Efficiency/equity analysis of water resources problems--a game theoretic approach

MISSING IMAGE

Material Information

Title:
Efficiency/equity analysis of water resources problems--a game theoretic approach
Physical Description:
xi, 160 leaves : ill. ; 28 cm.
Language:
English
Creator:
Ng, Elliot Kin, 1950-
Publication Date:

Subjects

Subjects / Keywords:
Water resources development -- Cost effectiveness -- Data processing   ( lcsh )
Information storage and retrieval systems -- Water resources development   ( lcsh )
Environmental Engineering Sciences thesis Ph. D
Dissertations, Academic -- Environmental Engineering Sciences -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1985.
Bibliography:
Bibliography: leaves 150-158.
Statement of Responsibility:
by Elliot Kin Ng.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000876133
oclc - 14706818
notis - AEH3714
sobekcm - AA00004882_00001
System ID:
AA00004882:00001

Full Text














EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES
PROBLEMS--A 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. Mun-Fong Lee, and

Mr. Robert Ryczak. I would also like to give special thanks

to Mr. Robert Dickinson for keeping an extra copy of the

LP-80 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

3-1 Example of Total Enumeration Procedure for
3-Node Digraph............................... 27

3-2 The Number of Independent Calculations to
Find the Costs of Spanning Directed Trees
for All Possible Subdigraphs................. 31

3-3 Summary of Computational Effort for Digraphs
Shown in Figure 3-4.......... ..... ............ 34

3-4 Efficiency Analysis of a Three-User Water
Supply Network with Nonlinear Cost Function
Using Lotus 1-2-3................ ............ 37

4-1 Projected Population Growth and Projected
Average Per Capita Demand.................... 41

4-2 The Costs and Percent Savings for All
Options............................................. 44

4-3 Cost Allocation Matrix....................... 50

4-4 Cost Allocation of Optimal Network Based on
Population.................................... 52

4-5 Cost Allocation of Optimal Network Based on
Demand ..................... ................... 53

4-6 Cost Allocation of Optimal Network with Use
of Facilities Method......................... 56

4-7 Cost Allocation for the Use of Facilities
Method ............... ........................ 57

4-8 Cost Allocation of Optimal Network with
Direct Costing/Equal Apportionment of
Remaining Costs Method....................... 60

4-9 Cost Allocation for Direct Costing/Equal
Apportionment of Remaining Costs Method...... 61


vii









4-10 Core Geometry for Three-Person Cost Game
Example......................................... 73

4-11 Cost Allocation for Three-County Example
Using the Shapley Value...................... 80

4-12 Cost Allocation for Three-County Example
Using the Nucleolus.......................... 85

4-13 Empty Core Solution Methods.................. 89

4-14 Cost Allocation for Three-County Example
Using the SCRB Method......................... 94

4-15 Cost Allocation for Three-County Example
Using the MCRS Method......................... 98

4-16 Nominal Versus Actual Core Bounds for
Optimal Network Game......................... 100

4-17 Cost Allocations for the Optimal Network and
the Second Best Network ($).................. 103

4-18 Cost Allocation for Option 3 as a Two-Person
Game Using the SCRB Method................... 109

4-19 Comparing Cost Allocations for Option 3 as
Two-Person Game and Three-Person Game Using
the SCRB and MCRS Methods.................... 111

4-20 Core Bounds for Option 3 as a Three-Person
Game......................................... 112

4-21 Core Bounds for Option 3 as a Three-Person
Game with County 1 as a Dummy Player......... 114

4-22 Core Bounds for Option 3 as a Two-Person
Game........................................... 116

4-23 Comparison of Methods Discussed for
Allocating Costs of Water Resources Projects. 117

5-1 Using Independent Calculations from the
Total Enumeration Procedure to Find c(i),
c(S), and c(N) for the Three-County Regional
Water Network Problem......................... 121

5-2 Efficiency/Equity Analysis of the Optimal
Network..... .......... ............. ............ 126


viii















LIST OF FIGURES


Figure Page

3-1 Types of Cost Functions....................... 13

3-2 Example Digraph Representing a Regional
Water Network Problem for Three Users ......... 18

3-3 Flow Diagram of Total Enumeration Procedure
for n-Node Digraph............................ 23

3-4 Examples of 3,4,5-Node Digraphs............... 33

4-1 Lengths of Interconnecting Pipelines.......... 43

4-2 Geometry of Core Conditions for Three-Person
Cost Game Example............................... 71

4-3 Core for the Optimal Network Game
(C(N) = $4,556,409)............................ 101

4-4 Core for the Second Best Network Game
(C(N) = $4,556,826)............................ 102

4-5 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
PROBLEMS--A 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 n-person 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 go-it-alone. 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, socio-political, 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 subsidy-free 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 cost-benefit 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 n-person 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 2n-1 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 2n-1 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

2n-1 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, S-shape, or irregular (see

Figure 3-1). 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







SS-Shape





d) S-Shape


e) Irregular


Figure 3-1. 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 fixed-charge cost

functions and apply a mixed-integer programming code. This

approach guarantees a globally optimal solution, but

standard mixed-integer 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 S-shape 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 state-of-the-art 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 site-specific 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 site-specific 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 socio-political 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 (3-1)



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 3-2) consisting of nodes to represent the

source and users, and directed arcs to represent all



























































Figure 3-2. 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

3-2. 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, "n-node" means the number

of nodes, not including the root node, is n; e.g., an i-node

digraph or subdigraph consists of i+l nodes if the root node

is counted. The total enumeration procedure for the n-node

digraph is summarized by the flow diagram shown in Figure 3-3.

Step 1 begins the procedure for evaluating all subdi-

graphs consisting of the root node and one other node, i.e.,

the 1-node subdigraphs.

Step 2 initializes a count of the number of combina-

tions of i-node subdigraphs evaluated.

Step 3 generates all possible combinations of i-node

subdigraphs from the n-node digraph. The number of possible

combinations is ( ). For example, the 3-node digraph shown

in Figure 3-2 has (2) or three possible 2-node 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 i-node 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 i-node 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 3-3. Flow Diagram of Total Enumeration Procedure
for n-Node Digraph









i-node subdigraph, i.e., not every node in the i-node

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 i-node 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 i-node subdigraph. This process

substantially reduces the effort involved in enumerating all

the spanning directed trees for an i-node subdigraph because

a large number of spanning directed trees are inessential.

If the i-node 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 i-node

subdigraph or an n-node 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 3-node digraph

in Figure 3-2 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

i-node 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 i-node subdigraph.









Step 7 checks the counter to see if all possible

combinations of i-node subdigraphs have been evaluated. If

not, Step 8 advances the counter by one before returning to

Step 4 to evaluate another i-node subdigraph. If all of the

possible combinations of i-node 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 n-node 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

3-1 using the regional water network problem modeled by the

3-node digraph shown in Figure 3-2.

During the first iteration all combinations of

1-node 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 2-node subdigraphs are evaluated. In

this case, each combination has two spanning directed trees,

but the cost of only one spanning directed tree needs









Table 3-1. Example of Total Enumeration Procedure for
3-Node Digraph



Spanning Directed Are Spanning
Iteration i-Node Trees for i-Node 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 i-Node Trees for i-Node 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 3-node digraph is being evaluated. This

3-node 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

3-node 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 3-1, 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 3-2 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 3-node 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 2n-2

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 3-2.


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 3-4. For the 3-node 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 2n-2 optimal solu-

tions. For the 4-node 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 2n-2 optimal solutions. Finally, for the 5-node

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 2n-2 optimal solutions.

As we can see, summarized in Table 3-3, a large number of

the spanning directed trees of a digraph are inessential.




























































Figure 3-4. Examples of 3,4,5-Node Digraphs.















en r-i

co E
0) -4 -H- 0




Z-N-H 4-
0) 0U (N 3


H U 1-) N E


4-IJ

() -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)
4-1




-H O
Q-4


En
03


ul o-\ I
-4


,-I












N










CN







r- LO r-i
r-4 re









Also, the percentage of independent calculations decreases

as the number of nodes for a digraph increases.

The 5-node digraph in Figure 3-4 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 2n-1 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









decision-making process. For instance, the procedure can

be implemented using the extremely "user friendly" Lotus

1-2-3 spreadsheet software package. Lotus 1-2-3 has the

mathematical functions to handle calculations involving

nonlinear cost functions or involving detailed cost

analysis. A sample Lotus 1-2-3 printout is shown in Table

3-4 for a hypothetical water network problem modeled by the

3-node digraph shown in Figure 3-2. This printout should be

self-explanatory. 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 1-2-3 allow

automatic ranking of all the feasible solutions according to

cost. Moreover, the Lotus 1-2-3 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 (3-1), 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 3-4. Efficiency Analysis of a Three-User Water
Supply Network with Nonlinear Cost Function
Using Lotus 1-2-3.

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 n-person 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

(1980-1985) of the plan recommends the use of groundwater

from existing and newly developed well fields to satisfy

water demands in the tri-county 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 4-1). 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 4-1. Projected Population Growth and Projected
Average Per Capita Demand.



Projected Projected Average Projected
County Population Growth Per Capita Demand Additional
(gal/cap-day) 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 4-1. 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 1-2-3). 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 4-2). 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 4-1. Lengths of Interconnecting Pipelines.









Table 4-2. 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. 220-221):

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 (4-1)



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 (4-2)
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 (4-3)

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 (4-4)



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 4-3). 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 4-3. 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 4-4 and Table 4-5):



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 go-it-alone

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


O-c

4J
O
r0
0) 0










0
U *



O





--I
r-I-

0
.r-,

4a










0
u1
+J
n3


0) 0 (


o Ln
o a>
o o










SL
Co


mo


CN 0 00
r-I %D CV












0 0 0



mV-I
00 00
r -


0

O



LO
m








0
Ln
o



Ln




























E-4
r-


























E^
















*H





1-1
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 r-r




1-1 in n

< co m















0 0 C0
1r- rQ fn
^ *





CM i-


I-i 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 (4-5)



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 4-6).

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 4-7 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 r-pI

I U
HO

0
U
0


0
I
I
I


























In
I
U')
r-l
o







o











0-1
0












4-
at 4
i-i


(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
i-Q)

S0 u -
-q V)


H

>-i 0


-- H
0 II


U O


0-
Oa-


;io




O II









Table 4-7. 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 3-4). 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 4-7 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 4-8). Table 4-9 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 go-it-alone 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







r-4 -

4J 0 -
OU
0










ro










CN

r-i









I







a)
C-
-,-

-,

At


I
I


















I







Ln

1-I





o
0
o










HD
r4-)












r4
-t






r-1.


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 4-9. 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 n-person 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 S-member coalition if the

N-S 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 N-S.

c2(S) = value of coalition to S if N-S is not
present,

c3(S) = value of coalition in a strictly competitive
game between coalition S and N-S, and

c4(S) = value of coalition to S if N-S is given
preference.


If c(S) can be defined as the least cost solution for

coalition S if N-S is not present, then the cost game is

naturally subadditive; i.e.,


c(S) + c(T) > c(SUT)


SnT = 0, S,TcN


(4-6)









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. (4-7)



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) (4-8)



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 go-it-alone cost, i.e.,



x(i) < c(i), V ieN, (4-9)



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). (4-10)
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 go-it-alone cost, or



Z x(i) < c(S), V ScN. (4-11)
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 (4-12)



or equivalently, convexity can be written as



c(SUi) c(S) > c(TUi) c(T) ScTcN {i}, ieN. (4-13)



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 three-person 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 4-2, 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 4-2. 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 4-2

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 two-person

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 4-10. When the costs for the two-person 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 three-person 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 4-10.


Core Geometry for Three-Person 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) (4-14)



Summing the subgroup rationality conditions gives



2-[x(1) + x(2) + x(3)] < c(12) + c(13) + c(23). (4-15)



If the group rationality conditions are substituted into the

above equation, then we have the following condition to

determine if a core exists:



2-c(123) < c(12) + c(13) + c(23). (4-16)



Therefore, in Table 4-10, the core exists as long as the sum

of the two-person coalitions is greater than 200 or twice

the value of the grand coalition. When the sum of the

two-person coalitions equals 200, the core reduces to a

unique vector, i.e., X = [24, 32, 44]. Finally, when the

sum of the two-person 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 three-person game are as follows:

(123) (213) (231)

(132) (312) (321)









Therefore, the Shapley value or the cost to player 1 for a

three-person 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)]. (4-17)



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})] (4-18)
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,

(s-1)! is the number of arrangements for those
players before S, and

(n-l)! 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). (4-19)
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 n-person game, the

Shapley value for each player requires the computation of

2 n-coefficients 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

eight-player 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

go-it-alone. 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 4-11. Appendix A also checks whether each Shapley

value satisfies core conditions. All of the network games

in this example are nonconvex. Table 4-11 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 4-11. Cost Allocation for Three-County 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). (4-20)



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. (4-21)
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 n-l

linear programs (Kohlberg, 1972; Owen, 1974, 1982), where

the first linear programming problem is



maximize e(l)


V iEN (4-22)


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 (4-22)

gives the results summarized in Table 4-12. 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 (4-22) is infeasible. Finally, Table

4-12 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 4-12. Cost Allocation for Three-County 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(N-i) E x(j)
d(i) = j -i (4-23)
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, N-S, 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 4-13 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 (4-1) and (4-2) 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 4-13. Empty Core Solution Methods.


Method Approach Source



1. Least Core or Relax c(S) Shapley and Shubik,
B-Core 1973; Young et al.,
1982; Williams, 1982,
1983


2. Weak Least Core Relax c(S) Shapley and Shubik,
or a-Core 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