WATER RESOURCES RESEARCH, VOL. 22, NO. 3, PAGES 339-344, MARCH

1986

Sharing Regional Cooperative Gains From Reusing Effluent for Irrigation ARIEL DINAR

AND DAN YARON

Center for Agricultural EconomicResearchand Department of Agricultural Economicsand Management, Hebrew University of Jerusalem, Rehovot, Israel YAKAR

KANNAI

Departmentof TheoreticalMathematics,WeizmannInstitute of Science,Rehovot,Israel This paper is concernedwith the allocation of cost and benefits from regional cooperation, with respectto reuse of municipal effluent for irrigation at the Ramla region of Israel. An efficientregional solution provides the maximal regional income which has to be redistributed among the town and severalfarms. Different allocationsbased on marginal cost pricing and schemesfrom cooperativegame theory like the core, Shapley value, generalizedShapley value, and nucleolusare applied. The town and farm A have the main additional gains accordingto all allocation schemespresented.Advantagesand disadvantagesof theseallocation schemesare examined in order to suggesta fair and acceptableallocation of the regional cooperative gains. Although no method has been preferred, the marginal cost pricing was found to be unacceptableby the participants. The conclusionis that the theory of cooperativ½gamesmay provide guidelineswhile comparingthe different solutions.

Regional wastewater treatment systemsand regional water projects offer economic returns to the usersand environmental benefits to the region as a whole. While a proposed regional solution may be economicallyefficient,it might be rejected by the prospective users unless an acceptable cost/benefit allocation

scheme is found.

Most of the studiesin this field assumean optimal regional solution and solve the problem of apportioning the common cost of water resource projects between the water consumers or the problem of minimizing the common cost of a regional wastewater plant [Giglio and Wrightington, 1972; Suzuki and Nakayama, 1976' Loughlin, 1977; Loehmanet al., 1979' Straffin and Heaney, 1981' Heaney and Dickinson, 1982' Young et al., 1982]. This paper is concerned with the allocation of cost and benefits from reusing municipal effluent for irrigation. The study was applied to the Ramla region on the coastal plain of Israel. The problem involves a town and several farms which differ in their cropping patterns, cropland area and freshwater allotments. An efficientregional solution which maximizes the region'sincome is provided by a mathematical programming model which was solved by Dinar and Yaron [1986]. In corre-

spondence with,the prevailingregulationsit is assumedthat freshwater from the farm's quotas is not transferable and the only way to compensatethe participants is through direct income transfers (side payments). Thus the problem falls within the category of a transferable utility situation with an important property' the possibility of first solving for the efficient solution (maximal regional income) and at a second stage redistributing income. Such a possibility reduces the conceptualand computational difficulties. In this paper, different cost/benefit allocation solutions are applied to the specificcasestudy and compared. Several allocation schemeslike marginal cost pricing, the core [Owen, 1982], the Shapley value [Shapley, 1953] the generalizedSha,pley value [Loehman and Whinston, 1976], and the nucleolus

Copyright 1986by the AmericanGeophysicalUnion.

[Schmeidler, 1969] are applied to the regional problem discussedby Dinar and Yaron [1986]. The results are evaluated from the viewpoints of fairnessand reasonableness. The following section describesthe nature of the potential regional cooperativeactionsin terms of game theory concepts. Next, several cost allocation schemesare applied to the case study considered.Finally, the results are compared and discussed.While some schemesseem to be preferable to others, the conclusion of the study is that game theoretical approaches do not provide indisputable solutions to the cost benefit allocation problem. TH•

NATURE

OF THE ALLOCATION

The region in the case study is the Ramla region on the coastal plain of Israel. It consistsof three farms (A, B, C), a town, and a potential plant as describedby Dinar and Yaron [1986]. Prevailing conditionsin the region suggestthat (1) no cooperation can occur without the town, (2) farm A must be included in each cooperative organization because of its layout in the region, and (3) no transfer of freshwater quotas among the farms is allowed. The participants in the allocation problem (game) are the town and the three farms A, B, and C. They will be referred to as players 1, 2, 3, and 4, respectively.Let N be the set of all the playersin the game, and S* the set of all feasiblecoalitionsin the game; s* (s*• S*) is a feasiblecoalition in the game if and only if s* _ N and 1, 2 • s*. The noncooperativecoalitionsare

{i), i = 1, 2, 3, 4, and the grandcoalitionis N. The numberoi the a priori feasiblecoalitionsin the region is eight (lessthar

thepotentialnumberof 2'• - 1)' Noncooperative coalitions

{1}, {2}, {3}, {4}

Partial

coalitions

{1, 2}, {1, 2, 3}, {1, 2, 4}

Grand

coalition

{1, 2, 3, 4}

An optimization model [Dinar and Yaron, 1986] is applied to any feasible coalition s* in the region in order to maximize (1)

f,, = _•x + • •,

Paper number 5W0832. 0043-1397/86/005W-0832505.00

ies* i-• l

339

PROBLEM

(1)

340

DINAR ET AL.: SHARING REGIONALGAINS

TABLE 1. GrossIncomeof Playersin the Optimal Solutionfor Different Coalition Combinations,the Value for the DifferentCoalitions,and the IncrementalIncomeValuesa($000) Incremental

Gross Income of Player

Value for

Income for

Coalitions*, Coalition

1

{1) {2) {3) (4) {1, 2) {1, 2, 3) { 1, 2, 4) {1, 2, 3, 4)

-368

2

3

4

1940 1285 440

--410 --498 -- 395 --497

2267 2275 2205 2266

1370 488 488

1365

•_•f•

Coalitions*,

fs.

i • s*

v(x*)

-368 1940 1285 440 1857 3147 2299 3622

-368 1940 1285 440 1571 2857 2011 3297

0 0 0 0 285 290 287 325

aRounded figures; values are constant October 1980 dollars.

Source:Solutionsfrom Dinar and Yaron[1986] for differentpossiblecoalitionsin the region. Column 6 is the summationof columns2-5. Column 7 is the summationof the s*'s players'income while they are acting alone. Column 8 is calculatedaccordingto equation 2 and is the differencebetween 7 and 6.

and to any noncooperative coalition{i), i = 1, 2, 3, 4 in order cooperativegame theory and economictheory. A comprehento maximizefi subjectto the availableproductionfactors, sivediscussionof gametheory can be found in works by Owen

giCeen technology, pricesandenvironmental regulations. Here, fs. is the value of the objectivefunction for coalition s*; •x is the town'sgrossexpenses; • is the ith farm'sgrossincome. Any cooperativeuse of municipal wastewaterin irrigation is linked to the incentive of increasingthe region's farms' income by augmenting their irrigation potential. Regional cooperation is economically feasibleif

E (•i _fi) > • _f•

s*6S*

i•s* i-• l

wherefi is the ith player'sincome(costfor the town) in the noncooperative situation. Income values for each of the possible coalitions and the noncooperative situation in the region are presented in Table 1, column 6. The regional optimization model (1) can be interpreted as a cooperative game with side payments and described in terms of a characteristic

function.

The

value

of the characteristic

function for any coalition expressesthe coalitional gains (in terms of incremental income; above the income in the case of

noncooperation), assuming that the coalition acts efficiently under a given set of constraints(see Table 1). Equation (2) definesthe characteristicfunction of a normalized game where the players have to allocate the additional income from coop-

[1982-1,Rapoport [1970] and Shubik [1982]. The conventional competitive rule that each user of effluent

should pay accordingto the marginal treatment cost might provide a possiblecharge scheme.However, it is not applicable in our case,since a) the number of participantsin the wastewater-effluent"market" is small; they are not anonymous, and agreementsand coalitions among some of the potential participantsare possible;and b) the treatment costsare subjectto increasingeconomiesand decreasingmarginal cost. In such a case,marginal cost pricing leads to a deficit which would need to be coveredeither by the usersor someexternal agency(e.g.,government),or both. The difficultiesencounteredby the marginal cost allocation (pricing) scheme lead to a search for solutions in axiomatic approaches.The essenceof these approachesis that once the usersof a public utility or the participants in a regional enterprise (as in our case) agree on certain axioms believed to satisfyselectedcriteria, such as efficiency,fairness,and reasonableness,the cost allocation solution is uniquely determined [e.g., Hazelwood, 1950-1951; Loehman and Whinston, 1971, 1974; Nash, 1950; $hapley, 1953; and $chmeidler,1969].

eration:

THE CORE

v(s*)-- fs. _ •fi

s*• S*

(2)

i•s*

Here v(s*) is the value of the characteristic function for coalition

s* in terms of incremental

income.

It is assumed that

v((i)) = Ofor i • N. Table 1 suggeststhat the incremental income of a coalition generated by players 1 and 2 is $285,000; the gross income generated by player 2 increasesfrom $1,940,000 to $2,209,000, while the treatment cost increasesby only $41,500. Players 3

and 4's individualentry to (1, 2) increasesthe incomeby a negligibleamount.When they both join (1, 2), the additional net income generated is only $40,000 (325,000- 285,000 in normalized values). These observations imply that farm A is the significantcontributor to the regional income, while farms B and C fall far behind. Common sensesuggeststhat this fact should be consideredin any income allocation scheme. ALLOCATION BACKGROUND

SCHEMES:

AND EMPIRICAL

RESULTS

This section provides the background for the application and comparison of several allocation schemesfrom the field of

The core of an N-cooperative game in the characteristic function form is the set of the game allocation gainswhich is not dominated by any other allocation set. The core fulfills the requirementsof individual rationality, group rationality, and joint efficiency. For details, see Friedman [1979], $hubik [1982], and Owen [1982]. Let a•i be player i's allocation of the incremental income, and let o• = (a• ..... a•n)be the vector of allocations.Using the above notation, the core is the set of income allocations satisfying the following conditions:

o•,> v({i})

• ah> v(s*)

for all it N

(3)

for all s*e S*

(4)

i•s*

=

(5)

ieN

It can be shownthat this regionalgame is convex[Shapley, 1971] and thus has a nonempty core (for details, see Dinar [1984]). Since the regional game has a nonempty core, one can refer to the set of core allocations.The core equations (3')-(5'), which are based on the different a priori feasibleco-

DINAR

ET AL.: SHARING REGIONAL GAINS

alitions and their values,have a solution. Results are presented in Table

1.

coi>0

i= 1,2,3,4

(Y)

co• + co: >_285

co• + co2 + co3 > 290

(4')

co• + co2 + c0½> 287

(D1 + 0)2 + (-O 3 + (-04 = 325

341

core," which is the intersection of all nonempty f• cores [Maschler et al., 1979], and if the core is nonempty, it still satisfiesthe conditionsof individual and group rationality. The regional game has a nonempty core, and it is too large. The least core can be generatedby the solution of (6) and (7),

whichcanbe interpretedas"subsidizing" thepartialcoalitions to make them more attractive, while not refuting the individual and the group rationality condition: Minimize fl

(5')

The core of this game is a four-dimensional polyhedron. A method of calculating the extreme points of the core of a convex game was used [Shapley, 1971]. The essenceof the method is to refer to a sequenceof establishmentof any feasible coalition and to allocate the incremental income generated by each player's entry into an existing coalition. Table 2 presentsthe extreme points of the core of the regional game. The first four columns indicate the contribution of each player when he joins an existing coalition, and these are in effect the maximal possible allocations payable to the players. The right-hand side of the table shows the correspondingcoalition formation sequences.Table 2 suggeststhat the negotiation set consistent with the assumptions underlying the core is quite large. Players 1 and 2 can each claim up to $325,000, the total additional regional income generated by the cooperation, because each is essentialto any coalition. On the other hand, the claims of players 3 and 4 consistentwith the core are bound to

(6)

subject to

• oo•-- f• > v(s*)

s*• S*, s* v•N

(7)

its*

•, ooi= v(N) ieN

f•>0

Note that alternative approaches to reducing the core and obtaining a unique solution can be formulated (Young et al., 1982). The least core solution of (6) and (7) for the regional problem studied is obtained for fl = fl* (seesolutions to (6') and (7') below); however, the optimal vector of allocations of

•o is not unique.Any convexcombinationof two alternative optimal solutions,•xo and •:o, is an optimal solutionalso becauseplayers 1 and 2 are "symmetrical," in the sensethat they both are essentialto any coalition formed and are jointly includedin each of (7)'srestrictions.Players 1 and 2 also gain

be modest,lessthan $38,000and $35,000for 3 and 4, respec-

the same income in the nucleolus solution.

tively. This is consistentwith their modest contributions to the regional cooperation under the present conditions. While logically and morally easy to accept, the core of the regional game is not conclusive because it provides only bounds on the claims of the participants in the regional game and therefore is only partly useful.It may serve as a starting point for the continuation of the analysis;a direct progression is the application of the least core and nucleolus concepts which are discussedin the following section.Also, the computations of the extreme points of the core are incorporated into the Shapley value, discussedlater.

The nucleoluscan be interpreted [Maschler et al., 1979] as the allocation, which minimizes the maximal possible objection to the allocations for any of the coalitions; it is a minimax solution. The extent of the objection is measuredby the

Tim

LEAST CORE AND THE NUCLEOLUS

The core of a cooperative game in the characteristic function form may be empty because certain coalitions provide greater incentives than the grand coalition. Conversely, conditions may arise where the core does exist but is too large and leavesthe cost/benefitallocation problem open for further bargaining (as in this study). A possibleapproach for the derivation of a unique solution in both casesis by reducing or expanding the core (f• core) and then obtaining the "least

TABLE 2. Maximum

Profit

Extreme Points of the Core in the Regional Gamea

2

s*) = Z

v(s*)

i•s*

where•q is the allocationto player i in solutionj to (6) and (7). By a lexicographical ordering, the nucleolus solution can be obtained.

For computational details of the nucleolus, the reader is referred to Maschler et al. [1979]. In the particular case considered here, the nucleolus of the regional game can be obtained immediately after solving for fl from (6) and (7), becauseof the symmetry of players 1 and 2. The linear programming problem to find the least core for the regional game is

Max fl

(6') coi

--f•>_0

(-D 1 --[-(-02

--•

(D1 -'[-(,02-'[-(-03

-- f• __> 290

(D1 -'[-(,02

i=

1,2,3,4

__> 285

(7')

-'[-roe --f• __> 287

Allocation

co• + c02+ c03 + c0½

to Player i ($000) 1

"excess" or the difference

3

4

0

285

5

35

0

285

38

2

0 0 0 285 285 290 325 287

290 325 287 0 0 0 0 0

0 0 5 5 38 0 0 38

35 0 0 35 2 35 0 0

Coalition Formation Sequences Leading to This Allocation 1234

= 325

the solution to which is f•* - 8. Since 1 and 2 are symmetrical players, c0x= co: in the second inequality, which yields the nucleolus:

1243 1324 1342 1423 2134 2143 2314 2341 2413

3124 1432 4123

3214 2431 4213

49.5, 149.5,13, 13) 3142

3412

4132

4312

SHAPLEY VALUE AND THE GENERALIZED

3241

3421

4231

aRounded figures; values are constant October 1980 dollars.

4321

SHAPLEY VALUE

The Shapley value is a uniquely defined solution to an Ncooperative game in the characteristic functional form [Shapley, 1953]. The solution allocation to each player 0• is the weighted average of his contributions to all possiblecoalitions and sequences:

342

DINAR ET AL.' SHARING REGIONALGAINS

and the allocationschemeaccordingto the generalizedShapley value is

•o= (142.7,142.7,21.4,18.2) Calculations were based on (9) and Figure 1. MARGINAL

COST PRICING

An alternative cost/benefit allocation schemeis the margin-

al costpricing.Marginal cost pricingof public utilitiesand other publiclysuppliedgoodshas beenextensivelydiscussed in the economic literature [e.g., Hotellin•7,1938; Ru•I•71es, 1949-1950]. Marginal cost pricing leads to an efficient allocation of resources,but, as indicated by Coase 1-1946'1, SamuelFig. 1. Coalition formation sequencesin the regional cooperative son (1964), and others, the problem .of covering the deficit game. Numbers in circlesdenote coalition and formation sequences. generated under the decreasingmarginal cost situation has Numbers along the branchesdenote the conditional probabilities of not been satisfactorilysolved. moving from one coalition to another. The shadow price of effluent is usedhere as being analogous

to marginalcostpricing.It can be shownthat chargingthe

O,=s,eS, • (n--Is*l)!(Is*l1)!Iv(s*)v(s* - {i})] n!

(8)

where 0i is the allocation to player i, n is the number of players in the game, and Is*l is the number of members in s*.

The Shapleyvalue assumesequal probability for the formation of any coalition of the same size which is theoretically possibleand also considersall the possiblesequencesof formation. That assumption has been criticized by Loehman and Whinston [1976]. As an alternative, they proposed the gener-

alized Shapley value, which differs from the original in two major aspects: 1. It refers only to coalitions that are practically possible, rather than possiblefrom a theoretical combinatorial point of view, and 2. The probability of a coalition occurringdependson the logical sequenceof its formation. In a manner similar to Shapley's, Loehman and Whinston [1976] assignto each player the weighted averageof his contributions to all coalitions realistically formed. The generalized

Shapley allocation to playeri is 0i:

O,= • P(s*,s* -- {i})[v(s*)- v(s*-- {i})]

i eN

(9)

s.eS* ies*

whereP(s*, s* -- {i})-- P(s*ls*-- {i})' P(s* - {i}) is the probability of player/joining coalition s* (seealso Figure 1). The allocation schemeaccordingto the Shapleyvalue is

0ø= (153, 153, 10, 9)

various

months

does not affect the values of the variables

the objective function. It has the effect of income redistribution only, in the form of payment by the farms to the town. Although efficiencyconditions are satisfied in the allocation solution according to the shadow price, there is no guarantee that the solution will be acceptedby the players. Specifically, group rationality is not guaranteed,and the solution might be outsideof the core. This indeed happensin the application of shadow cost pricing to the region studied, as shown below. Also, not all costsare covered (decreasingcost case). Table 3 presentsthe payments by the regional farms to the town when a regional solution is considered.The shadow prices of effluent minus the effluent transportation cost are multiplied by the monthly effluent demand of each farm to yield the farm's chargesfor using the town's effluents I-Dinar and Yaron, 1986]. The allocation scheme according to the shadow prices is calculated

in Table

3'

(188, 93, 14, 30) The allocation schemeaccordingto the shadowpricesof effluent discriminatesin favor of the town and against farm A and also assignshigher additional income to farm C than to farm B.

A strong point against shadow cost pricing is that the cost/ benefit allocation does not lie in the core, as it does not satisfy the conditionsof group rationality. The sum of players 1 and 2's additional income in the grand coalition is smaller than the

TABLE 3. Gross Income, Payment for Effluent, Net Income, and Cooperative Gains in the Regional Solution Accordingto the Shadow Price Allocationa ($000) Player Gross income

with

cooperation Payment for effluent Net

income

1

317

with

2

3

4

Region

2266

1365

488

3622

- 66

- 18

- 233 2032

1299

470

- 368

1940

1285

440

3297

188

93

14

30

325

cooperation Treatment costs Costs not covered

Income/cost with no cooperation

Net cooperativegainsb

497 180

aRoundedfigures' valuesare constantOctober 1980 dollars.

bNet cooperativegains equal grossincome at optimal solution minus payment for effluentminus income with no cooperation.

in

the optimal regional solution, nor does it affect the value of

ies*

coalition

farms for using effluent according to its shadow prices in the

DINAR ET AL.: SHARING REGIONAL GAINS

value of their partial coalition: 188 + 93 < v(1, 2)= 285. As shown by Table 3, the total payments amount to $317,000 out of $497,000 (the costs to the town after de-

ducting the 50% subsidyfrom the treatmentcost), leaving $180,000to be borneby the town.Recallthat the town's costswith no cooperationtotal $368,000;that is, the additional income of the town from cooperation is $188,000. This might be consideredby the town as a reasonablegain. On the other hand, the payment by farm A, $233,000, might be consideredtoo high in comparison to its contribution to the cooperation (Table 1). DISCUSSION AND COMPARISON OF THE ALTERNATIVE ALLOCATION

SCHEMES

A way to compare the different allocation solutionsfor the additional income is to test their efficiencyand acceptability to the players. The efficiencyof the solution is guaranteed becauseit was achieved as a solution to an economic system which optimizesthe use of limited resourcesand compensates the participantswith sidepayments.The core is a solution set which satisfiesindividual and group rationality; the extreme points of the core showthe maximum that any player can get,

343

If acceptabilitymeanssatisfyingthe assumptionsunderlying the core, and one tends to accept this interpretation, then all three allocations, nucleolus, Shapley, and generalized Shapley, are acceptable(the nucleolusby its computation, the Shapley and generalizedShapleybecausethe gameis convex). Regarding the criteria of fairness and reasonableness,no objective way to measure them is known [see also RawIs, 1971; Arrow, 1971; Yaari, 1981; Shapley, 1967]. The discussion will therefore be restricted to several observations

and

questions. The allocation based on shadow prices is quite divergent from the others. The allocation to the town is considerably higher, and to farm A, considerablylower. Furthermore, there is a large difference between the town's and farm A's allocations.In view of the symmetryof theseplayersand the large income generatedby farm A, such inequality might be con-

sidered as being unacceptableat least by farm A. Also, as previouslyshown, the shadow cost pricing allocation vector does not lie in the core, another reason for its rejection by players1 and 2. The

allocations

to the town

and farm

A in all the core

schemesare equal.This reflectstheir symmetryas playersand their bargaining power. However, the increasein income of and for players1 and 2 in somesolutionsit meansgainingall farm A ranges between 7.4 and 7.9% as compared to the the regional additional income. The solutionswhich are contained in the core are a convex combination of the extreme noncooperativesituation,while the decreasein the expensesof points and may serveas a guidelinewhile referringto other the town is in the range of 31-42%. The additional income to criteria. farm A is very small and farm A is much more exposed to uncertainty, using effluent in its production process,than the The four solutions, three of which are contained in the core, town in the wastewater treatment process.A fair allocation are summarized in Table 4. The first three systemsof allocating (Shapley, generalized Shapley, nucleolus) regional should provide a risk premium to farm A and discriminate in coat/incomewere derivedon the basisof a priori acceptable favor of farm A and against the town sincethey are not symaxioms. Do the resultsderived from these axioms (i.e., the metric players in the senseof risk. Other similar questions actual allocations) meet common criteria such as fairness, could be asked with no clear, sharp answers. To conclude, in the case studied, game theoretical apreaso:•ableness,and acceptability? proachesdo not provide an objective,indisputablesolutionto the income distribution problem. The various approachespreTABLE 4. Distribution of the Incremental Income Among the sented,and especiallythe core, provide (1) the bounds on the Participantsin the RegionalCooperationAccordingto Alternative claims by the participants and (2) a better understandingof Allocation Schemes ø the grounds for these claims and of possible objections to them. The marginal costpricing allocation doesnot reflect the Player exact contributions of the participants in the game. All the Allocation

Scheme

Noncooperativeincome Shaply Value Allocation, $000 % of regional incremental

1

2

3

4

-- 368

1940

1285

440

153 47

153 47

10 3

9 3

42 •

8

1

2

142.7 44

142.7 44

21.4 7

18.2 6

39•

7

2

1

149.5 46

149.5 46

13 4

13 4

41 •

8

1

3

188

93

14

30

58

29

4

9

51•

5

1

7

income*

% of income with no

cooperation* Generalized Shapley Value Allocation, $000 % of regional incremental income*

% of income with no

cooperation* Nucleolus

Allocation, $000 % of regional incremental

schemes allocate the main additional

Acknowledgments. This research was supported in part by grant 1-101-79 from BARD, The United States-Israel Agricultural Research and Development Fund. The authors are grateful for commentsfrom K. C. Knapp. REFERENCES

income*

% of income with no

cooperation* Shadow

Prices Procedure

Allocation, $000

% of regional incremental income*

% of income with no

cooperation*

øRoundedfigures,monetary values are constant October 1980 dollars.

•For the town it is the expenses decrementrate.

income to the town and

farm A accordingto their main contribution, farms B and C contributing less, which is reflected in the allocations. This information might be helpful in bargainingaimed at reaching an agreement.Quite similar findingsare also the result of the study by Heany and Dickinson[1982]. A good measure of discretion is needed while evaluating the moral qualifications againstsimplicityof the allocation schemes.

Arrow, K. J., A utilitarian approach to the concept of equality in public expenditures,Q. J. Econ.,85, 409-415, 1971. Coase,R. H., The marginal cost controversy,EconomicaNew Ser., 13, 169-182, 1946.

Dinar, A., Economic analysis of regional wastewater treatment and use of effluent in irrigation and related cost benefit allocation schemes(Hebrew), Ph.D. dissertation,190 pp., Hebrew Univ. of Jerusalem, Jerusalem, Israel, 1984.

Dinar, A., and D. Yaron, Treatment optimization of municipal wastewater and reusefor regional irrigation, Water Resour.Res., this issue, 1986.

Friedman, J. W., Oligopolyand the Theory of Games,Adv. Textbooks

344

DINAR ET AL.: SHARING REGIONAL GAINS

in Econ., vol. 8, edited by C. J. Bliss and M.D. Intrilligator, NorthHolland, Amsterdam, 1979.

Giglio, R. J., and R. Wrightington, Methods for apportioning costs among participantsin regional systems,Water Resour.Res., 8(5), 1133-1144, 1972.

Hazelwood, A., Optimum pricing as applied to telephoneservice,Rev. Econ. Stud., 18, 67-78, 1950-1951. Heany, J.P., and R. E. Dickinson, Methods for apportioningthe cost of a water resourceproject, Water Resour.Res., 18(3), 476-482, 1982.

Hotelling, H., The generalwelfarein relation to problemsof taxation and of railway and utility rates,Econometrica, 6, 242-269, 1938. Loehman,E., and A. Whinston,A new theoryof pricingand decision making for public investment,Bell J. Econ. Manage. Sci., 2, 606625, 1971.

Loehman, E., and A. Whinston, An axiomatic approachto cost allocation for public investment,Public Financ. Q., 2(2), 236-251, 1974. Loehman, E., and A. Whinston,A generalizedcost allocationscheme,

in TheoryandMeasurement ofEconomic Externalities, editedby A. Stevensand Y. Lin, Academic,Orlando, Fla., 87-101, 1976. Loehman, E., J. Orlando, J. Tschirhart, and A. Whinston, Cost allo-

cation for a regionalwastewatertreatmentsystem,Water Resour. Res., 15(2), 193-202, 1979.

Loughlin, J. C., The efficiencyand equity of cost allocation methods for multipurposewater projects,Water Resour.Res., 13(1), 8-14, 1977.

Maschler,M., B. Peleg,and L. S. Shapley,Geometricpropertiesof the kernel,nucleolusand relatedsolutionsconcepts, Math. Oper.Res., 4(4), 1979.

Nash,J. F., The bargainingproblem,Econometrica, 18, 155-162,1950. Owen, G., GameTheory,2nd ed., W. B. Saunders,Philadelphia,Pa., 1982.

Rapoport,A., N-PersonGameTheory,Universityof MichiganPress, Ann Arbor, 1970.

Rawls,J., A Theory of Justice,Harvard University Press,Cambridge, Mass., 1971.

Ruggles,N., Recentdevelopmentin the theory of marginalcostpricing, Rev. Econ. Stud., Ser. 1, 17, 29-46, 1949-1950.

Samuelson,P. A., The pure theory of public expenditures,Rev. Econ. Star., !964.

Schmeidler,D., The nucleolusof a characteristicfunction game, SIAM, J. Appl. Math., 17(6), 1163-1170, 1969.

Shapley,L. S., A Value for N-person games,Ann. Math. Stud.,2(28), 307-318, 1953.

Shapley,L. S., Utility Comparisons and the Theory of Games,The Rand Corporation, Santa Monica, Calif., 1967.

Shapley, L. S.,Cores ofconvex games, Int.J. Game Theory, 1,11•26, 1971.

Shubik,M., GameTheory in the SocialSciences:Conceptsand Solutions, MIT Press,Cambridge, Mass., 1982.

Straffin, P. D., and J.P. Heany, Game theory and the Tennessee Valley Authority, Int. J. GameTheory,10, 35-43, 1981. Suzuki,M., and M. Nakayama,The cost assignmentof the cooperative water resourcedevelopment:A game theoreticalapproach, Manage. Sci.,22(10), 1081-1086, 1976. Yaari, M. E., Rawls, Edgeworth,Shapley,Nash: Theoriesof distributivejusticere-examined,J. Econ.Theory,29, 1-39, 1981. Young, H. P., N. Okada, and T. Hashimoto, Cost allocation in water resourcesdevelopment,Water Resour.Res., 18(1), 463-475, 1982.

A. Dinar and D. Yaron, Levi EshkolSchoolof Agriculture,Hebrew University of Jerusalem,P.O. Box 12, Rehovot 76100, Israel. Y. Kannai, Department of Theoretical Mathematics, Weizman Institute of Science,Rehovot 76100, Israel. (ReceivedJuly 2, 1985;

revisedNovember5, 1985; acceptedNovember 8, 1985.)

1986

Sharing Regional Cooperative Gains From Reusing Effluent for Irrigation ARIEL DINAR

AND DAN YARON

Center for Agricultural EconomicResearchand Department of Agricultural Economicsand Management, Hebrew University of Jerusalem, Rehovot, Israel YAKAR

KANNAI

Departmentof TheoreticalMathematics,WeizmannInstitute of Science,Rehovot,Israel This paper is concernedwith the allocation of cost and benefits from regional cooperation, with respectto reuse of municipal effluent for irrigation at the Ramla region of Israel. An efficientregional solution provides the maximal regional income which has to be redistributed among the town and severalfarms. Different allocationsbased on marginal cost pricing and schemesfrom cooperativegame theory like the core, Shapley value, generalizedShapley value, and nucleolusare applied. The town and farm A have the main additional gains accordingto all allocation schemespresented.Advantagesand disadvantagesof theseallocation schemesare examined in order to suggesta fair and acceptableallocation of the regional cooperative gains. Although no method has been preferred, the marginal cost pricing was found to be unacceptableby the participants. The conclusionis that the theory of cooperativ½gamesmay provide guidelineswhile comparingthe different solutions.

Regional wastewater treatment systemsand regional water projects offer economic returns to the usersand environmental benefits to the region as a whole. While a proposed regional solution may be economicallyefficient,it might be rejected by the prospective users unless an acceptable cost/benefit allocation

scheme is found.

Most of the studiesin this field assumean optimal regional solution and solve the problem of apportioning the common cost of water resource projects between the water consumers or the problem of minimizing the common cost of a regional wastewater plant [Giglio and Wrightington, 1972; Suzuki and Nakayama, 1976' Loughlin, 1977; Loehmanet al., 1979' Straffin and Heaney, 1981' Heaney and Dickinson, 1982' Young et al., 1982]. This paper is concerned with the allocation of cost and benefits from reusing municipal effluent for irrigation. The study was applied to the Ramla region on the coastal plain of Israel. The problem involves a town and several farms which differ in their cropping patterns, cropland area and freshwater allotments. An efficientregional solution which maximizes the region'sincome is provided by a mathematical programming model which was solved by Dinar and Yaron [1986]. In corre-

spondence with,the prevailingregulationsit is assumedthat freshwater from the farm's quotas is not transferable and the only way to compensatethe participants is through direct income transfers (side payments). Thus the problem falls within the category of a transferable utility situation with an important property' the possibility of first solving for the efficient solution (maximal regional income) and at a second stage redistributing income. Such a possibility reduces the conceptualand computational difficulties. In this paper, different cost/benefit allocation solutions are applied to the specificcasestudy and compared. Several allocation schemeslike marginal cost pricing, the core [Owen, 1982], the Shapley value [Shapley, 1953] the generalizedSha,pley value [Loehman and Whinston, 1976], and the nucleolus

Copyright 1986by the AmericanGeophysicalUnion.

[Schmeidler, 1969] are applied to the regional problem discussedby Dinar and Yaron [1986]. The results are evaluated from the viewpoints of fairnessand reasonableness. The following section describesthe nature of the potential regional cooperativeactionsin terms of game theory concepts. Next, several cost allocation schemesare applied to the case study considered.Finally, the results are compared and discussed.While some schemesseem to be preferable to others, the conclusion of the study is that game theoretical approaches do not provide indisputable solutions to the cost benefit allocation problem. TH•

NATURE

OF THE ALLOCATION

The region in the case study is the Ramla region on the coastal plain of Israel. It consistsof three farms (A, B, C), a town, and a potential plant as describedby Dinar and Yaron [1986]. Prevailing conditionsin the region suggestthat (1) no cooperation can occur without the town, (2) farm A must be included in each cooperative organization because of its layout in the region, and (3) no transfer of freshwater quotas among the farms is allowed. The participants in the allocation problem (game) are the town and the three farms A, B, and C. They will be referred to as players 1, 2, 3, and 4, respectively.Let N be the set of all the playersin the game, and S* the set of all feasiblecoalitionsin the game; s* (s*• S*) is a feasiblecoalition in the game if and only if s* _ N and 1, 2 • s*. The noncooperativecoalitionsare

{i), i = 1, 2, 3, 4, and the grandcoalitionis N. The numberoi the a priori feasiblecoalitionsin the region is eight (lessthar

thepotentialnumberof 2'• - 1)' Noncooperative coalitions

{1}, {2}, {3}, {4}

Partial

coalitions

{1, 2}, {1, 2, 3}, {1, 2, 4}

Grand

coalition

{1, 2, 3, 4}

An optimization model [Dinar and Yaron, 1986] is applied to any feasible coalition s* in the region in order to maximize (1)

f,, = _•x + • •,

Paper number 5W0832. 0043-1397/86/005W-0832505.00

ies* i-• l

339

PROBLEM

(1)

340

DINAR ET AL.: SHARING REGIONALGAINS

TABLE 1. GrossIncomeof Playersin the Optimal Solutionfor Different Coalition Combinations,the Value for the DifferentCoalitions,and the IncrementalIncomeValuesa($000) Incremental

Gross Income of Player

Value for

Income for

Coalitions*, Coalition

1

{1) {2) {3) (4) {1, 2) {1, 2, 3) { 1, 2, 4) {1, 2, 3, 4)

-368

2

3

4

1940 1285 440

--410 --498 -- 395 --497

2267 2275 2205 2266

1370 488 488

1365

•_•f•

Coalitions*,

fs.

i • s*

v(x*)

-368 1940 1285 440 1857 3147 2299 3622

-368 1940 1285 440 1571 2857 2011 3297

0 0 0 0 285 290 287 325

aRounded figures; values are constant October 1980 dollars.

Source:Solutionsfrom Dinar and Yaron[1986] for differentpossiblecoalitionsin the region. Column 6 is the summationof columns2-5. Column 7 is the summationof the s*'s players'income while they are acting alone. Column 8 is calculatedaccordingto equation 2 and is the differencebetween 7 and 6.

and to any noncooperative coalition{i), i = 1, 2, 3, 4 in order cooperativegame theory and economictheory. A comprehento maximizefi subjectto the availableproductionfactors, sivediscussionof gametheory can be found in works by Owen

giCeen technology, pricesandenvironmental regulations. Here, fs. is the value of the objectivefunction for coalition s*; •x is the town'sgrossexpenses; • is the ith farm'sgrossincome. Any cooperativeuse of municipal wastewaterin irrigation is linked to the incentive of increasingthe region's farms' income by augmenting their irrigation potential. Regional cooperation is economically feasibleif

E (•i _fi) > • _f•

s*6S*

i•s* i-• l

wherefi is the ith player'sincome(costfor the town) in the noncooperative situation. Income values for each of the possible coalitions and the noncooperative situation in the region are presented in Table 1, column 6. The regional optimization model (1) can be interpreted as a cooperative game with side payments and described in terms of a characteristic

function.

The

value

of the characteristic

function for any coalition expressesthe coalitional gains (in terms of incremental income; above the income in the case of

noncooperation), assuming that the coalition acts efficiently under a given set of constraints(see Table 1). Equation (2) definesthe characteristicfunction of a normalized game where the players have to allocate the additional income from coop-

[1982-1,Rapoport [1970] and Shubik [1982]. The conventional competitive rule that each user of effluent

should pay accordingto the marginal treatment cost might provide a possiblecharge scheme.However, it is not applicable in our case,since a) the number of participantsin the wastewater-effluent"market" is small; they are not anonymous, and agreementsand coalitions among some of the potential participantsare possible;and b) the treatment costsare subjectto increasingeconomiesand decreasingmarginal cost. In such a case,marginal cost pricing leads to a deficit which would need to be coveredeither by the usersor someexternal agency(e.g.,government),or both. The difficultiesencounteredby the marginal cost allocation (pricing) scheme lead to a search for solutions in axiomatic approaches.The essenceof these approachesis that once the usersof a public utility or the participants in a regional enterprise (as in our case) agree on certain axioms believed to satisfyselectedcriteria, such as efficiency,fairness,and reasonableness,the cost allocation solution is uniquely determined [e.g., Hazelwood, 1950-1951; Loehman and Whinston, 1971, 1974; Nash, 1950; $hapley, 1953; and $chmeidler,1969].

eration:

THE CORE

v(s*)-- fs. _ •fi

s*• S*

(2)

i•s*

Here v(s*) is the value of the characteristic function for coalition

s* in terms of incremental

income.

It is assumed that

v((i)) = Ofor i • N. Table 1 suggeststhat the incremental income of a coalition generated by players 1 and 2 is $285,000; the gross income generated by player 2 increasesfrom $1,940,000 to $2,209,000, while the treatment cost increasesby only $41,500. Players 3

and 4's individualentry to (1, 2) increasesthe incomeby a negligibleamount.When they both join (1, 2), the additional net income generated is only $40,000 (325,000- 285,000 in normalized values). These observations imply that farm A is the significantcontributor to the regional income, while farms B and C fall far behind. Common sensesuggeststhat this fact should be consideredin any income allocation scheme. ALLOCATION BACKGROUND

SCHEMES:

AND EMPIRICAL

RESULTS

This section provides the background for the application and comparison of several allocation schemesfrom the field of

The core of an N-cooperative game in the characteristic function form is the set of the game allocation gainswhich is not dominated by any other allocation set. The core fulfills the requirementsof individual rationality, group rationality, and joint efficiency. For details, see Friedman [1979], $hubik [1982], and Owen [1982]. Let a•i be player i's allocation of the incremental income, and let o• = (a• ..... a•n)be the vector of allocations.Using the above notation, the core is the set of income allocations satisfying the following conditions:

o•,> v({i})

• ah> v(s*)

for all it N

(3)

for all s*e S*

(4)

i•s*

=

(5)

ieN

It can be shownthat this regionalgame is convex[Shapley, 1971] and thus has a nonempty core (for details, see Dinar [1984]). Since the regional game has a nonempty core, one can refer to the set of core allocations.The core equations (3')-(5'), which are based on the different a priori feasibleco-

DINAR

ET AL.: SHARING REGIONAL GAINS

alitions and their values,have a solution. Results are presented in Table

1.

coi>0

i= 1,2,3,4

(Y)

co• + co: >_285

co• + co2 + co3 > 290

(4')

co• + co2 + c0½> 287

(D1 + 0)2 + (-O 3 + (-04 = 325

341

core," which is the intersection of all nonempty f• cores [Maschler et al., 1979], and if the core is nonempty, it still satisfiesthe conditionsof individual and group rationality. The regional game has a nonempty core, and it is too large. The least core can be generatedby the solution of (6) and (7),

whichcanbe interpretedas"subsidizing" thepartialcoalitions to make them more attractive, while not refuting the individual and the group rationality condition: Minimize fl

(5')

The core of this game is a four-dimensional polyhedron. A method of calculating the extreme points of the core of a convex game was used [Shapley, 1971]. The essenceof the method is to refer to a sequenceof establishmentof any feasible coalition and to allocate the incremental income generated by each player's entry into an existing coalition. Table 2 presentsthe extreme points of the core of the regional game. The first four columns indicate the contribution of each player when he joins an existing coalition, and these are in effect the maximal possible allocations payable to the players. The right-hand side of the table shows the correspondingcoalition formation sequences.Table 2 suggeststhat the negotiation set consistent with the assumptions underlying the core is quite large. Players 1 and 2 can each claim up to $325,000, the total additional regional income generated by the cooperation, because each is essentialto any coalition. On the other hand, the claims of players 3 and 4 consistentwith the core are bound to

(6)

subject to

• oo•-- f• > v(s*)

s*• S*, s* v•N

(7)

its*

•, ooi= v(N) ieN

f•>0

Note that alternative approaches to reducing the core and obtaining a unique solution can be formulated (Young et al., 1982). The least core solution of (6) and (7) for the regional problem studied is obtained for fl = fl* (seesolutions to (6') and (7') below); however, the optimal vector of allocations of

•o is not unique.Any convexcombinationof two alternative optimal solutions,•xo and •:o, is an optimal solutionalso becauseplayers 1 and 2 are "symmetrical," in the sensethat they both are essentialto any coalition formed and are jointly includedin each of (7)'srestrictions.Players 1 and 2 also gain

be modest,lessthan $38,000and $35,000for 3 and 4, respec-

the same income in the nucleolus solution.

tively. This is consistentwith their modest contributions to the regional cooperation under the present conditions. While logically and morally easy to accept, the core of the regional game is not conclusive because it provides only bounds on the claims of the participants in the regional game and therefore is only partly useful.It may serve as a starting point for the continuation of the analysis;a direct progression is the application of the least core and nucleolus concepts which are discussedin the following section.Also, the computations of the extreme points of the core are incorporated into the Shapley value, discussedlater.

The nucleoluscan be interpreted [Maschler et al., 1979] as the allocation, which minimizes the maximal possible objection to the allocations for any of the coalitions; it is a minimax solution. The extent of the objection is measuredby the

Tim

LEAST CORE AND THE NUCLEOLUS

The core of a cooperative game in the characteristic function form may be empty because certain coalitions provide greater incentives than the grand coalition. Conversely, conditions may arise where the core does exist but is too large and leavesthe cost/benefitallocation problem open for further bargaining (as in this study). A possibleapproach for the derivation of a unique solution in both casesis by reducing or expanding the core (f• core) and then obtaining the "least

TABLE 2. Maximum

Profit

Extreme Points of the Core in the Regional Gamea

2

s*) = Z

v(s*)

i•s*

where•q is the allocationto player i in solutionj to (6) and (7). By a lexicographical ordering, the nucleolus solution can be obtained.

For computational details of the nucleolus, the reader is referred to Maschler et al. [1979]. In the particular case considered here, the nucleolus of the regional game can be obtained immediately after solving for fl from (6) and (7), becauseof the symmetry of players 1 and 2. The linear programming problem to find the least core for the regional game is

Max fl

(6') coi

--f•>_0

(-D 1 --[-(-02

--•

(D1 -'[-(,02-'[-(-03

-- f• __> 290

(D1 -'[-(,02

i=

1,2,3,4

__> 285

(7')

-'[-roe --f• __> 287

Allocation

co• + c02+ c03 + c0½

to Player i ($000) 1

"excess" or the difference

3

4

0

285

5

35

0

285

38

2

0 0 0 285 285 290 325 287

290 325 287 0 0 0 0 0

0 0 5 5 38 0 0 38

35 0 0 35 2 35 0 0

Coalition Formation Sequences Leading to This Allocation 1234

= 325

the solution to which is f•* - 8. Since 1 and 2 are symmetrical players, c0x= co: in the second inequality, which yields the nucleolus:

1243 1324 1342 1423 2134 2143 2314 2341 2413

3124 1432 4123

3214 2431 4213

49.5, 149.5,13, 13) 3142

3412

4132

4312

SHAPLEY VALUE AND THE GENERALIZED

3241

3421

4231

aRounded figures; values are constant October 1980 dollars.

4321

SHAPLEY VALUE

The Shapley value is a uniquely defined solution to an Ncooperative game in the characteristic functional form [Shapley, 1953]. The solution allocation to each player 0• is the weighted average of his contributions to all possiblecoalitions and sequences:

342

DINAR ET AL.' SHARING REGIONALGAINS

and the allocationschemeaccordingto the generalizedShapley value is

•o= (142.7,142.7,21.4,18.2) Calculations were based on (9) and Figure 1. MARGINAL

COST PRICING

An alternative cost/benefit allocation schemeis the margin-

al costpricing.Marginal cost pricingof public utilitiesand other publiclysuppliedgoodshas beenextensivelydiscussed in the economic literature [e.g., Hotellin•7,1938; Ru•I•71es, 1949-1950]. Marginal cost pricing leads to an efficient allocation of resources,but, as indicated by Coase 1-1946'1, SamuelFig. 1. Coalition formation sequencesin the regional cooperative son (1964), and others, the problem .of covering the deficit game. Numbers in circlesdenote coalition and formation sequences. generated under the decreasingmarginal cost situation has Numbers along the branchesdenote the conditional probabilities of not been satisfactorilysolved. moving from one coalition to another. The shadow price of effluent is usedhere as being analogous

to marginalcostpricing.It can be shownthat chargingthe

O,=s,eS, • (n--Is*l)!(Is*l1)!Iv(s*)v(s* - {i})] n!

(8)

where 0i is the allocation to player i, n is the number of players in the game, and Is*l is the number of members in s*.

The Shapleyvalue assumesequal probability for the formation of any coalition of the same size which is theoretically possibleand also considersall the possiblesequencesof formation. That assumption has been criticized by Loehman and Whinston [1976]. As an alternative, they proposed the gener-

alized Shapley value, which differs from the original in two major aspects: 1. It refers only to coalitions that are practically possible, rather than possiblefrom a theoretical combinatorial point of view, and 2. The probability of a coalition occurringdependson the logical sequenceof its formation. In a manner similar to Shapley's, Loehman and Whinston [1976] assignto each player the weighted averageof his contributions to all coalitions realistically formed. The generalized

Shapley allocation to playeri is 0i:

O,= • P(s*,s* -- {i})[v(s*)- v(s*-- {i})]

i eN

(9)

s.eS* ies*

whereP(s*, s* -- {i})-- P(s*ls*-- {i})' P(s* - {i}) is the probability of player/joining coalition s* (seealso Figure 1). The allocation schemeaccordingto the Shapleyvalue is

0ø= (153, 153, 10, 9)

various

months

does not affect the values of the variables

the objective function. It has the effect of income redistribution only, in the form of payment by the farms to the town. Although efficiencyconditions are satisfied in the allocation solution according to the shadow price, there is no guarantee that the solution will be acceptedby the players. Specifically, group rationality is not guaranteed,and the solution might be outsideof the core. This indeed happensin the application of shadow cost pricing to the region studied, as shown below. Also, not all costsare covered (decreasingcost case). Table 3 presentsthe payments by the regional farms to the town when a regional solution is considered.The shadow prices of effluent minus the effluent transportation cost are multiplied by the monthly effluent demand of each farm to yield the farm's chargesfor using the town's effluents I-Dinar and Yaron, 1986]. The allocation scheme according to the shadow prices is calculated

in Table

3'

(188, 93, 14, 30) The allocation schemeaccordingto the shadowpricesof effluent discriminatesin favor of the town and against farm A and also assignshigher additional income to farm C than to farm B.

A strong point against shadow cost pricing is that the cost/ benefit allocation does not lie in the core, as it does not satisfy the conditionsof group rationality. The sum of players 1 and 2's additional income in the grand coalition is smaller than the

TABLE 3. Gross Income, Payment for Effluent, Net Income, and Cooperative Gains in the Regional Solution Accordingto the Shadow Price Allocationa ($000) Player Gross income

with

cooperation Payment for effluent Net

income

1

317

with

2

3

4

Region

2266

1365

488

3622

- 66

- 18

- 233 2032

1299

470

- 368

1940

1285

440

3297

188

93

14

30

325

cooperation Treatment costs Costs not covered

Income/cost with no cooperation

Net cooperativegainsb

497 180

aRoundedfigures' valuesare constantOctober 1980 dollars.

bNet cooperativegains equal grossincome at optimal solution minus payment for effluentminus income with no cooperation.

in

the optimal regional solution, nor does it affect the value of

ies*

coalition

farms for using effluent according to its shadow prices in the

DINAR ET AL.: SHARING REGIONAL GAINS

value of their partial coalition: 188 + 93 < v(1, 2)= 285. As shown by Table 3, the total payments amount to $317,000 out of $497,000 (the costs to the town after de-

ducting the 50% subsidyfrom the treatmentcost), leaving $180,000to be borneby the town.Recallthat the town's costswith no cooperationtotal $368,000;that is, the additional income of the town from cooperation is $188,000. This might be consideredby the town as a reasonablegain. On the other hand, the payment by farm A, $233,000, might be consideredtoo high in comparison to its contribution to the cooperation (Table 1). DISCUSSION AND COMPARISON OF THE ALTERNATIVE ALLOCATION

SCHEMES

A way to compare the different allocation solutionsfor the additional income is to test their efficiencyand acceptability to the players. The efficiencyof the solution is guaranteed becauseit was achieved as a solution to an economic system which optimizesthe use of limited resourcesand compensates the participantswith sidepayments.The core is a solution set which satisfiesindividual and group rationality; the extreme points of the core showthe maximum that any player can get,

343

If acceptabilitymeanssatisfyingthe assumptionsunderlying the core, and one tends to accept this interpretation, then all three allocations, nucleolus, Shapley, and generalized Shapley, are acceptable(the nucleolusby its computation, the Shapley and generalizedShapleybecausethe gameis convex). Regarding the criteria of fairness and reasonableness,no objective way to measure them is known [see also RawIs, 1971; Arrow, 1971; Yaari, 1981; Shapley, 1967]. The discussion will therefore be restricted to several observations

and

questions. The allocation based on shadow prices is quite divergent from the others. The allocation to the town is considerably higher, and to farm A, considerablylower. Furthermore, there is a large difference between the town's and farm A's allocations.In view of the symmetryof theseplayersand the large income generatedby farm A, such inequality might be con-

sidered as being unacceptableat least by farm A. Also, as previouslyshown, the shadow cost pricing allocation vector does not lie in the core, another reason for its rejection by players1 and 2. The

allocations

to the town

and farm

A in all the core

schemesare equal.This reflectstheir symmetryas playersand their bargaining power. However, the increasein income of and for players1 and 2 in somesolutionsit meansgainingall farm A ranges between 7.4 and 7.9% as compared to the the regional additional income. The solutionswhich are contained in the core are a convex combination of the extreme noncooperativesituation,while the decreasein the expensesof points and may serveas a guidelinewhile referringto other the town is in the range of 31-42%. The additional income to criteria. farm A is very small and farm A is much more exposed to uncertainty, using effluent in its production process,than the The four solutions, three of which are contained in the core, town in the wastewater treatment process.A fair allocation are summarized in Table 4. The first three systemsof allocating (Shapley, generalized Shapley, nucleolus) regional should provide a risk premium to farm A and discriminate in coat/incomewere derivedon the basisof a priori acceptable favor of farm A and against the town sincethey are not symaxioms. Do the resultsderived from these axioms (i.e., the metric players in the senseof risk. Other similar questions actual allocations) meet common criteria such as fairness, could be asked with no clear, sharp answers. To conclude, in the case studied, game theoretical apreaso:•ableness,and acceptability? proachesdo not provide an objective,indisputablesolutionto the income distribution problem. The various approachespreTABLE 4. Distribution of the Incremental Income Among the sented,and especiallythe core, provide (1) the bounds on the Participantsin the RegionalCooperationAccordingto Alternative claims by the participants and (2) a better understandingof Allocation Schemes ø the grounds for these claims and of possible objections to them. The marginal costpricing allocation doesnot reflect the Player exact contributions of the participants in the game. All the Allocation

Scheme

Noncooperativeincome Shaply Value Allocation, $000 % of regional incremental

1

2

3

4

-- 368

1940

1285

440

153 47

153 47

10 3

9 3

42 •

8

1

2

142.7 44

142.7 44

21.4 7

18.2 6

39•

7

2

1

149.5 46

149.5 46

13 4

13 4

41 •

8

1

3

188

93

14

30

58

29

4

9

51•

5

1

7

income*

% of income with no

cooperation* Generalized Shapley Value Allocation, $000 % of regional incremental income*

% of income with no

cooperation* Nucleolus

Allocation, $000 % of regional incremental

schemes allocate the main additional

Acknowledgments. This research was supported in part by grant 1-101-79 from BARD, The United States-Israel Agricultural Research and Development Fund. The authors are grateful for commentsfrom K. C. Knapp. REFERENCES

income*

% of income with no

cooperation* Shadow

Prices Procedure

Allocation, $000

% of regional incremental income*

% of income with no

cooperation*

øRoundedfigures,monetary values are constant October 1980 dollars.

•For the town it is the expenses decrementrate.

income to the town and

farm A accordingto their main contribution, farms B and C contributing less, which is reflected in the allocations. This information might be helpful in bargainingaimed at reaching an agreement.Quite similar findingsare also the result of the study by Heany and Dickinson[1982]. A good measure of discretion is needed while evaluating the moral qualifications againstsimplicityof the allocation schemes.

Arrow, K. J., A utilitarian approach to the concept of equality in public expenditures,Q. J. Econ.,85, 409-415, 1971. Coase,R. H., The marginal cost controversy,EconomicaNew Ser., 13, 169-182, 1946.

Dinar, A., Economic analysis of regional wastewater treatment and use of effluent in irrigation and related cost benefit allocation schemes(Hebrew), Ph.D. dissertation,190 pp., Hebrew Univ. of Jerusalem, Jerusalem, Israel, 1984.

Dinar, A., and D. Yaron, Treatment optimization of municipal wastewater and reusefor regional irrigation, Water Resour.Res., this issue, 1986.

Friedman, J. W., Oligopolyand the Theory of Games,Adv. Textbooks

344

DINAR ET AL.: SHARING REGIONAL GAINS

in Econ., vol. 8, edited by C. J. Bliss and M.D. Intrilligator, NorthHolland, Amsterdam, 1979.

Giglio, R. J., and R. Wrightington, Methods for apportioning costs among participantsin regional systems,Water Resour.Res., 8(5), 1133-1144, 1972.

Hazelwood, A., Optimum pricing as applied to telephoneservice,Rev. Econ. Stud., 18, 67-78, 1950-1951. Heany, J.P., and R. E. Dickinson, Methods for apportioningthe cost of a water resourceproject, Water Resour.Res., 18(3), 476-482, 1982.

Hotelling, H., The generalwelfarein relation to problemsof taxation and of railway and utility rates,Econometrica, 6, 242-269, 1938. Loehman,E., and A. Whinston,A new theoryof pricingand decision making for public investment,Bell J. Econ. Manage. Sci., 2, 606625, 1971.

Loehman, E., and A. Whinston, An axiomatic approachto cost allocation for public investment,Public Financ. Q., 2(2), 236-251, 1974. Loehman, E., and A. Whinston,A generalizedcost allocationscheme,

in TheoryandMeasurement ofEconomic Externalities, editedby A. Stevensand Y. Lin, Academic,Orlando, Fla., 87-101, 1976. Loehman, E., J. Orlando, J. Tschirhart, and A. Whinston, Cost allo-

cation for a regionalwastewatertreatmentsystem,Water Resour. Res., 15(2), 193-202, 1979.

Loughlin, J. C., The efficiencyand equity of cost allocation methods for multipurposewater projects,Water Resour.Res., 13(1), 8-14, 1977.

Maschler,M., B. Peleg,and L. S. Shapley,Geometricpropertiesof the kernel,nucleolusand relatedsolutionsconcepts, Math. Oper.Res., 4(4), 1979.

Nash,J. F., The bargainingproblem,Econometrica, 18, 155-162,1950. Owen, G., GameTheory,2nd ed., W. B. Saunders,Philadelphia,Pa., 1982.

Rapoport,A., N-PersonGameTheory,Universityof MichiganPress, Ann Arbor, 1970.

Rawls,J., A Theory of Justice,Harvard University Press,Cambridge, Mass., 1971.

Ruggles,N., Recentdevelopmentin the theory of marginalcostpricing, Rev. Econ. Stud., Ser. 1, 17, 29-46, 1949-1950.

Samuelson,P. A., The pure theory of public expenditures,Rev. Econ. Star., !964.

Schmeidler,D., The nucleolusof a characteristicfunction game, SIAM, J. Appl. Math., 17(6), 1163-1170, 1969.

Shapley,L. S., A Value for N-person games,Ann. Math. Stud.,2(28), 307-318, 1953.

Shapley,L. S., Utility Comparisons and the Theory of Games,The Rand Corporation, Santa Monica, Calif., 1967.

Shapley, L. S.,Cores ofconvex games, Int.J. Game Theory, 1,11•26, 1971.

Shubik,M., GameTheory in the SocialSciences:Conceptsand Solutions, MIT Press,Cambridge, Mass., 1982.

Straffin, P. D., and J.P. Heany, Game theory and the Tennessee Valley Authority, Int. J. GameTheory,10, 35-43, 1981. Suzuki,M., and M. Nakayama,The cost assignmentof the cooperative water resourcedevelopment:A game theoreticalapproach, Manage. Sci.,22(10), 1081-1086, 1976. Yaari, M. E., Rawls, Edgeworth,Shapley,Nash: Theoriesof distributivejusticere-examined,J. Econ.Theory,29, 1-39, 1981. Young, H. P., N. Okada, and T. Hashimoto, Cost allocation in water resourcesdevelopment,Water Resour.Res., 18(1), 463-475, 1982.

A. Dinar and D. Yaron, Levi EshkolSchoolof Agriculture,Hebrew University of Jerusalem,P.O. Box 12, Rehovot 76100, Israel. Y. Kannai, Department of Theoretical Mathematics, Weizman Institute of Science,Rehovot 76100, Israel. (ReceivedJuly 2, 1985;

revisedNovember5, 1985; acceptedNovember 8, 1985.)