs consisting of the root node and one other node, i.e., the 1node subdigraphs. Step 2 initializes a count of the number of combina tions of inode subdigraphs evaluated. Step 3 generates all possible combinations of inode subdigraphs from the nnode digraph. The number of possible combinations is ( ). For example, the 3node digraph shown in Figure 32 has (2) or three possible 2node subdigraphs, i.e., subdigraphs consisting of the following sets of nodes {S,1,2), {S,1,3}, and (S,2,3}. Step 4 selects one inode subdigraph not previously selected and enumerates all of its spanning directed trees. A spanning directed tree may not exist in a case where a path does not exist from the root node to every node in the btep 4. Select one inode subdigraph not previously selected and enumerate all spanning directed trees Step 5. Calculate the cost of each spanning directed tree Step 6. Rank all spanning directed trees Yes Figure 33. Flow Diagram of Total Enumeration Procedure for nNode Digraph inode subdigraph, i.e., not every node in the inode subdigraph has an arc directed into it. Actually, only the essential spanning directed trees need to be enumerated. The enumeration of inessential spanning directed trees is simply done by finding all possible combinations of inode digraphs from the entire set of essential spanning directed trees enumerated previously, i.e., all essential spanning directed trees for all possible subdigraphs of the inode subdigraph. This process substantially reduces the effort involved in enumerating all the spanning directed trees for an inode subdigraph because a large number of spanning directed trees are inessential. If the inode subdigraph is unusually large and dense, algorithms are available in Chen and Li (1973), Christofides (1975), and Minieka (1978) for generating spanning directed trees. If necessary, a procedure in Chen (1976) can be used to compute the number of spanning directed trees of an inode subdigraph or an nnode digraph. A directed tree matrix, M, is defined for a digraph, where m.. equals the number of arcs directed into node i and m.. is equal to the negative of the number of arcs in parallel from node i to node j. The number of spanning directed trees rooted at node S for the digraph defined by M is given by the determinant of the minor submatrix resulting from deleting the Sth row and 25 column of M. Applying this procedure to the 3node digraph in Figure 32 gives the following directed tree matrix. S 1 2 3 S 0 1 1 1 1 0 1 1 1 2 0 0 2 1 3 0 0 0 3 The determinant of the minor submatrix resulting from delet ing the Sth row and column is six, so there are six spanning directed trees rooted at node S for this digraph. Step 5 calculates the cost of each spanning directed tree enumerated in Step 4. The cost for each essential spanning directed tree is calculated independently. How ever, the cost for each inessential spanning directed tree is simply calculated by summing the costs of essential spanning directed trees of subdigraphs calculated previously that are associated with the arcs emanating from the root node. For inessential spanning directed trees the costs can be calculated along with the enumeration process described in Step 4. Step 6 ranks all the spanning directed trees for the inode subdigraph according to cost. The minimum spanning directed tree is the least cost network for the users associated with the set of nodes for the inode subdigraph. Step 7 checks the counter to see if all possible combinations of inode subdigraphs have been evaluated. If not, Step 8 advances the counter by one before returning to Step 4 to evaluate another inode subdigraph. If all of the possible combinations of inode subdigraphs have been evaluated, the procedure goes to Step 9 and begins the evaluation of subdigraphs with i+1 nodes. Step 10 checks if the nnode digraph has been evalu ated. If not, the procedure returns to Step 2 and proceeds to evaluate the subdigraphs with i+l nodes; otherwise, the procedure terminates. The total enumeration procedure is illustrated in Table 31 using the regional water network problem modeled by the 3node digraph shown in Figure 32. During the first iteration all combinations of 1node subdigraphs are evaluated. For this simple case three combinations, i.e., (i) = 3, are evaluated. Further more, each combination has only one spanning directed tree, and the one spanning directed tree is essential. As a result, the cost of the spanning directed tree for each combination must be calculated. Obviously each spanning directed tree is the least cost network for the associated user. During the second iteration, three combinations, 3 i.e., (2) = 3, of 2node subdigraphs are evaluated. In this case, each combination has two spanning directed trees, but the cost of only one spanning directed tree needs Table 31. Example of Total Enumeration Procedure for 3Node Digraph Spanning Directed Are Spanning Iteration iNode Trees for iNode Directed Trees i Subdigraphs Subdigraph Essential? i=1 {S, 11} ) Yes {S, 21} ( Yes {S, 31} ) Yes i=2 {S,l,2} 2 S 1 {S,l,3} Yes No Yes No Yes No 3 S 1 {S,2,3} 3 S 2 s 1 2 0~0~0 S 1 3 Table 3.1. Continued. Spanning Directed Are Spanning Iteration iNode Trees for iNode Directed Trees i Subdigraphs Subdigraph Essential? {S,1,2,3} Yes Yes No No No No 3 S 2 1 to be calculated. The cost of the inessential spanning directed tree is simply found by summing the costs of the corresponding essential spanning directed trees calculated during the first iteration. The minimum spanning directed tree for each combination is the least cost network for the associated group of users. Finally, for the third itera tion, i.e., i=n, the 3node digraph is being evaluated. This 3node digraph has six spanning directed trees, and these six spanning directed trees can be enumerated by inspection. The four inessential spanning directed trees can be enumerated by simply finding all possible combinations of 3node digraphs from the essential spanning directed trees generated during the first and second iterations. Thus, only two independent calculations are necessary to find the costs of the essential spanning directed trees. Mean while, the cost of the four inessential spanning directed trees is calculated simply by summing the costs of essential spanning directed trees for subdigraphs previously calcu lated during the first two iterations. For example, in Table 31, the cost for the inessential spanning directed tree consisting of the set of arcs {(S,3), (S,l), (1,2)} is determined by summing the costs of the two essential spanning directed trees consisting of the sets of arcs {(S,3)} and {(S,1), (1,2)} associated with the two sub digraphs consisting of the sets of nodes {S,3} and {S,1,2}, respectively. Therefore, eight independent calculations are necessary to find the costs of the six spanning directed trees for the digraph, and only two of the six spanning directed trees are essential. In fact, the eight indepen dent calculations enable us to find all 2 1 or seven optimal solutions necessary to perform efficiency/ equity analysis. Table 32 shows that the number of independent calculations necessary to find the cost of all the spanning directed trees for all possible subdigraphs is simply equal to the number of independent calculations to find the cost of all the spanning directed trees for the digraph less the number of essential spanning directed trees for the digraph. Consequently, for our 3node digraph, six independent calculations are necessary to find the optimal solution for each user and each subgroup of users. For the balance of this chapter, the optimal solution for each user and each subgroup of users will be referred to as the 2n2 optimal solutions. Finally, all suboptimal solutions are enumerated for all possible subdigraphs as well as for the digraph. Computational Considerations Although the number of independent calculations neces sary to find the costs of all the spanning directed trees for all possible subdigraphs as well as the digraph is uniquely determined by the configuration of the digraph, we can get a sense of the computational effort by examining the Table 32. The Number of Independent Calculations to Find the Costs of Spanning Directed Trees for All Possible Subdigraphs. Is Independent Calcu Is Independent Calcu lation Used to Find lation Used to Find the Costs of Spanning the Costs of Spanning Independent Directed Trees for the Directed Trees for All Calculation Digraph? Possible Subdigraphs? Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Total Number of Yes 32 three digraphs shown in Figure 34. For the 3node digraph, six independent calculations are necessary to find the costs of the four spanning directed trees for the digraph, and only one of the four spanning directed trees is essential. More importantly, 12 calculations are necessary to find the seven optimal solutions, but only 6 of the 12 calculations (50%) are independent. Furthermore, only five independent calculations are necessary to find the 2n2 optimal solu tions. For the 4node digraph, 10 independent calculations are necessary to find the cost of the eight spanning directed trees for the digraph, and only one of the eight spanning directed trees is essential. For this digraph, 33 calculations are necessary to find the 15 optimal solutions, but only 10 of the 33 calculations (30%) are independent. Moreover, only nine independent calculations are necessary to find the 2n2 optimal solutions. Finally, for the 5node digraph, 19 independent calculations are necessary to find the costs of the 24 spanning directed trees for the digraph, but only 2 of the 24 spanning directed trees are essential. In this case, 109 calculations are necessary to find the 31 optimal solutions, but only 19 of the 109 calculations (17%) are independent. From these 19 independent calculations, only 17 are necessary to find the 2n2 optimal solutions. As we can see, summarized in Table 33, a large number of the spanning directed trees of a digraph are inessential. Figure 34. Examples of 3,4,5Node Digraphs. en ri co E 0) 4 H 0 ZNH 4 0) 0U (N 3 H U 1) N E 4IJ () H 'O 4) 04)0 0)0 r C r HU C .I O O P 0Z ,0 1. O 4Ho S0 i H 41 ,I H O C14 0 O 4J OC O NI 0 Cd 0U *rl C C & *rl a m c: ic P wl n, 0) 4)J O01 4 0) OC 0) 41 H O Q4 En 03 ul o\ I 4 ,I N CN r LO ri r4 re Also, the percentage of independent calculations decreases as the number of nodes for a digraph increases. The 5node digraph in Figure 34 shows that the actual number of independent calculations necessary to determine the 31 optimal solutions to perform efficiency/equity analy sis of a regional water network problem involving five users is rather small. In fact, a regional water network serving five users may be considered a fairly large network. As larger systems form, increases in transactions costs because of multiple political jurisdictions, growing administrative complexity, etc., may eventually offset the gains from a regional system. In any case, real regional water network problems probably involve fairly small and sparse networks. That is, large networks can usually be broken down into smaller networks for analysis based on natural geographical and hydrological features, political boundaries, etc. Also, in actual problems there may not be that many choices for routing pipelines. Thus, the number of independent calculations necessary to calculate the 2n1 optimal solutions for a realistic regional water network should not be unreasonable. One of the advantages of using this total enumeration procedure is that it can be accomplished on a personal computer using readily available software. Thus, decision makers involved with planning and negotiating a regional water network can have easy access to information to aid the decisionmaking process. For instance, the procedure can be implemented using the extremely "user friendly" Lotus 123 spreadsheet software package. Lotus 123 has the mathematical functions to handle calculations involving nonlinear cost functions or involving detailed cost analysis. A sample Lotus 123 printout is shown in Table 34 for a hypothetical water network problem modeled by the 3node digraph shown in Figure 32. This printout should be selfexplanatory. The top portion of the printout contains the data for the problem, and the bottom portion is the calculations associated with the total enumeration pro cedure. The sorting capabilities of Lotus 123 allow automatic ranking of all the feasible solutions according to cost. Moreover, the Lotus 123 electronic spreadsheet automatically recalculates all values associated with a formula whenever a new value is entered or an existing value is changed. This automatically gives the total enumeration procedure the capability for sensitivity analysis. For example, the set of all feasible solutions ranked according to cost can be evaluated as the economies of scale, as represented by the value of b in equation (31), is varied over a specific range of values. Thus, for a regional network problem of realistic size, all the feasible solutions can be enumerated using a spreadsheet software package. Table 34. Efficiency Analysis of a ThreeUser Water Supply Network with Nonlinear Cost Function Using Lotus 123. Data Distance : L(i,j) is the distance in feet from i to j L(S,1)= 17000 L(S,3)= 3025k L(1,3)= 19670 L(S,2)= 25(00 L(1,2)= 13100 L(2,3)= 15500 Demand : Q(i) is the demand in mgd for user i Q(1)= 1 Q(2)= 6 Q(3)= 3 Cost Function: a(Q^b)L a= 38 b= 0.51 Calculations With Total Enumeration Procedure C(i..j)[x]= Cost of network [x] for i..j ; C(i..j)= Least cost for i..j C(1) [Sl]= 646000 C(2) [S2]=2463843. C(3)[S3]=2312936. C(12)[Sl,12]= 2984140. C(12)= 2934140. C(12)[S1;S2]= 3109848. C(13)[S1,13]= 2618975. C(13)= 2613975. C(13)[S1;S3]= 2658986. C(23)[S2,23]= 4061294. C(23)= 4061294. C(23)[32;S3]= 4476835. C(123)[S1,12,23]= 4548439. C(123)[S1,12;S3]= 4997126. C(123)[S1,12,13]= 4640756. C(123)= 4647756. C(123)[S1,13;S2]= 52324. C(123)[S1;S2,23]= 4707294. C(123)[S1;S2;S3]= 5122835. Sort C(123) in ascending order Paths Cost C(123)[S1,12,13]= 4640756. C(123)[S1,12,23]= 4648439. 3E3T C(123)[S1;S2,23]= 4707294. C(123)= 4640756 C(123)[S1,12;S3]= 4997126. C(123)[S1,13;S2]= 5032824. C(123) [S1;S2;S3]= 5122835. Summary A total enumeration procedure for finding the optimal solutions necessary for efficiency/equity analysis of realistic regional water network problems is presented. The procedure can be easily understood and applied by engineers with little knowledge or experience in operations research techniques. Furthermore, the procedure allows the engineers to handle all problems regardless of the types of cost function involved or to perform detailed cost analysis. Finally, if the optimal solution is impractical for implementation, all suboptimal solutions ranked according to cost are readily available for consideration. CHAPTER 4 EQUITY ANALYSIS Introduction Proposed regional water resources systems involve multiple purposes and groups who must somehow share the cost of the entire project. The project may focus on construc tion of a large dam which serves numerous purposes such as water supply, flood control, and recreation. Also, canals from the dam direct the water to nearby users. A signifi cant portion of the total cost of this project may involve elements which serve more than one purpose and/or group. These costs are referred to as joint or common costs. In such cases, it is possible to find the optimal or the most economically efficient regional system, i.e., the one that maximizes benefits minus costs. However, a major effort remains to somehow apportion the project cost in an equitable manner. In fact, the importance of the financial analysis to apportion project cost is not limited to the optimal system but includes any other integrated systems being considered for implementation as well. This chapter examines principles of cost allocation using concepts from cooperative nperson game theory. An example regional water network is used to illustrate these principles. Cost Allocation for Regional Water Networks A hypothetical situation similar to options contained in the West Coast Regional Water Supply Authority's master plan for Hillsborough, Pasco, and Pinellas counties in Florida (Ross et al., 1978) is now considered. Phase I (19801985) of the plan recommends the use of groundwater from existing and newly developed well fields to satisfy water demands in the tricounty area. For this hypothetical problem, assume that an existing well field is the most high quality and cost effective water supply source (S) available for three counties (1, 2, and 3) with projected demands of 1, 6, and 3 million gallons per day (mgd), respectively. The demand for each county is based on projected population growth and average per capital demand over a period of 5 years (see Table 41). Assume that the existing well field is currently operating below its capacity of 20 mgd and can satisfy the additional 10 mgd demanded by the three counties. In addition, assume that no facility expansion is required except for a new regional water network. Further more, each county can be served directly by the well field, and engineering considerations, e.g., gravity flow, have determined that water can be sent from county 1 to both county 2 and county 3, and from county 2 to county 3. The Table 41. Projected Population Growth and Projected Average Per Capita Demand. Projected Projected Average Projected County Population Growth Per Capita Demand Additional (gal/capday) Demand (mgd) 1 8,000 125 1 2 40,000 150 6 3 18,750 160 3 Total 66,750  10 Weighted Average  150  42 lengths of all possible interconnecting pipelines are shown in Figure 41. For our hypothetical problem, assume that the total cost of constructing a pipeline has strong economies of scale and is C = 38Q'5L, where C is total cost of pipeline in dollars, Q is quantity of flow in mgd, and L is the length of pipeline in feet. Given the problem just described, the cost of a pipe line serving county 1 alone is $646,000; the cost of a pipeline serving county 2 alone is $2,420,095; and the cost of a pipeline serving county 3 alone is $1,990,992. The total cost for three individual pipelines is $5,057,087. However, when the costs for all the options available to these three counties are enumerated using the procedure outlined in the preceding chapter, we see that the counties can do better by cooperating (see calculations in Appendix A using Lotus 123). There may be a slight difference between the numbers used in the text and the numbers in Appendix A because of rounding off. Also, cost data are only significant to the nearest thousand dollars. If the three counties cooperate, they can construct the least cost or optimal network consisting of pipelines from the well field to county 1, from county 1 to county 2, and from county 2 to county 3 (see Table 42). This optimal network costs $4,556,409 and represents a savings of 9.9% or $500,678 when compared with the cost for three individual pipelines. Obviously, constructing the optimal network is 3 mgd 19,670 ft 1 mgd Figure 41. Lengths of Interconnecting Pipelines. Table 42. The Costs and Percent Savings for All Options. Option Cost ($) Savings (%) (Rank) 1 4,556,409 9.90 2 4,556,826 9.89 3 4,630,177 8.44 4 4,919,503 2.72 5 5,006,734 1.00 6 5,057,087 0 (1) (2) (3) (4) 45 in the best interest of the three counties, but to implement this least cost network, an equitable way to allocate the cost among the three counties must be found. This financial problem is known as a cost allocation problem. The complex ity is introduced because the counties share common pipes. Criteria for Selecting a Cost Allocation Method Several sets of criteria for selecting a cost alloca tion method are found in the literature. For the water resources field, criteria for allocating costs date back to the Tennessee Valley Authority (TVA) project in 1935 when prominent authorities were brought together to address the cost allocation problem. They developed the following set of criteria for allocating costs (Ransmeier, 1942, pp. 220221): 1. The method should have a reasonable logical basis. It should not result in charging any objective with a greater investment than the fair capitalized value of the annual benefit of this objective to the consumer. It should not result in charging any objective with a greater invest ment than would suffice for its development at an alternate single purpose site. Finally, it should not charge any two or more objectives with a greater investment than would suffice for alternate dual purpose or multiple purpose improvement. 2. The method should not be unduly complex. 3. The method should be workable. 4. The method should be flexible. 5. The method should apportion to all purposes present at a multiple purpose enterprise a share in the overall economy of the operation. 46 This set of criteria developed for the water resources field is similar to the following set of criteria proposed by Claus and Kleitman (1973) for allocating the cost of a network: 1. The method must be easy to use and under standable to users. They must be able to predict the effects of changes in their service demands. 2. The method must have stability against system breakup. It should not be an advantage to one or more users to secede from the system. Thus, there are limits to which a method can subsidize one user or class of user at the expense of others. 3. It is desirable, though not necessary, that the costing be stable under evolutionary changes in the system or under mergers of users. 4. It is again desirable that the method should preserve the substance and appearance of non discrimination among users. 5. If the method represents a change from present usage it is desirable that transition to the new method be easy. From these two sets of criteria, the most important criterion for selecting a method to allocate the cost of a regional water network is the method's ability to ensure stability or prevent breakup of the network. That is, the method should not allocate cost in a manner whereby an individual or a subgroup of individuals can acquire the same level of service by a less expensive alternative. Other wise, the individual or subgroup of individuals will con sider their allocated cost inequitable or unfair and secede from the regional network for a less expensive alternative. Heaney (1979) has expressed these fairness criteria for an equitable cost allocation mathematically as follows: 1) x(i) < minimum [b(i), c(i)] V iEN (41) where x(i) = cost allocated to individual i, b(i) = benefit of individual i, c(i) = the alternative cost to individual i of independent action, and N = set of all individuals; i.e., N = {l,2,...,n}. This criterion simply means that individual i should not be charged a cost greater than the minimum of individual i's benefit and alternative cost for independent action. 2) E x(i) < minimum [b(S), c(S)] V ScN (42) iES where c(S) = alternative cost to subgroup S of independent action, and b(S) = benefit of subgroup S. This second criterion extends the first criterion to include subgroup of individuals as well. These two fairness criteria are now used to evaluate some simple and seemingly fair cost allocation schemes for our regional water network problem. Throughout this chapter, we will assume for our regional water network problem that each county's and each 48 subgroup of counties' alternative cost of independent action is less than or equal to each county's and each subgroup of counties' benefits, respectively; i.e., c(i) = minimum [b(i), c(i)] V ieN, and (43) c(S) = minimum [b(S), c(S)] V ScN. Ad Hoc Methods Over the years, many ad hoc methods have been proposed or used to apportion the costs of water resources projects (Goodman, 1984). In general, ad hoc methods used in the water resources field for allocating costs can be described as follows: allocate certain costs that are considered identifiable to an individual directly and prorate the remaining costs, i.e., total project cost less the sum of all identifiable costs, among all the individuals in the project by some physical or nonphysical criterion. Mathe matically, this can be expressed as follows: x(i) = x(i)id + 4(i)rc (44) where x(i) = cost allocated to individual i, x(i) = costs identifiable to individual i, 0(i) = prorating factor for individual i, and rc = remaining costs, i.e., c(N) Z x(i)d. ieN Furthermore, the requirement that Z p(i) = 1.0 should be iN obvious. James and Lee (1971) summarize 18 ways for allocating the costs of water projects depending on the definition of identifiable costs and the basis for prorating the remaining costs (see Table 43). Basically, the differences among these 18 methods are the following three ways of defining identifiable costs: 1) zero, 2) direct or assignable costs, or 3) separable costs; and the following six ways of prorating remaining costs: 1) equal, 2) unit of use, 3) priority of use, 4) net benefit, 5) alternative cost, or 6) the smaller of net benefit or alternative cost. The next two sections analyze the effects of defining identifiable costs as either zero or direct costs. A detailed treatment of separable costs, i.e., the difference between total project costs with and without an individual, is given in the section on the separable costs, remaining benefits method. Defining Identifiable Costs as Zero The simplest way to allocate costs is to define identi fiable costs as equal to zero and prorate total project cost by some physical or nonphysical criterion. For example, population and demand are two ways to prorate total project Table 43. Cost Allocation Matrix. Definition of Identifiable Cost Basis for Prorating A. B. C. Remaining Costs Zero Direct Separable Cost Cost a. Equal Aa Ba Ca b. Unit of Use Ab Bb Cb c. Priority of Use Ac Bc Cc d. Net Benefit Ad Bd Cd e. Alternative Cost Ae Be Ce f. Smaller of d. or e. Af Bf Cf Source: Modified from James and Lee, 1971, p. 533. 51 cost (Young et al., 1982). Using these two ways to prorate the cost of the optimal network for our regional water network problem gives the following cost allocations (see calculations in Table 44 and Table 45): Proportional to Population County 1 $ 546,769 County 2 2,733,845 County 3 1,275,795 $4,556,409 Proportional to Demand County 1 $ 455,641 County 2 2,733,845 County 3 1,366,923 $4,556,409 Although these cost allocations are simple to calculate and easy to understand, they fail to implement the optimal network because county 2 considers these cost allocations unfair. In contrast to counties 1 and 3, county 2 loses money by being allocated a cost in excess of its goitalone costs using either of these two methods. Consequently, county 2 would rather acquire a pipeline by itself than cooperate with counties 1 and 3 to construct the optimal network. The principal failure with these proportionality .I U M V H 0 0) c  .I HO I0 O U I O .4 ,I m x O 0 0 Oc 4J O r0 0) 0 0 U * O I rI 0 .r, 4a 0 u1 +J n3 0) 0 ( o Ln o a> o o SL Co mo CN 0 00 rI %D CV 0 0 0 mVI 00 00 r  0 O LO m 0 Ln o Ln E4 r E^ *H 11 C, Ov H I ,. 0 O a0 4J i)  O 4  0r 0Q m a) ci0 i P4 0 01 0 U o m ci 0 0) > z > o n CM o o a 0 C) m CM rr 11 in n < co m 0 0 C0 1r rQ fn ^ * CM i Ii CM n I I I 0 L o Ln, o to rd 0 'i 4J 0 methods is that they do not recognize explicitly each individual's contribution to total project cost. Defining Identifiable Costs as Direct Costs A way to recognize each individual's contribution to total project cost is by defining identifiable costs as those costs that can be directly assigned, and prorating the remaining costs by some physical or nonphysical criterion such as use or number of individuals; i.e., x(i) = x(i)direct + p(i)rc (45) where (i)direct = direct cost or assignable cost to individual i. Although this direct costing approach intuitively seems fair, inequitable and unpredictable cost allocations can result. To illustrate, two direct costing methods are applied to our regional water network problem. A common approach to allocating remaining costs is by some physical measure of each individual's use of the common facilities; this method is generally referred to as the use of facilities method (Loughlin, 1977; Goodman, 1984). This traditional method is easy to understand and apply because quantitative information on a physical measure of use is generally available. In the water resources field, use can be measured in terms of the storage capacity and/or the quantity of water flow provided by the common facilities. For our regional water network problem, the flow to each county is the obvious measure of use to apportion the costs of common pipelines since the assumed cost function depends on the flow. In the case of the optimal network, the only direct cost is the cost of the pipeline from county 2 to county 3 serving county 3, and the use of facilities method gives the following cost allocation (see calculations in Table 46). County 1 $ 204,283 County 2 2,221,299 County 3 2,130,827 Total $4,556,409 Unfortunately, this cost allocation does not implement the optimal network because county 3 can do substantially better by going alone, i.e., $1,990,992 versus paying $2,130,827. In addition to giving an inequitable cost allocation for the optimal network, the use of facilities method can promote noncooperation if other networks are also being considered. Table 47 shows the cost allocations for all possible options available to the three counties using the use of facilities method. Suppose the "second best" network or option 2 is also being considered by the counties. The second best network consists of the pipelines from the well field to county 1, from county 1 to county 2, and from county 1 to county 3. This second best network costs a, H 0 I rpI I U HO 0 U 0 0 I I I In I U') rl o o 01 0 4 at 4 ii (N o' I C) m a, CN o,1 0 Io 0 oC co ,a C, 00 co I o o  0 0 Cr 0 o m Nm O 0 0 C N Nrv a, o ( 0 cN 0) c *<1 O iQ) S0 u  q V) H >i 0  H 0 II U O 0 Oa ;io O II Table 47. Cost Allocation for the Use of Facilities Method. Cost Allocation to County i ($) Is Cost Ex(i) Allocation Option County 1 County 2 County 3 ($) Equitable? (Rank) x(l) x(2) x(3) 1 204,283 2,221,299 2,130,827 4,556,409 No x(3)>c(3) 2 204,283 2,445,055 1,907,488 4,556,826 No x(2)>c(2) 3 646,000 1,976,000 2,008,177 4,630,177 No x(3)>c(3) 4 244,165 2,684,346 1,990,992 4,919,503 No x(2)>c(2) 5 323,000 2,420,095 2,263,639 5,006,734 No x(3)>c(3) 6 646,000 2,420,095 1,990,992 5,057,087 (1) (2) (5) (6) $4,556,826 or $417 more than the optimal network; so, both networks are essentially comparable in cost, and either network might be considered the least cost network. In fact, the second best network becomes the optimal network if the economies of scale or the value of b in the cost function is .51 instead of .50 (see Table 34). Never theless, applying the use of facilities method to this second best network gives the following cost allocation. County 1 $ 204,283 County 2 2,445,055 County 3 1,907,488 $4,556,826 In this case, the cost allocation fails to implement the second best network because county 2 is better off going alone, i.e, paying $2,420,095 rather than $2,445,055. Furthermore, if we examine the cost allocation for the optimal network and the second best network, another problem is evident. Although the costs for the two networks are $417 apart, the difference in costs between the two networks for county 2 and county 3 is enormous. Consequently, this cost allocation method imposes another obstacle for the counties to cooperate and implement either one of the two networks. County 2 strongly opposes the second best network because of its substantially higher cost while county 3 strongly opposes the optimal network for the same reason. This problem is even more serious when more options are considered by the counties. Table 47 indicates tremendous differences in allocated cost for each county depending on the network, thereby making cooperation very difficult. This situation shows the danger for individuals to simply accept the least cost network without carefully examining all of their options if the use of facilities method for allocating costs is chosen. Another simple way to prorate the remaining costs is to divide it equally among the individuals associated with the common facilities (see calculations for optimal network in Table 48). Table 49 shows the cost allocations using this egalitarian approach and indicates that none of the cost allocations for options with savings are equitable. At first glance, the cost allocation for option 5 appears equitable because each county is charged a cost less than or equal to its goitalone cost. However, closer examina tion reveals that counties 1 and 2 can do better as a coalition. They can construct a pipeline from the well field to county 1 and from county 1 to county 2, i.e., option 4, for $2,928,511 rather than pay the sum of their costs for option 5, i.e., $3,066,095. Unfortunately, a transition from option 5 to option 4 causes county 1 to lose money, i.e., $854,577 for option 4 versus $646,000 for option 5. To further complicate matters, option 5 only gives a 1% savings and requires county 1 to cooperate with county 3 to build a pipeline without getting any savings. I to W H1 Q 4 o 0 0 S0 rcd *H O o 04 c O 0 S0 I ,i r * a) cs 0 I 4 , 1 4) U)  H 0 U I U HOU 0 U r4  4J 0  OU 0 ro CN ri I a) C , , At I I I Ln 1I o 0 o HD r4) r4 t r1. CO' o CM 0 o\' crl r 1 co N 0 or r 01 o o CC cO II 0 I Ln o N a% 0 r> m 0 0 CN SW 'W 0 H 0N 01 P4U  0 '0 4o  O z t z r 0 0 11 U U 0O t.o C I O > ^ Table 49. Cost Allocation for Direct Costing/Equal Apportionment of Remaining Costs Method Cost Allocation to County i ($) Is Cost Ex(i) Allocation Option County 1 County 2 County 3 ($) Equitable? (Rank) x(l) x(2) x(3) 1 680,944 1,427,644 2,447,821 4,556,409 No x(l)>c(1) x(3)>c(3) 2 680,944 1,900,300 1,975,582 4,556,826 No x(1)>c(l) 3 646,000 1,482,000 2,502,177 4,630,177 No x(3)>c(3) 4 854,577 2,073,934 1,990,992 4,919,503 No x(l)>c(l) 5 646,000 2,420,095 1,940,639 5,006,734 No x(1)+x(2)> c(12) 6 646,000 2,420,095 1,990,992 5,057,087 ( )3 3 3 S 2 S 2 S 2 1 1 1 (1) (2) (3) (5) (4) (6) Given these observations, the stability of option 5 as a regional water network is at best questionable. Again, if the allocated costs for counties 2 and 3 for the optimal network are compared to the second best network, a similar situation like the one discussed for the use of facilities method exists. That is, counties 2 and 3 face substantially different costs for these two networks with comparable costs. Thus, assigning direct costs does not help eliminate inequitable cost allocations. In fact, direct costing methods can impose additional obstacles to cooperation. This occurs because the assignment of direct costs depends on the configuration of the facilities. For instance, the cost of the pipeline from county 2 to county 3 for our regional water network problem can be a direct cost or a joint cost depending on the network. The cost of the pipe line is a direct cost for county 3 if the second best network, i.e., option 2, is being considered; yet, the cost of the pipeline is a joint cost for counties 2 and 3 if the optimal network, i.e., option 1, is being considered. These changes in the cost classification for the pipeline from county 2 to county 3 contribute to the tremendous difference in the cost allocations for counties 2 and 3 for the two comparable cost networks. This situation indicates an additional criterion not addressed by Claus and Kleitman (1973) for selecting a procedure to allocate network cost. The cost allocation procedure should be independent of network configuration; otherwise, the cost allocation pro cedure can promote noncooperation if more than one network is being considered. In summary, two approaches for allocating costs in the water resources field have been examined: 1) allocate total project cost in proportion to a physical or nonphysical criterion; or 2) allocate assignable costs directly and prorate the remaining costs by a physical or nonphysical criterion. In general, these two approaches are simple to apply and easy to understand. In fact, these two approaches are currently accepted cost allocation methods used in accounting (Kaplan, 1982). However, these two approaches are unable to consistently provide an equitable cost allocation when an equitable cost allocation exists, i.e., sometimes these methods work and sometimes they fail. Furthermore, methods attempting to assign costs directly may be influenced by the configuration of the facilities and may discourage cooperation when more than one configuration is being considered. This is particularly evident for our regional water network problem. For a theoretically sound method that is able to find an equitable cost allocation if one exists and is not influenced by the configuration of the facilities, concepts from cooperative nperson game theory are necessary. Cooperative Game Theory Game theory has been with us since 1944 when the first edition of The Theory of Games and Economic Behavior by John Von Neumann and Oskar Morgenstern appeared. In particular we are interested in games wherein all of the players voluntarily agree to cooperate because it is mutually bene ficial. Furthermore, games are studied in three forms or levels of abstraction. The extensive form requires a com plete description of the rules of a game and is generally characterized by a game tree to describe every player's move. A game in normal form condenses the description of a game into sets of strategies for each player and is represented by a game matrix. However, most efforts in cooperative game theory have been with games in charac teristic function form whereby the description of a game is in terms of payoffs rather than rules or strategies. The characteristic function form appears to be the most appropriate for studying coalition formation which is an essential feature in cooperative games. Also, cooperative games can be of three types depending on whether the game is defined in terms of costs, savings, or values. To keep the notation as simple as possible, only cost games will be discussed. Introductory and intermediate material on coop erative game theory can be found in Schotter and Schwodiauer (1980), Jones (1980), Luce and Raiffa (1957), Lucas (1981), Rapoport (1970), Shubik (1982), and Owen (1982). Concepts of Cooperative Game Theory Let N = {l,2,...,n} be the set of players in the game. Associated with each subset of S players in N is a charac teristic function c, which assigns a real number c(S) to each nonempty subset of S players. For cost games, the characteristic function, c(S), can be defined as the least cost or optimal solution for the Smember coalition if the NS member or complementary coalition is not present. However, depending on how the problem is defined, alterna tive definitions for c(S) may be required. For example, Sorenson (1972) presents the following four alternative definitions for the characteristic cost function: cl(S) = value to coalition if S is given preference over NS. c2(S) = value of coalition to S if NS is not present, c3(S) = value of coalition in a strictly competitive game between coalition S and NS, and c4(S) = value of coalition to S if NS is given preference. If c(S) can be defined as the least cost solution for coalition S if NS is not present, then the cost game is naturally subadditive; i.e., c(S) + c(T) > c(SUT) SnT = 0, S,TcN (46) where 0 is the empty set; and S and T are any two disjoint subsets of N. Subadditivity is a natural consequence of c(S) because the worst S and T can do as a coalition is the cost of independent action; i.e., c(S) + c(T) = c(SUT) SnT = 0, S,TcN. (47) A coalition in which the players realize no savings from cooperation is said to be inessential. General reasons why subadditivity exists are discussed by Sharkey (1982a). The primary reason why subadditivity exists for our regional water network problem is because of the economies of scale in pipeline construction cost. For a single output cost function, C(q), economies of scale is defined by C(Xq) < XC(q) (48) where q = output level, and for all X such that 1 < A < 1 + E, E is a small positive number. This definition means that the average costs are declining in the neighborhood of the output q. From Sharkey (1982a), economies of scale is sufficient but not a necessary condi tion for subadditivity. Subadditivity is a more general condition which allows for both increasing marginal cost and increasing average cost over some range of outputs. Solution concepts for cooperative cost games involve the following three general axioms of fairness (Heaney and Dickinson, 1982; Young et al., 1982): 1) Individual Rationality: Player i should not pay more than his goitalone cost, i.e., x(i) < c(i), V ieN, (49) where x(i) is the allocated cost or the charge to player i. 2) Group Rationality: The total cost of the grand coalition, c(N), must be apportioned among the N players; i.e., Z x(i) = c(N). (410) ieN 3) Subgroup Rationality: This final axiom extends the notion of individual rationality to include subgroups, i.e., no subgroup or subcoalition S should be apportioned a cost greater than its goitalone cost, or Z x(i) < c(S), V ScN. (411) ieS The set of solutions or charges satisfying the first two axioms is called the set of imputations, while the additional restriction of the third axiom defines what is known as the core of the game. For subadditive cost games the set of imputations is not empty, but the core may be empty. Shapley (1971) has shown that the core always exists for convex games. A cost game is convex if c(S) + c(T) > c(SUT) + c(SAT) SnT d 0, V S,TCN (412) or equivalently, convexity can be written as c(SUi) c(S) > c(TUi) c(T) ScTcN {i}, ieN. (413) Convexity simply means the incremental cost for player i to join coalition T is less than or equal to the incremental cost for player i to join a subset of T. This notion of convexity is analogous to economies of scale and implies the game has a particular form of increasing returns to scale in coalition size. As will be shown, the more attractive the game, i.e., larger savings in project costs, the greater the chance that the game is convex; whereas, if the game is less attractive, i.e., lower savings in project costs, the poten tial for a nonconvex game or an empty core game is greater. To illustrate the concept of the core, assume a three person cost game with the following characteristic function values: c(1) = 35 c(12) = 66 c(2) = 45 c(13) = 75 c(123) = 100 c(3) = 50 c(23) = 87 This game is subadditive so each player has an incentive to cooperate; i.e., c(1) + c(2) + c(3) > c(123) c(1) + c(23) > c(123) c(2) + c(13) > c(123) c(3) + c(12) > c(123) c(1) + c(2) > c(12) c(l) + c(3) > c(13) c(2) + c(3) > c(23). Furthermore, this game is convex; i.e., c(12) + c(13) > c(123) + c(1) c(12) + c(23) > c(123) + c(2) c(13) + c(23) > c(123) + c(3). Using the three general axioms of fairness, the core conditions are as follows: x(1) < 35 x(2) < 45 x(3) < 50 x(1) + x(2) < 66 x(1) + x(3) < 75 x(2) + x(3) < 87 x(l) + x(2) + x(3) = 100. The first three conditions determine the upper bounds on x(i), i = 1,2,3, while the last four conditions determine the lower bounds on x(i), i = 1,2,3, i.e., c(123) c(23) = 13 < x(l) < 35 = c(l) c(123) c(13) = 25 < x(2) < 45 = c(2) c(123) c(12) = 34 < x(3) < 50 = c(3) For a threeperson game, graphical examination of the core conditions and the nature of the charge vectors is possible using isometric graph paper (Heaney and Dickinson, 1982). As shown on Figure 42, each player is assigned a charge axis. The plane of triangle ABC, with vertices (100,0,0), (0,100,0), and (0,0,100), represents points satisfying group rationality (axiom 2); whereas, the smaller triangle abc represents the set of imputations satisfying both individual rationality (axiom 1) and group rationality (axiom 2). The vertices a, b, and c represent the charge vectors: [35, 15, 50], [5, 45, 50], and [35, 45, 201, respectively. Line ab represents the upper bound for player 3, i.e., x(3) = c(3), where c(123) c(3) is allocated between players 1 and 2. As we move along line ab from point a to point b, the allocation to player 1 decreases from c(l) to c(123) c(2) c(3), i.e., from 35 to 5, while the allocation to player 2 increases from c(123) c(l)  c(3) to c(2), i.e., from 15 to 45. Similar explanations can be given for lines bc and ac. A more restrictive set of solutions satisfying subgroup rationality (axiom 3), x(3) C (0,0,100) 13 < x (1) < 35 25 < x(2) < 45 ,/ ^34 < x(3) < 50 ih A (0,100,0) (100,0,0) B x(1) x(2) Figure 42. Geometry of Core Conditions for Three Person Cost Game Example. the shaded area on triangle abc, is the core for this game. The geometry of the core for this convex game is a hexagon. Line de represents the lower bound for player 2 or the set of charges where c(13) is allocated between player 1 and player 3 with the remainder, c(123) c(13), going to player 2. Similar explanations can be given for lines fg and hi which are the lower bounds for players 1 and 3, respectively; and for lines id, gh, and ef which are the upper bounds for players 1, 2, and 3, respectively. If an allocation lies outside the core, an inequitable situation prevails. For instance, point Z in Figure 42 allocates player 2 a cost less than its lower bound, c(123)c(13), which means c(13) increases or the cost allocated to players 1 and 3 increases. Clearly, player 1 and player 3 can do better by forming their own twoperson coalition rather than subsidizing player 2. As mentioned earlier, the convexity of a game and its attractiveness are related. This relationship is illustrated in Table 410. When the costs for the twoperson coalitions progressively decrease, there is less incentive for forming the grand coalition so the core becomes progressively smaller and the game becomes progressively more nonconvex. As a consequence of the core conditions for a threeperson sub additive cost game, a condition can be derived to determine if a core exists. From subgroup rationality and group rationality, we have the following conditions: Table 410. Core Geometry for ThreePerson Cost Game Example. Characteristic Function c(1) = 35, c(2) = 45, c(3) = 50, c(123) = 100 Geometry _c(ij) of Core c(12) c(13) c(23) 228 Hexagon 220 215 208 Pentagon Pentagon /l A Trapezoid Triangle 200 Point 192 x(2) x(3) Empty Source: Modified from Fischer and Gately, 1975, p. 27a. x(1) + x(2) < c(12) x(1) + x(3) < c(13) x(2) + x(3) < c(23) x(l) + x(2) + x(3) = c(123) (414) Summing the subgroup rationality conditions gives 2[x(1) + x(2) + x(3)] < c(12) + c(13) + c(23). (415) If the group rationality conditions are substituted into the above equation, then we have the following condition to determine if a core exists: 2c(123) < c(12) + c(13) + c(23). (416) Therefore, in Table 410, the core exists as long as the sum of the twoperson coalitions is greater than 200 or twice the value of the grand coalition. When the sum of the twoperson coalitions equals 200, the core reduces to a unique vector, i.e., X = [24, 32, 44]. Finally, when the sum of the twoperson coalition is less than 200, then the core is empty. Unfortunately, for larger games there is no simple condition for checking the existence of a core; however, as we will see later, a check can be made using linear programming. Unique Solution Concepts The three axioms of fairness defining the core of the game significantly reduce the set of admissible solutions. Unless the core is empty or is a unique vector, an infinite number of possible equitable charge vectors remain to be considered, so additional criteria are needed to select a unique charge vector. Numerous methods are available for selecting a unique charge vector; but the two most popular methods discussed in the literature are the Shapley value (Shapley, 1953; Heaney, 1983b; Shubik, 1962; Heaney et al., 1975; Littlechild, 1970) and the nucleolus (Schmeidler, 1969; Kohlberg, 1971; Suzuki and Nakayama, 1976). Shapley value. The Shapley value for player i is defined as the expected incremental cost for the coalition of adding player i. Thus, each player pays a cost equal to the incremental cost incurred by the coalition when that player enters. Since the coalition formation sequence is unknown, the Shapley value assumes an equal probability for all sequences of coalition formation, i.e., the probability of each player being the first to join is equal, as are the probabilities of joining second, third, etc. For an n person game there are n! orderings. The six sequences of coalition formation for a threeperson game are as follows: (123) (213) (231) (132) (312) (321) Therefore, the Shapley value or the cost to player 1 for a threeperson game is 0(1) = 1/3 c(l) + 1/6 [c(12) c(2)] + 1/6 [c(13 c(3)] + 1/3 [c(123) c(23)]. (417) Player 1 has 1/3 probability of entering the coalition as the first player and 1/3 probability of entering the coalition as the last player. In addition, player 1 has 1/6 probability of entering the coalition after player 2 and 1/6 probability of entering the coalition after player 3. Notice that [c(S+i) c(S)] is the incremental cost of adding player i to the S coalition. The general formula for the Shapley value for player i is $(i) = ~ a. (S) [c(S) c(S {i})] (418) ScN (s i)! (n s)! where ai(S) = n! s is the number of players in coalition S, n! is the total number of possible sequences of coalition formation, (s1)! is the number of arrangements for those players before S, and (nl)! is the number of arrangements for those players after S. For example, for i = 1, n = 3: al(l) = 0!2!/3! = 1/3 al(12) = 1!1!/3! 1/6 al(13) = 1!1!/3! = 1/6 a (123) = 2!0!/3! = 1/3 Total 1.0 Note that Z 0(i) = c(N). (419) iEN Furthermore, if the game is convex, the Shapley value lies in the center of the core (Shapley, 1971). The Shapley value is criticized for several reasons. It may fall outside the core for nonconvex games, and it may be computed even when the core does not exist (Hamlen, 1980). Furthermore, the Shapley value is computationally burdensome for large games. For an nperson game, the Shapley value for each player requires the computation of 2 ncoefficients and incremental costs. For example, an eight player game requires 128 coefficients and incremental costs to calculate the charge for each player. Loehman and Whinston (1976) attempted to reduce the computational burden of the Shapley value by relaxing the assumption that all sequences of coalition formation are equally likely. This generalized Shapley value allows using a priori information to eliminate impossible sequences of coalition formation. Unfortunately, when Loehman et al. (1979) applied the generalized Shapley value to an eightplayer regional wastewater management problem, they got a solution outside the core (Heaney, 1983a). Littlechild and Owen (1973) developed the simplified Shapley value for games wherein the characteristic function is a cost function with the property that the cost of any subcoalition is equal to the cost of the largest player in the subcoalition. Although Littlechild and Thompson (1977) demonstrated the computational ease of the simplified Shapley value in their case study of airport landing fees consisting of 13,572 landings by 11 different types of aircraft, the use of the simplified Shapley value is restricted to games with these special properties. Before calculating the Shapley value for our regional water network problem, the total enumeration procedure described in the preceding chapter is used to find the following characteristic cost function values (see Appendix A): c(l) = 646,000 c(2) = 2,420,095 c(3) = 1,990,992 c(12) = 2,928,511 c(13) = 2,586,638 c(23) = 3,984,177 and c (123) = 4,556,409 c (123) = 4,556,826 c3(1,23) = 4,630,177 c (12,3) = 4,919,503 c5(13,2) = 5,006,734 cw(1,2,3) = 5,057,087 where ck(hi,j) is the cost of the kth best regional water network consisting of pipelines from the well field to county h, from county h to county i, and from the well field to county j. Also, cW(1,2,3) is the cost for each county to goitalone. The cost allocation associated with the kth best regional water network, i.e., the kth network game, is simply found by setting c(N) equal to ck(N). The Shapley values for all options available to the three counties are calculated in Appendix A and summarized in Table 411. Appendix A also checks whether each Shapley value satisfies core conditions. All of the network games in this example are nonconvex. Table 411 shows that the cost allocations for the optimal and the second best networks, i.e., the first two options, satisfy all core conditions; therefore, these cost allocations are in the core and are considered equitable. Furthermore, unlike the cost allocations using the direct costing methods discussed earlier, the cost allocations for these two comparable cost Table 411. Cost Allocation for ThreeCounty Example Shapley Value. Using the Cost Allocation to County i Is Cost ($) Allocation Ex(i) In Core? Option County 1 County 2 County 3 ($) (From Ap (Rank) x(l) x(2) x(3) pendix A) 1 590,087 2,175,905 1,790,417 4,556,409 Yes 2 590,226 2,176,044 1,790,556 4,556,826 Yes 3 614,677 2,200,494 1,815,006 4,630,177 No 4 711,119 2,296,936 1,911,448 4,919,503 No 5 740,196 2,326,013 1,940,525 5,006,734 No 6 646,000 2,420,095 1,990,992 5,057,087 (1) (2) (3) (5) networks are nearly identical. The Shapley value divided the additional $417 for the second best network equally among the counties. Option 3 illustrates the failure of the Shapley value to consistently give a core solution for nonconvex games. As shown in Appendix A, the cost alloca tion for option 3 fails to satisfy subgroup rationality for the coalition consisting of county 2 and county 3; i.e., x(2) + x(3) > c(23). (420) Moreover, options 4 and 5 illustrate Shapley values for games with an empty core. The nonexistence of the core for network games with options 4 and 5 can be determined by using other game theory methods, e.g., nucleolus. A close examination of the core conditions for network games with options 4 and 5 reveals these games are no longer subaddi tive. By defining c(N) as c (N), c(N) is no longer the least cost or optimal solution for the grand coalition. Consequently, the kth best network game is not naturally subadditive even though ck (N) may be less expensive than c (N). In any event, because network games with options 4 and 5 are not subadditive, there is no incentive to cooperate. Therefore, options 4 and 5 no longer need to be considered by the counties. Nucleolus. The other popular method to obtain a unique charge vector is to find the nucleolus. For a cost game, the fairness criterion used by the nucleolus is based on finding the charge vector which maximizes the minimum savings of any coalition. For each imputation in the core of a cost game, a vector in R2 is defined. The components of this vector are arranged in increasing order of magnitude and are defined by e(S) = c(S) Z x(i) V ScN. (421) ieS n 2 The imputation whose vector in R is lexicographically the largest is called the nucleolus of the cost game. Given two vectors, X = (Xl,...,xn) and Y = (yl,...yn), X is lexi cographically larger than Y if there exists some integer k, 1 < k < n, such that x. = y. for 1 < j < k and xk > yk (Owen, 1982). Basically, e(S) represents the minimum savings of coalition S with respect to charge vector X. Obviously, the coalition with the least savings objects to charge vector X most strongly, and the nucleolus maximizes this minimum savings over all coalitions. The nucleolus can be found by solving at most nl linear programs (Kohlberg, 1972; Owen, 1974, 1982), where the first linear programming problem is maximize e(l) V iEN (422) subject to e(l) + x(i) < c(i) e(l) + Z x(i) < c(S) V S N ieS Z x(i) = c(N) iEN x(i), e(l) > 0 The nucleolus is calculated by sequentially solving for e(l), then e(2), e(3), etc., where e(i) is the ith smallest savings to any coalition. Unlike the Shapley value, the nucleolus always is in the core for games with nonempty core. In fact, the nucleolus is always unique. However, the nucleolus is criticized because it cannot be written down in explicit form (Spinetto, 1975), and that it is difficult to compute and use in practice (Gugenheim, 1983). Probably the most difficult problem with using the nucleolus is the acceptance of its notion of fairness as opposed to other prevailing notions of fairness without generating unending controversies and debates. The nucleolus is generally considered to be analogous to Rawls' (1971) welfare criteria: the utility function of the least well off individual is maximized. Other notable notions of fairness include (1) Nozick's (1974) procedural approach to justice, and (2) Varian's (1975) or Baumol's (1982) definition of equitable distribution whereby no one prefers the consumption bundle of anyone else. Calculating the nucleolus for our regional water network problem using the linear programming problem (422) gives the results summarized in Table 412. Equitable cost allocations are given for the first three options, and the cost allocations for the optimal and the second best networks are essentially the same. The additional $417 for the second best network is apportioned as follows: County 1 $209 County 2 104 County 3 104 Total $417 No cost allocations are given for options 4 and 5 because these network games have empty cores. That is, the linear programming problem (422) is infeasible. Finally, Table 412 reveals that each of the three counties has an incentive to cooperate in order to implement the cheapest regional water network. Propensity to disrupt. Another unique solution concept worth mentioning because of its intuitive appeal is the concept of an individual player's "propensity to disrupt." Gately (1974) defined an individual player i's propensity to disrupt as a ratio of what the other players would lose if player i refused to cooperate over how much player i would lose by not cooperating. Mathematically, player i's Table 412. Cost Allocation for ThreeCounty Example Using the Nucleolus. Cost Allocation to County i ($) Is Cost Ex(i) Allocation Option County 1 County 2 County 3 ($) In Core? (Rank) x(1) x(2) x(3) 1 609,116 2,144,583 1,802,710 4,556,409 Yes 2 609,325 2,144,687 1,802,814 4,556,826 Yes 3 646,000 2,163,025 1,821,152 4,630,177 Yes 6 646,000 2,420,095 1,990,992 5,057,087 5      6 646,000 2,420,095 1,990,992 5,057,087  (2) (5) (6) propensity to disrupt, d(i), a charge vector, X = [x(l),...,x(n)], which is in the core is c(Ni) E x(j) d(i) = j i (423) c(i) x(i) The higher the propensity to disrupt, the greater a player's threat to the coalition; e.g., d(i) = 10 implies player i could impose a loss of savings to the other players 10 times as great as the loss of savings to player i. As an illustration, the propensity to disrupt is calculated for each of the counties using the nucleolus for the optimal network of our regional water network problem: X = [609,116; 2,144,583; 1,802,710]. d(l) =_ c(23 x(2) x(3) 1.0 c(l) x(l) d(2) c(13) x(l) x(3) c(2) x(2) d(3) c(12) x(1) x(2) d(3) = = 93 c(3) x(3) The calculations show that none of the counties have a strong threat against the other two counties with the nucleolus charge vector. County 1 could impose a loss to the other two counties which equals the loss imposed on itself, while, county 2's or county 3's departure would hurt the departing county more than it would hurt the remaining two counties. 87 Gately suggested equalizing each player's propensity to disrupt as a final cost allocation solution. Subsequently, Littlechild and Vaidya (1976) have generalized Gately's concept of an individual player's propensity to disrupt to include a coalition S's propensity to disrupt. That is, a coalition S's propensity to disrupt is defined as the ratio of what the complementary coalition, NS, stands to lose over what the coalition S itself stands to lose for a given charge vector. More recently, Charnes et al. (1978) and Charnes and Golany (1983) refined these propensity to disrupt concepts into a unique solution concept which appears to have some empirical support. Finally, Straffin and Heaney (1981) have shown that Gately's propensity to disrupt is exactly the alternative cost avoided method first proposed during the TVA project in 1935. The alternative cost avoided method is discussed in the section on the separable costs, remaining benefits method. Empty Core Solution Concepts Examining games with an empty core is an active area of research. An empty core implies that no equitable cost allocation exists, and results from games wherein the addi tional savings from forming the grand coalition is rela tively small. That is, the savings resulting from forming smaller coalitions are almost as much as the savings from forming the grand coalition. Therefore, proposed solution concepts generally seek to relax the bounds on subgroup rationality until a "quasi" or "anti" core is created. Table 413 lists four methods for finding a charge vector for games with an empty core. In any case, given the modest amount of economic gain for games with an empty core, it may be more advantageous to forego the grand coalition in favor of smaller coalition formations as suggested by Heaney (1983a). Furthermore, engineering projects tend to have a large proportion of the costs common to all participants; consequently, one would expect these games to be very attractive and games with an empty core to be fairly rare. Nevertheless, the game theory approach does alert us that a problem exists in allocating costs for such cases. Cost Allocation in the Water Resources Field Straffin and Heaney (1981) showed that the criteria of fairness as expressed by equations (41) and (42) associated with cost allocation proposed by the TVA experts in the 1930's paralleled the development of the concepts of individual and subgroup rationality found in cooperative game theory. Given that full costs have to be recovered, the core conditions are equivalent to the fairness criteria for allocating cost originally proposed by the TVA experts. Therefore, current practice for allocating costs in the water resources field should require the solution be in the core of a game. Table 413. Empty Core Solution Methods. Method Approach Source 1. Least Core or Relax c(S) Shapley and Shubik, BCore 1973; Young et al., 1982; Williams, 1982, 1983 2. Weak Least Core Relax c(S) Shapley and Shubik, or aCore 1973; Young et al., 1982; Williams, 1982, 1983 3. Minimum Cost, Relax c(S) Heaney and Dickinson, Remaining Savings 1982 4. Homocore Relax c(N) Charnes and Golany, 1983 