Mechanisms for Allocation of Environmental Control Cost: Empirical Tests of Acceptability and Stability

Journal of Environmental Management (1997) 49, 183–203

Mechanisms for Allocation of Environmental Control Cost:Empirical Tests of Acceptability and Stability

Ariel Dinar∗ and Richard E. Howitt†

∗Agriculture and Natural Resources Department, The World Bank, 1818 H Street,Room 5-8039, Washington, D.C. 20433, U.S.A., and †Department of AgriculturalEconomics, University of California, Davis, California 95616, U.S.A.

Received 2 November 1995; accepted 14 December 1995

This paper evaluates schemes for allocation of joint environmental control costamong polluters using, as an example, the drainage pollution problem of theSan Joaquin Valley of California. The analysis is conducted by comparing costallocation schemes’ performance in a regional context. Different schemes(proportional allocation, Nash–Harsanyi, allocation according to marginal cost,Shapley, the nucleolus and the Separable Cost Remaining Benefit principle) areused to allocate regional joint environmental control costs under two extremestates of nature scenarios, resulting in different pollution flows. The resultsprovide clear empirical evidence that regional arrangements may vary withstate of nature. Another important result is that the different allocationschemes have different outcomes in terms of their acceptability to the players,and in terms of their derived stability, as measured by the Shapley–ShubikPower Index, and by the Propensity to Disrupt. Therefore, implementation ofenvironmental policy should also be examined based on its long-termsustainability, and taking into consideration only the cost recovery aspect. 1997 Academic Press Limited

Keywords: allocation schemes, game theory allocations, drainage waterpollution, San Joaquin Valley.

1. Introduction

Implementing environmental policy often requires overlaying or modifying existingresource management arrangements with additional regulatory or incentive structures.The boundaries required for environmental policy are rarely consistent with the existingresource management boundaries, thus the introduction of efficient environmentalpolicies takes on the characteristics of a transboundary problem on a local scale. Whileit is conceptually possible to impose direct regulatory control of environmental impactson local resource management districts, such high-handed policies come at significantpolitical and transaction costs. An example of such a problem is the new complexscheme that was imposed on various existing arrangements for the management of the

0301–4797/97/020183+21 $25.00/0/ev950088 1997 Academic Press Limited

Mechanisms for allocation of environmental control cost184

Murray–Darling River Basin which is contained within four states of Australia (Mus-grave, 1995). A preferable approach to centrally directed environmental policies isto find allocation rules that are transboundary in nature, but amenable to localcooperation.

The literature on methods for evaluating alternative policy mechanisms for en-vironmental control (e.g. Portney, 1991; Russell and Shogren, 1992) often assumes thatthe producing (polluting) units can be dealt with individually, or that different solutionscan be tailored to different operational units. However, in many cases joint action ispreferable. There are two reasons why joint action may be desirable to controlenvironmental effects at practical costs to the producers (polluters). The first is basedon the economies of scale that are found in many methods of abatement or treatmentof environmental damage. It is unlikely that the political unit that controls resources,such as a local water or air quality control district, will be of the optimal size to fullyuse such economies of scale. The second reason is that there are often physical,asymmetric (externality) linkages between the environmental impacts of the decisionunits, or of the region. Internalization of these externalities by joint action can oftengreatly reduce the unit cost of environmental control. In some cases, such as theempirical example analysed in this paper, of agricultural irrigation districts, sub-surfaceflows and drainage restrictions, joint action between the districts may be required toreduce the cost of drainage control to the level that is compatible with continuedcommercial agriculture and sustainable environment.

However, the physical and economic differences between the parties that have toact jointly also mean it is very unlikely that there is a uniquely efficient cost allocationscheme for all parties involved. Invariably, different cost allocation schemes will favorgroups with different scales, financial structures or geographic locations. The resultingcost allocation method will follow a negotiation process between the parties actingjointly.

Borrowing from Schmiesing (1989), the resulting cost allocation method should beviewed by the players as fair. It must also be flexible enough to respond to externalregulations and changes in the state of nature that may affect the resulting pollution.Loehman and Dinar (1994) have shown, for an irrigation externality problem, thatsuperior cost allocation rules may result from the solution of a cooperative gamebetween the parties that are subject to environmental regulation. Rinaldi et al. (1979)examined stable taxation schemes to achieve regional environmental management policywith the existence of an environmental authority which is a stakeholder by itself. Usinga concave environmental damage function with respect to pollution with a known meanand variance, they conclude that if the variance of the environmental cost damage isbig enough, there is no environmental control policy that can result in any stabletaxation scheme. Therefore, a decentralized regional pollution reduction policy wherethe polluters set the institution for their collaboration might be socially preferable.

This paper addresses the problem of defining acceptable mechanisms of en-vironmental control cost allocation among politically independent units when they haveto act jointly to meet environmental standards. In addition, the empirical examplerefers to different states of nature. Since the example discusses water-related pollution,and since pollution from irrigation is highly correlated with water supply, the regionaljoint allocation problem is analysed in two separate sub-problems, each correspondingto a different state of nature.

We use the problem of allocating the fixed and variable costs of treating agricultural

A. Dinar and R. E. Howitt 185

drainage water by several water distribution districts1 in California. The paper empiricallyexamines how different allocation schemes for joint cost to comply with environmentalconstraints fulfill core conditions, and tests their efficiency and stability. The effect onstability of stochastic drainage flows that result from changes in precipitation and waterallocations is also analysed. The empirical results compare standard cost allocationinstitutions to game theory allocations.

2. Allocation approaches

The empirical focus of this paper is based on the problem of complying with en-vironmental standards at a minimum cost, and thus takes the cost of meeting thestandard and its enforcement as given. The boundaries of the environmental policiesand the input cost allocation among different resource management units (water districts)must be resolved by individual action or joint agreement. The prior establishment ofresource management regions means that any collective action will occur as the resultof gaming between the regions. While the game will center on the cost allocationmethod and its effect on district compliance cost, the boundaries of the environmentalmanagement regions will simultaneously be determined by the different coalitions ofresource management regions that emerge from the game.

Three standard schemes are considered for the allocation of the joint costs ofpollution (drainage) control among resource management regions: namely, allocationbased on annual drainage flows, a marginal cost-based allocation and the separablecost remaining method. The likelihood of resource regions forming stable coalitionsfor joint pollution reduction can be measured by comparing the empirical attributesof the standard allocation schemes with game theoretic allocation schemes whoseacceptability and stability can be measured.

Given the stochastic nature of resource availability and pollution control re-quirements, acceptability of the expected levels of pollution control and cost allocationis a necessary, but not sufficient, condition for successful collective action. The coalitionfor reducing environmental compliance cost must also be stable over anticipatedfluctuations of the pollution reduction requirements. That is, the cost allocation schememust be one that does not destabilize the cooperation due to stochastic resource flows,despite differences in size, location and economic values between regions.

2.1.

There are a wide variety of cost allocation schemes for collectively operated facilitiesproposed in the accounting and engineering literature. Biddle and Steinberg (1984)provide a comprehensive review, but for this empirical example we compare the threemain types. An engineering approach where the cost allocation is proportional to thephysical use of the facility, marginal cost analysis based on economic efficiency principlesand the Separable Cost Remaining Benefit (SCRB) principle, where the allocation of

1 While Rosen and Sexton (1993) were interested in the process of decision-making within a water district,the nature of the problem addressed in this paper is the inter-district action. Evidence suggests that waterdistricts’ managers in the Grasslands region in the San Joaquin Valley of California have been authorizedby their districts’ boards to consider, among other actions, a joint inter-district solution (including regionaldrainage water treatment) to drainage regulations. The managers set their own monthly meeting forinformation and data sharing, planning and responding to state and federal regulations.


the fixed investment is based on an equitable division of the cost. In the followingsection, the terms “player” and “user” are used interchangeably.

2.1.1. Allocation based on drainage generation

This allocation scheme simply suggests that each user of the joint facility will be chargedin proportion to the volume of pollution it generates. Thus, the cost to user j is

pj=f N . qj

]jvN

qj ,(1)

where pj is the cost allocated to user j ; f N is cost of the joint facility; and qj is thequantity of pollution generated by user j. This scheme allocates all the joint cost amongall N users.

2.1.2. Allocation based on marginal cost

Allocation on the basis of the marginal cost of the joint facility takes into account themarginal quantities generated by each potential user. Since economies of scale in thejoint cost function exist, the revenues generated by this allocation scheme will not coverthe total cost. Therefore, an additional procedure is necessary to account for theremaining uncovered costs. Usually this can be done using any proportional rule (suchas pollution volume, or volume of production). The formula for this scheme is

bj=∂ f N

∂qj

.]jvN

qj+G f N−C]jvN

∂ f N

∂qj

. f NDH . qj

]jvN

qj ,(2)

where bj is the allocation of the joint cost to user j ; ∂ f N/∂qj is the marginal cost associatedwith user j ; and

G f N−C]jvN

∂ f N

∂qj

.f NDHis the remaining uncovered cost.

2.1.3. Separable Cost Remaining Benefit (SCRB)

The separable cost of user jvN is the incremental cost mj=f N−f N−{j}. The alternate costfor j is the cost f {j} it bears while acting alone, and the remaining benefit to j (afterdeducting the separable cost) is rj=f {j}−mj . The SCRB assigns the joint cost accordingto the following formula (Young, 1985)

jj=mj+rj

]jvN

rjC f N−]

jvN

mjD . (3)


In other words, each user pays their separable cost, and the “non-separable costs”f N−RN mj are then allocated in proportion to the remaining benefits, assuming that allremaining benefits rj are non-negative for each player.

2.2.

Given the initial conditions of voluntary collective action, and the prior establishmentof independent regional resource management institutions, the problem of allocatingthe joint costs of a regional drainage water treatment facility is modelled as a gameamong the regional units. The approach is empirically general for other environmentalcontrol situations, such as toxic effluent cleanup and regional air pollution control.Based on the empirical situation, we assume that the environmental regulations facingthe region are already in place and that the water districts in the region need to complywith them by treating the drainage water produced on each district’s land2. If a waterdistrict chooses not to cooperate (not to participate in the investment and the operationof a treatment facility), it faces a certain outcome resulting from the operation of aprivate treatment facility or alternative measures needed to meet the regulations. If thewater districts choose to cooperate, they may benefit from economies of scale embodiedin a larger capacity joint treatment facility with lower average treatment costs comparedto private actions. Some water districts may cooperate while others may choose not tocooperate, depending on the degree to which they can reduce their treatment cost. Thelarger the economies of scale, the bigger the incentive for cooperation.

Let N be the set of all water districts (players) in the region, S (SgN ) the set ofall feasible coalitions in the game, and s (svS ) a feasible coalition in the game. Thenon-cooperative coalitions are { j}, j=1, 2, . . . , n, and the grand coalition is {N}.

Assuming that the players’ objective is to minimize their treatment cost, let f s bethe treatment cost of coalition s, and f {j} be the treatment cost of the j ’s member innon-cooperation. A necessary condition for regional cooperation is that the jointtreatment cost will be less than the sum of the individual treatment costs

f s≤]jvs

f {j} ! svSgN. (4)

The regional savings that are allocated among the players are

]jvs

f {j}−f s≥0 ! svSgN. (5)

Equation (5) can be interpreted as a cooperative game, with side payments, and bedescribed in terms of a characteristic function. The value of a characteristic functionfor any coalition expresses the coalition expenses

v(s)=f s ! svSgN. (6)

Four game theory allocation schemes will be considered: the core, the Shapleyvalue, the nucleolus and the Nash–Harsanyi.

2 Several combinations exist for each district to comply with these regulations. They may include reductionof the agricultural activity, improvement of the water-related technologies in the production process and on-farm treatment facilities.


2.2.1. The core

The core of an n-cooperative game in the characteristic function form is a set of gameallocation gains that is not dominated by any other allocation set. The core providesa bound for the maximum (or minimum in terms of cost) allocation each player mayrequest. In this respect, it is an overall solution for many allocation schemes that arecontained within the core. The core fulfills requirements for individual and grouprationality, and for joint efficiency (Shubik, 1982).

The game theory analysis presented here is conducted under the assumption thatthe polluters (players) in the game are economically rational. This means that thedecision of each player to join a given coalition is voluntary, and is based on theminimal cost they bear by joining that coalition. Let xj be player j ’s core allocation ofthe cost from the regional game. The core equations (for the case of a cost allocationgame) are

xj≤v({ j} ! jvN, (7)

]jvs

xj≤v(s) ! svS, (8)

]jvN

xj=v(N ). (9)

Equation (7) fulfills the conditions for individual rationality—that the cooperativesolution for each player is preferred to the non-cooperation case; Equation (8) fulfillsthe group rationality conditions—that the cooperative allocation to any combinationof players is preferred to any allocation in any sub-coalition they can establish; andEquation (9) fulfills the efficiency condition—that the cost of the joint facility will befully covered by the grand coalition participants. The system [Equations (7)–(9) ] hasmore than one allocation solution. A method of calculating the extreme points of thecore (Shapley, 1971) provides the incremental contributions of each player when joiningany existing coalition, and assigns these contributions to that player.

2.2.2. The nucleolus

The nucleolus (Schmeidler, 1969) is a single point solution that always exists (if thecore is non-empty) and minimizes the dissatisfaction of the most dissatisfied coalition.To obtain the nucleolus, we define the e-core of the game v to be the set of allocationsthat would be in the core if each coalition were given a subsidy at the level of e. Byvarying e one finds the smallest non-empty e-core (called the least core). The least coreis the intersection of all e-cores. The least core for a cost allocation game satifies

min e,

subject to

]jvs

xj≤v(s)+e ! svS, (10)


]jvN

xj=v(N ), (11)

e≤≥0. (12)

The solution to Equations (10)–(12) may provide the nucleolus (as a single solution)but it may also provide several individual cost allocations xj for the same value of efor each coalition s. In this case, we define the excess function e(x, s) for each s (thatmeasures how much less it costs a coalition to act alone) and in a lexicographicalprocess (Schmeidler, 1969) obtains the nucleolus, for which the value of the smallestexcess e(x, s) is as large as possible.

The interpretation of e is interesting. It can be used as a tax or a subsidy to changethe size of the core. If the core is empty then e (e<0) is an “organizational fee” for theplayers in sub-coalitions, so they prefer the grand coalition. If the core is too big, emight reduce it (e>0) by subsidizing sub-coalitions. The nucleolus is always in the coreif it exists.

2.2.3. The Shapley value

The Shapley value (Shapley, 1953) scheme allocates hj to each player based on theweighted average of their contributions to all possible coalitions and sequences. In thecalculation, an equal probability is assigned for the formation of any coalition of thesame size, assuming all possible sequences of formation.

The Shapley value is calculated as

hj=]svS

,jvs

(n−|s | )!( |s |−1)!n!

[v(s)−v(s−{ j})] ! jvN, (13)

where n is the number of players in the game, and |s | is the number of members incoalition s.

2.2.4. The Nash–Harsanyi (N–H) solution

The N–H solution (Harsanyi, 1959) to an n-person bargaining game is a modificationto the 2-player Nash solution (Nash, 1953). This solution concept maximizes the productof the grand coalition members’ additional utilities (income, or savings in this paper)from cooperation compared to the non-cooperation case, subject to core conditions,by equating the utility gains of all players. The N–H solution satisfies the Nash axioms(Nash, 1953), it is unique and it is contained in the core (if it exists). The solutionmight provide unfair allocations if there are big utility differences between the players(e.g. very rich player and very poor player).

max\jvN

( f {j}−hj ), (14)


subject to the core conditions

hj≤f {j} ! jvN, (15)

]jvs

hj≤f s ! svS, (16)

]jvN

hj=f N, (17)

where hj is the N–H allocation that satisfies efficiency and individual rationalityconditions.

2.3.

The fulfillment of the core conditions for an allocation scheme is a necessary conditionfor its acceptability by the players. Thus, solutions not included in the core are alsonot stable. Although an allocation scheme may fulfill the core requirements for theregional game, it still may not be accepted by some players that might view it asrelatively unfair compared to another allocation. Allocations which are viewed as unfairby some players are less stable. Some players might threaten to leave the grand coalitionand form sub-coalitions because of their critical position in the grand coalition. Thestability of any solution is important given the existence of fixed investments, and amore stable solution might be preferred even if it is harder to implement.

A simple measure for stability of an allocation scheme is the number of playersthat prefer it over other schemes. In the cost allocation game, a player will prefer ascheme that assigns her or him the least cost.

Shapley and Shubik (1954) suggest a method of measuring power in voting games:“. . . the power of an individual member depends on the chance he has of being criticalto the success of a winning coalition”. Loehman et al. (1979) used an ex-post approachto measure power in a cooperative game. Their index can be interpreted in a similarway to the Shapley–Shubik Power Index. The power index (ai ) compares the gains toa player with the gains to the coalition. The power index (ai ) is

ai=xi−v({i})

]jvN

(xj−v({ j })), ivN ;]

ivN

ai=1, (18)

where xi is the solution allocation for player i. This index is also used in Willliams(1988) in the equation that calculates the allocation using the Alternative Cost AvoidedMethod (Ransmeier, 1942).

The ai power index can also be used to calculate a measure of stability for thevarious solutions. The power index is calculated separately for each state of the nature.If the power is distributed more or less equally among the players, then the coalitionis more likely to be stable. The stability measure (Saora/a) is simply the coefficient ofvariation calculated over all players in a given allocation solution (0≤Sa≤1) and stateof the nature. The greater the value of Sa the larger the instability of the allocationsolution.


In the following section, the concept of adding a cooperatively financed and manageddrainage water treatment plant to a regional water system is empirically tested. In theregion water resource allocation is managed by four distinct and independent watersupply districts. The three cost allocation methods are tested for acceptability by theirpresence or absence in the core solution, and for stability by comparison of gametheoretic measures under the extremes of high and low drainage flows.

Gately (1974) introduced the concept of “propensity to disrupt” the grand coalitionas the ratio of how much the other players would lose if player i refuses to cooperateto how much it would lose if it refuses to cooperate. This concept was also applied byGately (1974) to a problem of investment in electric power in India, and modified andapplied (for N>3) by Straffin and Heaney (1981) to the case of the Tennessee Valley.

According to Straffin and Heaney, the propensity of i to disrupt an allocation (x1,. . . , xn) is:

Gi=]jri

xj−v(N−i )

xi−v({i})=

v(N )−v(N−i )xi

−1, ivN. (19)

Gi measures the loss to members in coalition N−i compared to player i ’s loss, if playeri disrupts the grand coalition. Negative values reflect enthusiasm for the allocation.When this ratio is positive and large, player i has a powerful threat to disrupt the grandcoalition unless his allocation is improved. Using this concept one can reduce the setof mutually acceptable allocations by eliminating any core imputations for which anyplayer’s propensity to disrupt is higher than a certain value.

3. An empirical application to regional water quality control

3.1.

In many locations, drainage water resulting from irrigated agriculture has elevatedsalinity or selenium levels that are hazardous to the environment and human beings(SJVDP, 1990). Reduction of such pollution problems can be addressed at field, farmand regional levels and include, among many other measures, the biological andchemical treatment of the drainage water.

Treatment facilities display significant economies of scale, which, in addition to theinternalization of externalities within the polluting water districts, provide incentivesfor regional cooperation in the construction and operation of these facilities. Recently,treatment processes have been technically evaluated by several water districts and urbancommunities in an area affected by drainage problems in the San Joaquin Valley ofCalifornia. Dinar and Howitt (1993) used the information accumulated in these evalu-ations to estimate cost functions for these treatment facilities (CH2MHILL, 1986; seealso Ergas et al., 1990; Gerhardt and Oswald, 1990; Hanna and Kipps, 1990). The costfunction for the Algal treatment process is used here

C=e0·377Q0·789, (20)

where C is the total annual cost, e0·377 is the fixed cost component and Q is the annualcapacity of the plant.

The cost function has strong economies of scale in the capacity of the treatment


T 1. Range of annual drainage flow of four waterdistricts in the San Joaquin Valley (1986–1991)

Water district Lowa High(AF) (AF)

A 2 500 4 500B 3 000 5 000C 2 000 4 000D 14 000 35 000Total 21 500 48 500

a Affected mainly by drought conditions and reduced surfacewater allotments.

Source: Interviews with water districts’ managers.

facility, thus regional cooperation in the form of joint drainage water treatment iseconomically preferable to individual actions (given that all collective costs are embodiedin C ). For purposes of this analysis it is assumed that the incremental coordinationcost of the regional cooperative solution is negligible.

3.2.

Four water districts on the west side of the San Joaquin Valley that face severe drainageproblems and drainage disposal restrictions were included in the regional game. Thesedistricts comprise 114 000 irrigated acres (1 acre=0·4 hectare) of which 37 000 aredrained. In a normal water supply year they purchase, through long-term contracts,287 000 acre feet (1 acre foot=1235 m3) of water from the Bureau of Reclamationwhich is used on 246 farms producing a variety of agricultural crops. In the agriculturalproduction process, while irrigation water is applied, drainage water is generated dueto a partially impermeable clay layer that prevents the deep percolation of excess water.The water table then rises and needs to be managed in order to allow continuedproduction from the fields. Therefore, drain tiles are installed and the drainage wateris collected and handled by each district separately. With all other factors constant, theamount of drainage water generated on a field is determined by the amount of waterapplied (rainfall is negligible). The annual quantity of drainage water disposed of fromthese districts (Table 1) varies with the irrigation water that is available to the districts.In a normal year when surface water supply is at the contract level, drainage water isabout 20–25% of the irrigated water.

In the last 6 years the water districts faced cuts of up to 50% of their annual watersupply allotment because of the drought situation in California. As a result, drainagegeneration has fluctuated and the total annual amount of drainage disposed variedfrom 20 000 AF to 50 000 AF (Table 1). The range of drainage water volumes, reportedby the districts, reflects the administrative cuts in surface water supply to the regionfollowing the drought. Changes in water supply generate different quantities of drainage,and affect the economies of scale of a possible regional treatment facility. Therefore,changing water volumes will be used to evaluate the effect of high and low drainageflows on the stability of the cooperative solution.

To comply with drainage reduction regulations imposed by state and federal agencies,several operational arrangements including a regional joint treatment facility must be


T 2. Annual treatment cost values for different coalitional settings and drainage flows

High drainage flow Low drainage flow

Annual drainage Annual drainageCoalition volume Annual cost volume Annual cost

(AF) (000 $) (AF) (000 $)

{A} 4 500 1 112 2 500 699{B} 5 000 1 208 3 000 807{C} 4 000 1 013 2 000 586{C} 35 500 5 610 14 000 2 722{AB} 9 500 2 005 5 500 1 303{AC} 8 500 1 836 4 500 1 112{AD} 39 500 6 172 16 500 3 099{BC} 9 000 1 921 5 000 1 208{BD} 40 000 6 233 17 000 3 173{CD} 39 000 6 110 16 000 3 025{ABC} 13 500 2 645 7 500 1 664{ABD} 44 500 6 780 19 500 3 536{ACD} 43 500 6 660 18 500 3 392{BCD} 44 000 6 720 19 000 3 464{ABCD} 48 500 7 257 21 500 3 820

considered by the water districts. Agreement on the allocation of the joint costs of sucha treatment plant, and its optimal size and siting is a major obstacle to optimal pollutioncontrol. The possible coalitions in the regional game are all possible permutations ofcoalitions among water districts. The coalitions of districts and the associated annualcosts are presented in Table 2 for the two drainage flow scenarios. The characteristicfunction values for all possible coalitions in the case of high and low drainage flowsare presented in Table 3. The core of the regional game will be presented first since allthe allocation solutions are evaluated with regard to the core allocations.

3.2.1. The core of the regional game

The core equations for the regional game in the case of high drainage flow (coreequations for the low drainage flow are not presented, but can be derived from datain Table 3) are

xA≤1112,

xB≤1208,

xC≤1013,

xD≤5610,

xA+xB≤2005,

xA+xC≤1836,

xA+xD≤6172,

xB+xC≤1921,


xB+xD≤6233,

xC+xD≤6110,

xA+xB+xC≤2645,

xA+xB+xD≤6780,

xB+xC+xD≤6720,

xA+xC+xD≤6660,

xA+xB+xC+xD=7257.

T 3. Characteristic function values for different coalitions

Incremental cost per player Incremental(000 $) Value for gains for

coalition s coalition sCoalition A B C D f s v(s)]

jvs

f j

High drainage flow

{A} 1112 1112 1112 0{B} 1208 1208 1208 0{C} 1013 1013 1013 0{D} 5610 5610 5610 0{AB} 1112 1208 2005 2320 315{AC} 1112 1013 1836 2125 289{AD} 1112 5610 6172 6722 550{BC} 1208 1013 1921 2221 300{BD} 1208 5610 6233 6818 585{CD} 1013 5610 6110 6623 513{ABC} 1112 1208 1013 2645 3333 688{ABD} 1112 1208 5610 6780 7930 1150{BCD} 1208 1013 5610 6720 7831 1111{ACD} 1112 1013 5610 6660 7735 1075{ABDC} 1112 1208 1013 5610 7257 8943 1686

Low drainage flow

{A} 699 699 699 0{B} 807 807 807 0{C} 586 586 586 0{D} 2722 2722 2722 0{AB} 699 807 1303 1506 203{AC} 699 586 1112 1285 173{AD} 699 2722 3099 3421 322{BC} 807 586 1208 1393 185{BD} 807 2722 3173 3529 356{CD} 586 2722 3025 3308 283{ABC} 699 807 586 1664 2092 428{ABD} 699 807 2722 3536 4228 692{BCD} 807 586 2722 3392 4115 723{ACD} 699 586 2722 3464 4007 543{ABDC} 699 807 586 2722 3820 4814 994


T 4. Extreme points of the core in the regional game

Maximum incremental cost allocation to player j(000 $) Coalition

formationHigh drainage flow Low drainage flow sequences

leading toA B C D A B C D this allocation

1112 893 640 4612 699 604 303 2156 ABCD1112 893 477 4775 699 604 284 2233 ABDC1112 809 724 4612 699 552 413 2156 ACBD1112 597 724 4824 699 428 413 2280 ACDB1112 608 477 5060 699 437 284 2400 ADBC1112 597 548 5060 699 428 293 2400 ADCB797 1208 640 4612 496 807 361 2156 BACD797 1208 477 4775 496 807 284 2233 BADC721 1208 713 4612 456 807 401 2156 BCAD537 1208 713 4799 356 807 401 2256 BCDA547 1208 477 5025 363 807 284 2366 BDAC537 1208 487 5025 356 807 291 2366 BDCA823 809 1013 4612 526 552 586 2156 CABD823 597 1013 4824 526 428 586 2280 CADB724 908 1013 4612 456 622 586 2156 CBAD537 908 1013 4799 356 622 586 2256 CBDA550 597 1013 5097 367 428 586 2439 CDAB537 610 1013 5097 356 439 586 2439 CDBA562 608 477 5610 377 437 284 2722 DABC562 482 603 5610 377 428 293 2722 DACB547 623 417 5610 363 451 284 2722 DBAC672 623 487 5610 356 451 291 2722 DBCA550 597 500 5610 367 428 303 2722 DCAB537 610 500 5610 356 439 303 2722 DCBA

This system of inequalities may have more than one solution (allocation). Onepossible solution that satisfies all constraints for the case of high drainage flow is x=(797, 1208, 477, 4775). A method to calculate the extreme points of the core thatcontains all possible core solutions (Shapley, 1971) was applied for the cases of highand low drainage flows (Table 4). The two sets of four left-hand columns in Table 4are the maximum possible allocations to the players in the case of high and low drainageflows, respectively, while the right-hand side of the table shows the coalition formationsequences leading to these allocations. In some games, however, there are severalcoalition formation sequences leading to the same cost allocation.

The extreme points of the core are calculated in the following way (for example,see the sequence ABCD): player A joins coalition {£} (the empty set) and createscoalition {A} associated with cost of 1112, hence xA=1112. Then player B joins {A}to create coalition {AB} whose cost is v(A, B)=2005. Therefore, xB=v(A, B)−v(A)=2005−1112=893. Player C then joins coalition {A, B} and forms {A, B, C} whose costis v(A, B, C)=2645. Therefore, player C’s share in the joint cost will be v(A, B, C)−v(A,B)=2645−2005=640. Player D joins coalition {A, B, C} to create the grand coalition{A, B, C, D} whose cost is v(A, B, C, D)=7252. D’s share in the joint cost is thereforev(A, B, C, D)−v(A, B, C)=7257−2645=4612. Notice that the allocation suggested by


T 5. Allocation of the joint cost based on marginal cost

Player A B C D Total

High drainage flow

Q [AF/yr] 4 500 5 000 4 000 35 000 48 500MC [000 $/AF] 0·195 0·190 0·199 0·126Q×MC [000 $] 877·0 950·0 796·0 4 410·0 7 033·0Additional chargea to cover 27·9 30·3 25·3 140·5 224·0

differenceFinal allocation [000 $] 904·9 980·3 821·3 4 550·0 7257·0

Low drainage flow

Q [AF/yr] 2 500 3 000 2 000 14 000 21 500MC [000 $/AF] 0·220 0·212 0·231 0·153Q×MC [000 $] 551·7 637·0 462·6 2 147·8 3 799·1Additional chargea to cover 2·4 2·6 2·8 13·2 20·9

differenceFinal allocation [000 $] 554·1 639·6 465·4 2 161·0 3 820·0

a See text for explanation.

the solution to the equation system above matches the point resulting from the sequenceBACD.

The extreme points of the core provide an ordering of the players’ preferences.Player A is likely to prefer allocations closer to 537 than to 1112. Player B will preferallocations closer to 482 than to 1208. Player C will prefer allocations closer to 477than to 1013 and player D will prefer allocations closer to 4612 than to 5610. A similarprocedure for the case of low drainage flow yields the allocation x=(699, 604, 361,2156).

3.2.2. Allocation based on the annual drainage flow

This simple allocation scheme to calculate the joint cost share uses each player’s sharein the annual drainage volume. Player A’s cost will be 7257×[4500/48 500]=673·3;player B’s cost will be 7257×[5000/48 500]=748·2; player C’s cost will be 7257×[4000/48 500]=598·5; and player D’s cost will be 7257×[35 000/48 500]=5237·0. In the caseof high drainage flow, the allocation based on proportional use of the treatment facility,p=(673·3, 748·2, 598·5, 5237·0), covers the joint cost and is also in the core. However,there is no guarantee, in general, that allocation based on proportional use of a jointfacility will satisfy core requirements. The allocation in the case of the low drainageflow is p=(444·2, 533·0, 355·3, 2487·5).

3.2.3. Allocation based on marginal cost

Using the cost function for the Algal process, and the annual drainage volume by eachuser from Table 2, a marginal cost value of the treatment facility can be calculated foreach player (Table 5). It is clear that the users will not be willing to pay a share higherthan the marginal cost. When the marginal cost function is decreasing as in the caseof the treatment facility (because of economies of scale), cost charges based on the


T 6. Allocation of the joint cost based on the SCRB method

Player A B C D Total

High drainage flow

Alternate cost ( f j ) 1112 1208 1013 5610 8943Separable cost (mj ) 537 597 477 4612 6223Remaining benefits (rj ) 575 611 536 998 2720Allocation of non-separable 218·6 232·3 203·7 379·4 1034

costFinal allocation [000 $] 755·6 829·3 680·7 4991·4 7257

Low drainage flow

Alternate cost ( f j ) 699 807 586 2722 4814Separable cost (mj ) 356 428 284 2156 3224Remaining benefits (rj ) 343 379 302 566 1590Allocation of non-separable 128·6 142·1 113·2 212·1 596

costFinal allocation [000 $] 484·6 570·1 397·2 2368·1 3820

marginal cost will not cover the joint cost. In this case an additional charge is neededif full coverage is required. Results in Table 5 show a difference of $224 000 betweenthe joint cost and the charges to the users. Additional charges can be made usingvarious schemes. We allocated the uncovered cost in proportion to the existing shares.The adjusted marginal cost allocation b=821·3, 904·9, 980·3, 4550·5) now covers thejoint cost but does not fulfill one of the core conditions (xA+xB+xC=2716·5>2645·0).A similar procedure yields the allocation b=(554·1, 639·6, 465·4, 2161·0) for the lowdrainage flow, which is included in the core.

3.2.4. Allocation based on the Separable Cost Remaining Benefit (SCRB) method

Using the definitions for the alternate cost and the separable cost, the remaining benefitcan be calculated for each player (Table 6). Then the remaining non-separable cost iscalculated using the second term of Equation (3). In this high flow case, f N−RN mj=1034 (=7257−6223) is the total non-allocated separable cost; and rj /RN rj is the ratioof the j th player in the remaining benefits. The allocation according to the SCRBmethod j=(755·6, 829·3, 680·7, 4991·4) does cover the joint cost and fulfills the coreconditions. A similar procedure yields the allocation j=(484·6, 570·1, 397·2, 2368·1)for low drainage flow.

3.2.5. The nucleolus

The nucleolus of the regional game is obtained by calculating the e-core—by applyingEquations (10)–(12) to the specific core equations of the high drainage flow scenarioand the low drainage flow scenario. In the case of high drainage flows the nucleolusallocation is l=(795, 855, 735, 4870), with e=258. In the case of the low drainageflows, the nucleolus allocation is l=(561, 552, 417, 2289) and e=133.


3.2.6. The Shapley value

Using Equation (13) and the results shown in Table 3, the Shapley allocation of theregional game in the case of high drainage flows is h=(745·7, 822·1’, 627·7, 5061·5). Inthe case of low drainage flows, the Shapley allocation (using the results in Table 3) ish=(502·9, 543·6, 389·3, 2384·2). The Shapley allocation satisfies the core requirements,and is fair and efficient.

3.2.7. The Nash–Harsanyi (N–H) solution

Using Equations (14)–(17) and the data in Table 3 we obtain the non-linear programmingproblem

max ( f {1}−h1) . ( f {2}−h2) . ( f {3}−h3) . ( f {4}−h4),

subject to the core conditions as specified in Equation (21) where h replaces x. Thesolutions for the high and low drainage flows (Table 3) are, respectively, h=(690·5,786·5, 591·5, 5188·5) and h=(450·5, 558·5, 337·5, 2473·5).

3.3.

Three conventional allocation methods and four game theory solutions were appliedto the problem of allocating the joint cost of a regional water treatment facility. Theobjective is to allocate the joint cost such that the solution will be acceptable to allplayers (users) and that the efficiency condition will hold, resulting in a stable collectiveagreement. Table 6 summarizes the allocation schemes.

The Shapley value and the SCRB suggest allocations that are very much similar inthe case of high drainage flow scenarios, and the Shapley and the proportional allocationsuggest similar values in the case of of low drainage flow scenarios. Using theseallocations, players A, B and C pay very similar shares of the joint cost and theirrelative reduction in cost compared to the non-cooperation situation is also similar.Player D is probably less satisfied with the Shapley and proportional use allocations.This player “gains” only 7% in lower fees from cooperation compared to 40% for eachother player in the game. Player D would prefer the marginal cost allocation wherethe contribution to the game is more substantial.

Table 7 presents the ratios of the cost allocation in the grand coalition comparedto the non-cooperation. It is clear from the results in this table that players A, B andC gain more from cooperation, in terms of percentage cost reduction, than player Din all allocation schemes, except the marginal cost. So, player D, who contributes tothe cooperative gains may consider defection. Comparison of the high and low drainageflow scenarios (Table 7) suggest that player D’s share in the joint cost always increaseswhen capacity increases. For players B and C it happened in three out of five allocations,and player A’s share decreases as capacity increases.

3.3.1. Acceptability and stability

Testing for core conditions (Table 7) reveals that the marginal cost allocation is notincluded in the core in the case of high drainage flows (the condition xA+xB+xC≤2645does not hold). All allocation solutions fulfill core requirements in the case of low


T 7. Comparison of allocation schemes used in the regional game (in parentheses are costshares relative to non-cooperation)

Player

Scheme A B C D In core

High drainage flow

Shapley 745·7 (66) 822·1 (68) 627·7 (62) 5061·5 (90) YesNucleolus 797·0 (71) 855·0 (71) 735·0 (72) 4870·0 (87) YesNash–Harsanyi 690·5 (62) 786·5 (65) 591·5 (58) 5188·5 (92) YesProportional use 673·3 (60) 748·2 (62) 598·5 (59) 5237·0 (93) YesMarginal costa 904·9 (81) 980·3 (81) 821·3 (81) 4550·5 (81) NoSCRB 755·6 (68) 829·3 (69) 680·7 (67) 4991·4 (89) Yes

Low drainage flow

Shapley 502·9 (72) 543·6 (66) 389·3 (66) 2384·2 (87) YesNucleolus 562·0 (80) 552·0 (68) 417·0 (71) 2289·0 (84) YesNash–Harsanyi 450·5 (64) 558·5 (69) 337·5 (57) 2473·5 (91) YesProportional use 444·2 (63) 533·0 (66) 355·3 (60) 2487·5 (91) YesMarginal costa 554·1 (79) 639·6 (79) 465·4 (79) 2161·0 (79) YesSCRB 484·6 (69) 570·1 (70) 397·2 (68) 2368·1 (86) Yes

a With adjustments.

drainage flow. Using the core argument, the marginal cost allocation will not be acceptedby the players.

Comparing the different schemes for the reduction in the cost share of each playersuggests that in the case of high drainage flow, players A, B and C prefer the proportionalallocation scheme for its least cost share (60, 62 and 59%, respectively, of the costcompared to non-cooperation), and player D prefers the marginal cost allocation (81%compared to non-cooperation). In the case of low drainage flows, players A andC prefer the proportional allocation (63 and 60%, respectively, compared to non-cooperation), player B is indifferent between the Shapley and the proportional (66%compared to non-cooperation) and player D prefers the marginal cost (79% comparedto non-cooperation).

Table 8 presents the power indexes for the players, and the stability index for thedifferent allocation schemes. In both the high and low drainage flow scenarios, theleast stable scheme is the marginal cost and the most stable scheme is the proportionalallocation. The other schemes consistently keep the same order of stability over bothdrainage flow scenarios, with Shapley, SCRB and the nucleolus being in the middlebetween the proportional allocation and the marginal cost allocation.

Table 9 presents the propensity to disrupt calculations for each player using the sixscheme allocations. In general, all players have an interest in keeping the grand coalition.Among all four players, D is the one with the highest propensity to keep the coalitionunder all allocation schemes and drainage flow scenarios. Other players vary in theirstability of propensity to disrupt index. Player C has the lowest values of propensityto disrupt, so this player is more likely to consider defection under certain conditions.

Figure 1 compares the changes in stability for the two drainage flow scenarios. Anypoint below the 45° line indicates a decreased stability as capacity increases, and pointsabove it indicate decreased stability as capacity decreases. It is interesting that the


T 8. Power and stability indexes

Power indexes

PlayerStability

Scheme A B C D indexa

High drainage flow

Shapley 0·217 0·229 0·228 0·325 0·175Nucleolus 0·188 0·209 0·165 0·439 0·427Nash–Harsanyi 0·250 0·250 0·250 0·250 0·000Proportional use 0·260 0·273 0·246 0·221 0·077Marginal costa 0·122 0·135 0·113 0·628 0·876SCRB 0·211 0·224 0·197 0·366 0·268

Low drainage flow

Shapley 0·197 0·265 0·198 0·339 0·234Nucleolus 0·138 0·256 0·170 0·435 0·461Nash–Harsanyi 0·250 0·250 0·250 0·250 0·000Proportional use 0·256 0·276 0·232 0·236 0·070Marginal costa 0·145 0·168 0·122 0·565 0·730SCRB 0·216 0·238 0·190 0·356 0·254

a See text for explanation.

T 9. Propensity to disrupt values as calculated for the various allocation schemes

Scheme

Proportional MarginalPlayer Shapley Nucleolus Nash–Harsanyi use cost SCRB

High drainage flow

A −0·30 −0·28 −0·17 −0·15 −0·36 −0·24B −0·25 −0·28 −0·22 −0·10 −0·37 −0·26C −0·15 −0·27 −0·09 −0·10 −0·35 −0·21D −0·80 −0·79 −0·81 −0·81 −0·78 −0·80

Low drainage flow

A −0·46 −0·52 −0·40 −0·39 −0·51 −0·44B −0·18 −0·19 −0·28 −0·16 −0·30 −0·21C −0·22 −0·27 −0·10 −0·15 −0·35 −0·24D −0·76 −0·75 −0·77 −0·77 −0·73 −0·76

proportional allocation’s stability is quite insensitive to the capacity of the joint facility.The two game theory allocations (Shapley and nucleolus) increase in their stability ascapacity increases, and the other engineering schemes’ (marginal cost and SCRB)stability decreases as capacity increases.


0.9

0.9

Stability index for the high flow game

Sta

bili

ty in

dex

for

the

low

flo

w g

ame

0.3

Proportional

Nucleolus

Shapley

Nash–Harsanyi

0.7

0.5

0.3

0.1

0.1 0.5 0.7

SCRB

–0.1

MarginalCost

45°

Figure 1. Stability index of various solutions in the high flow game and low flow game.

4. Discussion

The ability to meet existing or future environmental standards at an acceptable costoften requires regional approaches due to externalities or treatment economies ofscale. Cooperative solutions to regional pollution problems have lower political andtransaction costs than non-cooperative solutions. Several questions need to be addressedin the case of regional actions to meet environmental regulations. First, the technicalfeasibility of each cooperative arrangement; second, the economic feasibility comparedto other alternatives; and, third, the regional setting in which this solution will operate.Assuming the existence of technical and economic feasibility, acceptability and stabilityof the cooperative solutions are the conditions for their success.

This paper attempts to define acceptable and stable mechanisms of environmentalcontrol cost allocations among established resource management regions. Differentschemes and states of nature that may affect the regional arrangements are tested,using a drainage problem in the San Joaquin Valley of California. In this paper wedemonstrated, employing the best available data from that region, that, in order tosolve a regional water quality problem, several necessary conditions for cooperationare met.

In our analysis we considered the variable nature of the drainage flows. For example,the continued drought and the reduction of surface irrigation water to the regiondiminished the amount of drainage water generated and disposed. In long-term planningof a treatment facility, the changes in water quantities for treatment need to beincorporated into the allocation scheme, or the solution will be unstable over time andcooperation is discouraged.

The analysis provides clear empirical evidence that the different allocation schemeshave different outcomes in terms of their acceptability to the players, and the derivedstability. The allocation schemes can be ranked by the players for their fairness indifferent ways. However, the Nash–Harsanyi and proportional allocation are alwaysranked first and the marginal cost is always ranked last. The regional problem has alsobeen analysed for two representative state of nature drainage flow scenarios. Amongthe allocation schemes, Nash–Harsanyi, proportional, Shapley and SCRB were found


to be more stable in both drainage scenarios (Sa<0·25) while the nucleolus and themarginal cost allocations were less stable (0·40<Sa<0·90).

The stability of game theory allocations (Shapley, nucleolus) increases as drainageflows increase; however, the stability of the traditional cost allocation methods (SCRB,marginal cost) decreases as drainage flows increase. The proportional allocation methodand the Nash–Harsanyi solution have the same degree of stability in both drainageflow scenarios analysed.

The propensity to disrupt index suggests that the players in the game do not considerdefection from the grand coalition. Several players have higher levels of (negative)propensity to disrupt values than others, but in general it can be said that all sixallocation schemes are stable.

The research leading to this paper was funded by grants from the University of California Centerfor Cooperatives, and U.S Geological Survey (Grant #14-08-001-G2084). The views expressedare those of the authors and should not be attributed to the World Bank.

References

Biddle, G. C. and Steinberg, R. (1984). Allocation of joint and common costs. Journal of Accounting Literature3, 1–45.

CH2MHILL. (1986). Reverse Osmosis Desalting of the San Luis Drain—Conceptual Level Study, Reportprepared for the San Joaquin Valley Drainage Program.

Dinar, A. and Howitt, R. E. (1983). The Economics of Water Quality Cooperatives: Joint Treatment Facilitiesfor Drainage Water. Report to the University of California Center for Cooperatives, Davis.

Ergas, S. R., Pfeiffer, L. W. and Schroeder, E. (1990). Microbial Process for Removal of Selenium fromAgricultural Drainage Water. Report prepared for the San Joaquin Valley Drainage Program, 67 pp.

Gately, D. (1974). Sharing the gains from regional cooperation: a game theoretic application to planninginvestment in electric power. International Economic Review 15, 000–000.

Gerhardt, M. B. and Oswald, W. J. (1990). Microbial–Bacterial Treatment for Selenium Removal from SanJoaquin Valley Drainage Waters. Final Report prepared for the San Joaquin Valley Drainage Program,236 pp.

Hanna, G. P., Jr. and Kipps, J. A. (1990). Agricultural Drainage Treatment Technology Review. MemorandumReport prepared for the San Joaquin Valley Drainage Program.

Harsanyi, J. C. (1959). A bargaining model for the cooperative n-person game. In Contributions to the Theoryof Games, vol. 4 (A. W. Tucker and D. R. Luce, eds), pp. 324–356. Princeton, NJ: Princeton UniversityPress.

Loehman, E. and Dinar, A. (1994). Cooperative solution of local externality problems: a case of mechanismdesign applied to irrigation. Journal of Environmental Economics and Management 26, 235–256.

Loehman, E., Orlando, J., Tschirhart, J. and Winstion, A. (1979). Cost allocation for a regional wastewatertreatment system. Water Resources Research 15, 193–202.

Musgrave, W. (1995). Interstate water management: the case of the Murray–Darling Basin in Australia. InWater Quantity/Quality Management and Conflict Resolution (A. Dinar and E. Loehman, eds), pp. 93–104.Connecticut: Praeger Publishing.

Nash, J. F. (1953). Two Person Cooperative Game. Econometrica 21, 128–140.Portney, P. R. (ed.). (1991). Public Policies for Environmental Protection. Washington, D.C.: Resources for

the Future.Ransmeier, J. (1942). The Tennessee Valley Authority: a Case Study in the Economics of Multiple Purpose

Stream Planning. Nashville, Tennessee.Rinaldi, S., Soncini-Sessa, R. and Whinston, A. R. (1979). Stable taxation schemes in regional environmental

management. Journal of Environmental and Economic Management 6, 29–50.Rosen, M. D. and Sexton, R. J. (1993). Irrigation districts and water markets: an application of cooperative

decision-making theory. Land Economics 69, 39–53.Russell, C. and Shogren, J. (eds). (1992). Theory, Modeling, and Experience in the Management of Nonpoint-

Source Pollution. Boston, MA: Kluwer Academic Publishers.Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics

17, 1163–1170.Schmiesing, B. (1989). Theory of Marketing Cooperatives and Decision Making. In Cooperatives in Agriculture

(D. Cotia, ed.). Englewood Cliffs, NJ: Prentice Hall.Shapley, L. S. (1953). A value for n-person games. Annals of Mathematics Studies 28, 307–318, and in


Contributions to the Theory of Games, vol. II (H. W. Kuhn and A. W. Tucker, eds). Princeton: PrincetonUniversity Press.

Shapley, L. S. (1971). Cores of convex games. International Journal of Game Theory 1, 11–26.Shapley, L. S. and Shubik, M. (1954). A method for evaluating the distribution of power in a committee

system. American Political Science Review 48, 787–792.Shubik, M. (1982). Game Theory in the Social Sciences. Cambridge, MA: MIT Press.SJVDP (San Joaquin Valley Drainage Program). (1990). A Management Plan for Agricultural Subsurface

Drainage and Related Problems on the West San Joaquin Valley. Final Report. Sacramento, CA.Straffin, P. D. and Heaney, J. P. (1981). Game theory and the Tennessee Valley Authority. International

Journal of Game Theory 10, 35–43.Williams, M. A. (1988). An empirical test of cooperative game solution concepts. Behavioral Science 33,

224–237.Young, P. H. (ed.). (1985). Cost Allocation: Methods, Principles, Applications. Amsterdam: Elsevier Science

Publishers.

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Mechanisms for Allocation of Environmental Control Cost: Empirical Tests of Acceptability and Stability