game theory analyses applied to water resource problems

003~121181/0301~1ou)2.00/0 Pergamon Press Ltd.

So&-Eron Plan 5-c;. Vol. 15, pp. IO!-118. 1981 Printed in Great Britain

GAME THEORY ANALYSES APPLIED TO WATER RESOURCE PROBLEMS

MICHAEL SHEEHAN

Institute of Urban and Regional Research, The University of Iowa, Iowa City, IA 52242, U.S.A.

and

K. C. KOGIKU

Department of Economics, University of California, Riverside, California, U.S.A.

(Received 5 March 1980)

Abstract-Most economists are more or less familar with game theoryt still, applications of game theory have been limited, for the most part, to: the operations research literature on war gamest (including differential or pursuit games); k Some scattered works on bargaining, labor negotiations,( etc. and finally some recent literature on game theory applied to water resource problems of various sorts.

It is this latter category which will be subject of this paper. Our concern is with the theory of cooperative games involving two or more actors. The practical problem dealt with is how to apportion “fairly” the reduced costs and increased benefits which arise when individual actors band together in order to reap the gains of collective action.

The study of these problems is of considerable potential imporance as most of the larger water resource development projects are collective endeavors. Furthermore, a large proportion of waste treatment projects involve at least two major actors. In addition, in the last several years many water resource development situations have attracted the attention of a third party in the form of the Environmental Protection Agency (and/or its cousins on the State/Regional level: e.g. the Porter-Cologne Water Quality Control Boards in California). This intervention opens up a range of new bargaining outcomes wherein standard-setting government agencies allot themselves a share of the benefits of cooperation which would normally have accrued solely to the local participants!’

In the first section below we give a brief description of cooperative games. In the second section we use simplified, though likely, scenarios to illustrate the concepts we have developed, along with some of the pitfalls and complexities involved in even the simplest situations. In the final section we outline four real world situations where cooperative game theory has provided a useful framework for analysis.

1. COOPERATIVE GAMES Perhaps the best known of all cooperative games is the Prisoner’s Dillema. Imagine a situation where two petty criminals have been arrested on burglary charges. The district attorney knows that he doesn’t have enough evidence to convict either one of them without the testimony of the other. Being astute, the prosecutor separates the two and gives each the following information: If criminal A is silent but B squeals, then A goes to jail for ten years while B goes free. On the other hand, if B is silent and A turns state’s evidence, A goes free and B gets a ten year sentence. If, however, both confess, they both go to jail for eight years (because the judge will assume that they confess from remorse). Only in the situation where they both keep silent will they both escape lengthy incarceration, receiving only two-year sentences for possession of burglary tools.

Setting up the game in its typical form we have:

Criminal “B” Confess Confess Silence

Criminal “A”

Silence 1 10,2 1 22

(Note: The first number in each box is the “sentence” given A for each combination of strategies. The second number

is B’s sentence.)

Because the principals are not allowed to communicate so as to reach an agreement, we find that A, if he acts to

minimize his years in jail, will always choose to confess. Glancing at the chart we find that if B has confessed, A is better off to confess; and even if B has not confessed, A is better off to confess. The same logic applies to B. (In game theory it would be said that the strategy “confess” dominates the strategy “silence” for both players.) The result is that both criminals will confess and both will go to jail for eight years.

So much for the game played without cooperation. If, however, players are allowed to make binding agreements among themselves then, of course, they would both elect to remain silent and go to jail for only two years. I Note that in this situation there is no question of how

to divide the gains from cooperation, i.e. if they cooperate to keep silent each saves himself six years in jail-they do not save a total of twelve years and then decide among themselves who should serve which portion. In the more interesting applications, this question of the division of the spoils is paramount.

Let us assume, for example, that three farmers with fields adjoining the same irrigation canal have several alternatives for pumping water.Each could buy his own pump, which would cost farmer 1 $1800, farmer 2 $1600

?See, e.g. Ferguson([l], chapt. II), Chiang ([2], chapt. 21). lntriligator (131, chapt. 6 and 15) Shubik[4], and Newmann and Morgenstern[S].

SSee the Lanchester literature in volumes l-12 of Operations

Research (1952-64) §Intriligator ([13], chapt. 15). YSchelling [6]. IKneese and Bower ([7], chapt. I I).

109

110 MICHAEL SHEEHAN and K. C. K~GIKU

and farmer 3 $1300. Alternatively, if farmers 1 and 2 bought a pump jointly it would cost $3000; farmers 2 and 3 together could by a pump for $2400; and farmers 1 and 3 could buy a pump costing $2500. On the other hand, if they all got together to buy one big pump it would only cost $3300. In this situation it is pretty clear that our farmers have an incentive to combine to buy the largest pump, as in the aggregate they should save $1400 over the sum of their separate costst

The problem of course, is that most players will usually have an argument over how to divide the gains. In the case, for example, farmer 3 pays a share of $1100 for the new pump and saves $200 over his separate costs, while farmer 1 pays $1100 but saves $700. We can almost hear farmer 3 arguing for a division where the net savings, not the final costs, are shared equally. And there are a number of other systems all with a different impact on income distribution and all with their own special appeal to some doctrine of fairness.+ We can eliminate a certain portion of the distribution of gains resulting from the various rules without serious consideration, as they conflict with some simple rules of common sense. None of our farmers, for example, would accept membership in a coalition where distribution of benefits would leave him worse off than if he had acted alone. Nor would any subgroup of farmers remain in larger coalition if they could do better in a separate group.8 Following these two rules we may find either that no possible distribution fits

tin the jargon of game theory this $1400 is called the “characteristic function” for the coalition, farmer 1, farmer 2, farmer 3, and is usually written t~(lr, fi, fs). The characteristic function tells the amount of the saving gained by joining a specific coalition, over the aggregate costs of all the members going it alone.

fAn elaboration of a number of these systems or rules of “fair” division is found in the next section.

BSee Suzuki and Nakayama[8], Gately[9] or Intriligator ([3], chapt. 6).

Wee Lute and Raiffa ((lo], p. 61 ff). [‘See Gately (A, p. 199 f).

the bill or that many are acceptable. All the distributions which conform to these two rules are called, for the sake of “elegance,” imputations. For any coalition an imputation gives a feasible pattern of distribution for the amount given in that coalition’s characteristic function. For example (see Table l), u(F,, Fz, F,)= $1400; the imputation based on equal distribution of benefits would be I($466, $466, $466); the imputation based on equal division of final costs would be [($700, $500, $200) and so on.

Other elements of cooperative games which will form the building blocks of our analysis are the ideas of “security level,” “ threat value,” and the “propensity to disrupt a coalition.” As might be imagined, a player’s “security level” is the amount he can count on receiving even under the most perverse set of actions of the other participants. A player, in conformity with our first rule of common sense, would always insist that his return from any proposed cooperative action should at least have to pay/receive if he acted independently).7

To convey the ideas of threat values and propensity to disrupt, let us consider the following hypothetical situation: Town A can build a waste treatment plant for $2m. Town B, which is also required to have one, can also build its own for $2m. As the two towns are adjacent, however, it would be less costly to build a joint facility which would cost only $3m. Note that each would choose to build its own plant rather than build the joint plant alone.

Suppose B proposes to A that A chip in $2m toward the construction of the joint facility and B will pay the remainder, on the logic that A would have to pay the $2m anyway. In this situation A would have a very large “threat value;” refusal to agree to B’s proposal costs him very little (here, nothing-in coalition with B he pays $2m and his security level is $2m) while B would then have to pay $2m instead of $lm. Alternatively, using Gately’s idea of the “propensity to disrupt”” given as the ratio of B’s net saving from joint action to A’s net saving from joint action- we find that A’s propensity to disrupt the agreement (in this case his refusal to join the coalition under B’s terms) approaches infinity as A’s net return approaches zero.

Now let’s complicate the situation to illustrate another point. Assume both these towns have been dumping wastes into the local river unbeknownst to local water quality officials. “A” dumps more than “B.” If this dumping were reported to the authorities, new require- ments would be imposed which would raise the cost of constructing A’s plant to $2.6m, while increasing the cost of B’s separate plant option to $2.4m. The cost of the joint plant would remain the same, $3m. The question

Table 1. Characteristic functton and costs of hypothetical coalitions

Characteristic cost to Each Farmer If

Coaliti0” Cost F”llctiWl Costs Divided Evenly

1 1800 0 1800

2 1600 0 1600

3 1300 0 1300

192 3000 400 1500

!,3 2500 600 1p0

2,3 2400 500 1200

1.2,) 3300 1400 1100

Game theory analyses applied to water resource problems ill

then is, does B have an incentive to report the illegal dumping? (Because of the situation, B must also report itself if it wants to report A.) The answer is yes. The probability of A refusing the $2-lm split under the first set of conditions was 100%. But if A’s position deteriorates relative to B’s position, A will be more likely to accept the agreement. Here, for example, A’s new propensity to disrupt would be ($lm-$2.4m/$2m-$2.6m = 2.33) which as these things go, is quite reasonable in any case much lower than under the first set of conditions. That is to say, A would be much more likely to accept the offer under the revised conditions. This leads us to a general guideline for behavior: it pays any player to increase the costs of conflict as long as the costs of the other players increase in greater proportion. Con- versely, it pays a player to reduce the other players’ costs of conflict as long as his own costs fall in greater proportion.~

2. SEVERAL IDEAS OF FAfRNESS AKD SEVERAL SYSTEMS FOR APPORTIONING COSTS AND BENEFITS

Two areas that water resource negotiators should be thoroughly familiar with are concepts of fairness and applications of the law of liability. Concepts of fairness range from “equal sharing is fair” and “might makes right” to Rawl’s Theory of JusticeS and on to such esoterica as Shapely values and nucleoli. Adventuresome beginners may even be interested in learning “How to Cut a Cake Fairly”§ using such simple tools as “par- titions of nonatomic probability measures.” Other principles of fairness are more homely; for example, it was generally thought, before the advent of the Coase

tSchelling [6]. SRawls [ll]. §Dubins and Spanier [ 121. Koase [I9 NPosner[f4, l5]. ttCox and Walker[~6]. S$The Japanese have adopted an imaginative policy based par-

tilaly on the “do unto others.. .” idea: industrial polluters are required to dump their wastes upstream of their water intake!

IKox and Walker([l6], p. 252). (lSee[17-211.

FACTORIES

TREATMENT

Theoremq and the works of Posner,” that innocent parties ought to be held harmless from the purpositive activities of others.?? Even farther back we find, “Do Unto Others . . . ,” an ancient canon of a defunct moral- ity, echoes of which persist atavistically even today.9

Legally, one of the prime manifestations of the idea of fairness is the law of liability. It is important to be familiar with its major doctrines, especially in water pollution cases, as liability often institutionally delimits the bargaining space. Magnitudes of “security” and “threat” values are often determined by such principles as the juxtaposition of negl~en~e, contributory negligence, and whether the local courts believe in “strict liability” or require a showing of “unreasonable activity” on the part of the person(s) causing the damage.98 This area over the years has been a tangle, but the increasing local application of federal laws like NEPA, the FWQCA and the Refuse Act of 1899 have taken some of the burden off quirky regional inter- pretations of liability laws.$fi

As an illustration of some of these problems, consider the following situation, as shown in Fig. 1. We have two polluters, factory 1 and factory 2, both of which have decided to adopt new techniques which produce wastes very damaging to water quality. Downriver is a residen- tial district populated by wealthy, riverfront homeowners. Currently the river is unpolluted, a condition which the river front residents value to the tune of $2m per year. At this point we make the important assumption that the River Front Association (RFA) could prevent the factories from dumping into the river by going to court. However, and this is the first point of the story, it is impor~nt that we assume that the court would issue an injuncfion against the firms and not simply assess permanent damages. If the factories were to be confronted with the alternative of not being allowed to dump at all, they have an incentive to bargain with the RFA and the RFA will probably receive in excess of the $2m. That is to say, the RFA would receive a share in the net gains of polluting the river and would be fully compensated for damages. On the other hand, if the court only assesses damages the factories will con- tinue to pollute the river (we assume that it is profitable to do so) and divide the net gains between themselves.

Fig. I. Factory polluters and alternatives for waste disposal.

112 MICHAEL SHEEHAN and K. C. K~GIKU

The factories have the following alternatives for dis- posing of the wastes:

1. They can truck the wastes to a nearby lake and dump them. Because of the local political situation, lake residents cannot muster the clout to interfere, so no damages need to be paid. However, this strategy would cost factor 1, $1.8 m in trucking and pumping costs and factory 2, $1.4m. This differential in costs is the result of factory 1 producing 1.5 times as much waste as factory 2 (the actual dollar amount is less than 50% greater due, let us say, to economies of scale);

2. The two factories could construct a waste treatment plant jointly at total yearly costs of $2.6m; or

3. The factories could bribe the RFA not to object to the dumping of the wastes. Note again that it is only because we have given the RFA recourse to a sure injunction that we have a problem of any interest. If our hypothetical locale were in any area of the U.S. (leaving aside federal law for the moment) where the rule of negligence applied, the polluters would be able to dump wastes in the river without having to compensate the RFA. This is because the “social costs” (i.e. the com- bined costs to the polluters) of preventing the pollution outweight damages, hence, legally (in some places) the polluters are not negligent, and hence not liable.? A second point which might play an important part in some situations is that in many states it would be necessary to prove exactly how much of the degradation of the water quality is due to the wastes from each factory. If the RFA cannot do this then it cannot collect against either factory 1 or factory 2, even though everyone agrees that together the factories produce all of the pollution..$

Using this example as our model, we shall elaborate several methods for allocating the gains of cooperation which would accrue from different sorts of agreements. However, before going into the various methods, let us narrow the field by being explicit about which courses of action would normally be unacceptable.

tCox and Walker[l6]. $Cox and Walker ([16], p. 256f) and Sheehan[ZZ]. §Note that this imputation conforms to our two rules of

common sense: no gains from individual action exceed the corresponding payment in the imputation; an no subgroup, e.g. F, and F2, building the treatment plant has a characteristic function exceeding the aggregate value of the subgroup’s payments in the imputation.

I/The propensity to disrupt is defined as (D, = R(j) t R(K) - u(i.k)/R(i)-o(i) where R( ) is the payment to player i under the

‘_‘.‘. coahtton being disrupted and where j,.. , k are the other players in the coalition. v(j,k) is the aggregate return to the other players in the coalition were they to be left by themselves.

First, it is a step in the right direction for everyone concerned if the factories discard their first two alternatives and concentrate on making an agreement with the RFA which will provide for larger gains for everyone.

We have the following information: the security of each player is the amount they would receive if they participated in no joint action: F, = -$1.8m, Fz = -$1.4m and RFA = $2m. We also have the information shown in Table 2.

Rule I The simplest rule of division would entail the parti-

cipants acting together to produce the cheapest joint solution and then sharing the gains equally. (Note that equal sharing of the final costs would not be a viable alternative, as it would violate the first common sense rule because the RFA would receive only $0.67m from each factory and hence would be worse off if it had not entered the agreement.) Thus, if the factories agree to bribe RFA, the total gains would be a reduction in total costs of $1.2m below the individual action solution. Equal sharing would then entail factory 1 paying RFA $1.8m less $0.4m, or $1.4m; factory paying $1.4m less $0.4m, or $lm; and RFA receiving a total of $2.4m, which is $0.4m in excess of the damages from having the river polluted. We would write the characteristic function as v(F,, Fz, F3) = $1.2m and the imputation, i.e. distribution of gains, for this rule as 1($0.4m, $0.4m, $0.4 m).Q

It should be noted that in this example, as in most of the situations we will encounter, we have bargaining on two fronts: the factories bargain to get the best deal possible between themselves jointly and RFA, while at the same time each factory tries to get the better of the other factory in the division of whatever they are able to get from their dealings with RFA. In this situation a bargain might be struck where two factories agree to pay RFA a total of $2.3m, which would leave $0.9m to divide among themselves. Factory 2 would, no doubt, argue for a division based inversely on the volume of wastes dumped. This would give factory 1 40% of $0.9m, or $0.36m, and factory 2 would receive $0.54m, for an overall imputation of 1($0.36 m, %0.54 m, $0.3 m). Let us compute the propensities to disrupt for this distribution, as well as for an equal distribution of gains.

For the imputation 1($0.36m, $0.54m, $0.3m) we have Q, = 21/3, D,, = 1.22 and Dti. = 1.7 For the imputation 1($0.4m, $0.4m, $0.4m) based on an equal distribution of gains, we find Q, = 2, Dfz = 2 and D,,, = 0.05, which

Table 2. Characteristic function and costs of coalitions of factories and homeowners (RFA) --

CharacteriStic COalition Decision cost FUnCtiO”

F1 dump in lake $1.8 million 0

FZ dump in lake 1.4 million 0

FI build treatment plant 2.6 million -0.8 millfan

F2 build treatment plant 2.6 million -1.2 million

F1 6 F2 build treatment plant 2.6 million 0.6 million

F1 6 RFA bribe RFA at least 2 million at least -0.2 million

F2 & RFA bribe RFA at least 2 million at least -0.6 million

F,,F2 6 RFA bribe RFA 2 million 1.2 million

Game theory analyses applied to water resource problems 113

gives some idea of the changes in relative dissatisfaction, go to him. RFA and F,, by the same logic, would receive even though all the values are quite low. nothing.

Rule II As a variation of Rule I we might find that after

agreeing with RFA, F, and F2 divide their aggregate costs (the amount of the payment that they make to RFA, in this case again assume $2.3m) according to the ratio of costs they would have paid under self-suficiency. This would have F, paying (1.8/3.2=56%) of costs, or $1.29m, with F2 paying the difference = $l.Olm. The imputation for this distribution would then be Z($OSlm, $0.39m, $0.3m), with “propensities to disrupt” of 0(1.35, 2.08, l.O), apparently a more satisfactory distribution for factory 1.

On the other hand, if F2 arrives first, he is entitled to nothing, u(F2) = 0. When F, joins Fz, FI “should” receive u(F,, F2) = $0.6m. Finally, when RFA joins the other two, v(F,, F2, RFA) = $1.2m, and RFA should get the difference = $0.6m, etc.

Rule III Coming from a different perspective, one or another of

the participants might argue for that distribution of benefits which is most likely to result from a stable coalition, i.e. for that distribution which equalizes the “propensity to disrupt” of all the participants.t To compute these values we form a simultaneous system made up of the three “propensity to disrupt” equations along with a fourth equation which has all of the individual rewards summing to the value of the characteristic function (i.e. $1.2m). Doing this we obtain, 1($0.48m, $0.48m, :\24m), with the equalized “propensity to disrupt” at

Rule IV A more complicated system is based on what is known

as the Shapley va1ue.S This system was developed to get around the fact that the order in which the grand coalition is formed also establishes specific patterns of equity based on the intuitively pleasing idea that if the addition of another player to the coalition results in a large incremental gain, then the new participant ought to be entitled to the lion’s share of the gain because he is the “proximate cause.” To illustrate, let us assume that we begin with factory 1 initially convincing RFA to join a tentative coalition. By the logic of “proximate cause” RFA would be “entitled” to zero reward as his arrival at this point “produced” no coalition benefit, i.e. u(FI, RFA) = 0. However, if after F, and RFA form a coalition, Fz joins up, the characteristic function for our new coalition jumps to $1.2m, which F2 would argue ought to

Shapley’s system avoids this tangle by calculating benefits under the assumption that each possible sequence of coalition construction is equally likely. The values he obtains are the average (expected) btnefits from all the ways of reaching the final coalition.

In the literature the Shapley method is given some consideration as being “fairer” than other methods; however, there is no basis for this. Some people gain by using this system and other lose and there is, of course, no objective reason to prefer the Shapley distribution over any other. On the other hand, because it is con- sidered to be “fairer,” players who would benefit from its adoption have a head start in arguing for it.

To compute the Shapley value for each participant, first list all the ways the coalition could have been formed, noting what the marginal contribution of the particular player would have been had he joined at each point. Then take the expected value of all the marginal benefits, assuming that each possibility is equally likely. For example, Shapley would say that RFA is entitled to $0.2m, calculated in Table 3. Doing the same for factory 1 we have the calculation in Table 4. Given that factory 1 and factory 2 are symmetrically situated, factory 2’s Shapley value is also $OSm. The Shapley imputation is then 1($0.5m, $O.Sm, $0.2m) with corresponding propensities to disrupt 0(1,4, 1,4, 2).

Rule V In order to determine which distribution of gains will

arise if the distribution is based on the “power” of the participants, we use a rule based on the “kernel” of the game. This rule tells us that players share in proportion to the viability of their alternatives. He who has no alternative does not need to be bought-off and would not receive anything. This rule provides good service in situations where the “core,” i.e. the set of all those imputations which conform to our rules of common sense, is empty. The rule of the “kernel” will always yield a solution. Within this system there are five “cases” (for the three-pacticipant game) among which any game situation falls.5 We will use two examples to illustrate the operation of the kernel relative to our other rules.

tGately ([9]. p. 205). SLuce and Raiffa ([lo], p. 247ff). as well as Rapoport, (KU, P.

104ff .). §Rapport ([23], 13Off).

The first example uses the data of our last example: u(F,) = 0; u(FJ = 0; u(RFA) = 0; v(F,, RFA) = 0; u(F2, RFA) =O; u(F,, F2) =0.6; and u(F,, F2, RFA)= 1.2. Arranging the two-person characteristic functions in

Table 3. Calculation of shapley values for homeowners (RFA) Marginal Weighted

"*der Probability Contribution VZ3lWS

WA. Fl. F2 l/6 0 0

RFA, F2. PI l/6 0 0

F1, WA, F2 l/6 0 0

Fls F2, RFA l/6 $0.6 million $0.1 million

F2, WA, F1 l/6 0 0

F2. F1, RFA l/6 SO.6 million $0.1 million

Total $0.2 million

114 MICHAEL SHEEHAN and K. C. KOCIKU

Table 4. Calculation of shapley values for factory 1

n¶erg*ne1 Weighted Order Probability Contribution Values

RFA, F,, F2 l/6 80 0

WA, F2. F1 l/6 $1.12 million $0.2 million

F1> WA. F2 l/6 0 0

F1, P2s RFA l/6 0 0

F2. WA. Fl l/6 $1.2 million $0.2 million

F2, F1, RFA l/6 $0.6 million $0.1 million

Total $0.5 million

ascending order we have IQ,, RFA)s v(F2, RFA)s v(Fl, F2). Scanning the five different “cases,” which are in reality tests for assessing relative power, we find that the data place this example clearly under Rapoport’s case II i.e. the conditions,

and

u(F,, RFA) + u(F,, RFA) < u(F,, FJ

VU% Fz) 5 VU?, Fz, RFA) 5 3u(F,, Fz)

are met. Once these conditions are found to be satisfied, we can apply an algorithm for determining the payoff for each member of the coalition. Performing these operations gives us Z(0.45 m, 0.45 m, 0.30 m).t

Now let us consider a more complicated example in Table 5.

Investigating, we find that obedience to our common sense rules leaves us without a solution; we cannot distribute the gains of 700 from the grand coalition in such a fashion that at least two participants would not find it more profitable to pull out and form their own

tThe return to RFA is given by RFA = 1/2[u(F,, I$, RFA)- u(F,, F2)J = 0.3, which is to say that RFA receives half of what he brings into the grand coalition. To compute the rewards of the factories, calculate the following numbers:

W,, = MAX[O, o(F,, RFA) - RRFA] = 0 W,> = MAXIO, u (F2, RFA) - RaFA] = 0.

Let C = 1/2[u(F,, F2, RFA) t u(F,, F2)] = 0.9. The rewards to F, and Fz are then:

and

R,, = W,,t 1/2(C- W,, - W,J=O.45,

Rr2= W,>t 1/2(C- W,, - W,J= 0.45.

SYet these lesser coalitions would also be unstable. The odd man out would always be able to offer a large enough inducement to break one of the erstwhile partners away.

§As follows: Solve W, f W, = 400 w,tw,=500 WA+ WC=600

which gives W. = 250, W, = 150, WC = 350, and C W = 750. Th, rewards are then given by:

R * = w A

tu(AJ,0-XW=233. R = w 3 Y B B

+v(A,&CPW 3

= 133

and

R c = w A +G4=)-ZW=333, 3

For a more detailed explanation, see Rapoport (1231, p. 133). flhis is a free adaption of the situation facing the town of Ody

Cuyama as presented in ([24], pp. 45-49).

coalition. For example, say we decide upon an equal A B C

distribution, 1(233,233,233); would it not make sense for A and C to pull out, for AC, and share 600? Could not B and C also do better?4 Because of this contrariness in the structure of the problem our usual Rules (I-IV) will not apply, and it is necessary to have recourse to the “kernel,” or some similar rule, for a solution.

Checking our”cases” we find that this coalition structure falls into case III, as the complicated looking condition

2u(A, C) - v(A, B) - v(B, C) 5 u(A, B, C,) I 2u(A, b) t 2v(B, C) - o(A, C)

is met; i.e. 2(600) - 400 - 500 I 700 5 2(400) t 2(500) - 600 300~700~ 1200. Calculating with the algorithm for case III, we obtain the distribution: Z(233, 133, 333).§

After these exercises let us turn to a more realistic situation.7 Consider the case where community “M” has a water quality problem involving excessive amounts of total dissolved solids (TDS). Investigating possible reme- dies, “M” found that any one of four strategies would provide a solution. In the first, “M” could build a water treatment plant at a capital cost of $100,000, which would resolve the problem by providing fresh water at a cost of $1 per thousand gallons. Second, “M” could purchase low- salt water from an adjacent community “N” which is capable of supplying “M with water from its pre-existing plant at a constant incremental cost (to “N”) of 804 per thousand gallons. The third alternative would entail buying water from yet another community, “P”, which would require plant expansion costing $60,000 and deliver water at SO+ per thousand. Finally, “M" has the option of convincing “N” and “P“’ to join “M” to build a new facility for $200,000, which would deliver water to all communities at 204 per thousand. Let us assume simplicity that all facilities have a life span of 30 years

Table 5. Example of the distribution of gains based on the kernel of a game

Characteristic C0aliti0ll Costs F"llcti0"

A 1800 0

B 1600 0

C 1300 0

AL! 3000 400

BC 2400 500

AC 2500 600

ABC 4000 700

Game theory analyses applied to water resource problems

Table 6. Data for a distribution of gains exercise with a regional authority

:

w 1OOK $1.00 200 200 $3.1?4H 0

N _- .80 180 400 2.21313 0

P _- .50 150 200 1.153n 0

MN -- .80 380 400 4.672M S0.715M

HP KOK .50 350 400 2.750H 1.577H

NNP ZOOK .20 530 600 1.829M 4.711M

NP 0 __ __ _- _- 0

115

from the effective data of whichever coalition comes to fruiti0n.f Table 5 summarizes the relevant data.

Now let’s introduce regional authority invested with the power to tax beneficiaries to provide funds to build and operate facilities. (Examples of real world agencies of this sort are found in ORSANCO on the Ohio River and Genossenschaften in Germany.S) In this case the Regional Authority (RA) is constrained to allocate benefits so as to prevent any of its constituent muni- cipalities from repudiating its authority or causing too much trouble in the courts. To do this the RA must distribute the gains so that our common sense constraints are satisled. Within these constraints, however, we will assume that the RA, malevolent creature that it is, will attempt to maximize its own residual gains. It does this by taxing the participants in the grand coalition the amount they would have paid if they had had to act individually and then, after setting aside enough to build the necessary facilities, redistributing just enough of the residual, R, so as to just satisfy all the participants. This will, hopefully (from the perspective of the RA), leave money left over to build parks, stock rivers, etc.

In order to compute the proper magnitudes of these various payments, the RA, in our hypothetical rational world, solves a linear programming problem which minimizes the sum of the necessary rebates to the players subject to our cqmmon sense constraints in mathematical form.

In the present example the regional authority would tax “M” $3.174m, “N” %2.213m and “P” $l.l53m. (We abstract from the time dimension of these charges for the sake of simplicity.) The RA then sets aside $1.829m to build the plant and supply “free” water to “M,” “N” and “P” and itself.

The linear programming problem takes the following form:

Minimize R = R,,, t R, t R, s.t. R, f R, -cO.l1Sm

R, + RD 5 1 S77m R,, R,Rp SO.

tAssume that the discount rate is 5%. $Kneese and Bower (171, chapt. 8 and 131. $See Giglio and Wrightington ([25], p. 1140 ff). Tit might be noted, however, that their propensities to disrupt

are very high and unless there are some isiitutional constraints they are likely to cause trouble out of pure chagrin.

(ISee Gately, [91.

Solving, we obtain R, = $1.577m. R, = 0, and R, = 0 and the regional authority pockets the difference: $4.711m- $1.577m=$3.134m.

Notice that in this distribution of gains M’s payment is just large enough so that he has no incentive to go off with “N” or “P”. At the same time, because “M” has been awarded the full value of the characteristic function of the coalition MP, “P” cannot offer him an amount large enough to induce him to form the coalition MP. This logic applies, consequently in the case of “N,” in relation to the coalition MN. Notice that “N” and “p” each receive nothing because NP is not allowed coalition; and hence, by the idea of the kernel, they have no alternative, no power and no rebate.91

3. RECENT APPLICATIONS OF GAhfE THEORY After Section 2s emphasis on technique, it is ap-

propriate that the reader should be reminded that the role of game theory is to serve as a structure for researching a bargaining problem-the analytical aspects of game theory being insufficiently appealing to the common man to expect their explicit appearance in the conference room. The skillful negotiator would surely avoid the unravelling of great rolls of formulae in expectation of convincing the opposition through the pure force and grandeur of mathematics. Rather, game theory’s place is found in the preparatory stage, in identifying likely strategies and counter-strategies and in the facilitation of the pre-negotiation computation of the costs-and-benefits implications of the various specific, normally verbal, arguments. The negotiator should be aware, e.g. that the offhand acceptance of some apparently innocuous prin- ciple of equity may well result in a material catastrophe for the “good guys,” while arguing for an equally pleas- ant sounding variant may result in smiles all around back home.

Having cautioned the reader, we now proceed to a brief description of four game theory applications taken from several recent articles.

The first concerns the Sotithern Electricity District in India, which is made up of the four states which com- prise the southern third of that nation.11 Each state needs to generate more electric power. The bargaining situation arises because each has a different endowment of hydro-electric sites which makes it unlikely that the optimal regional development plan would coincide with policies of self-sufficiency on the part of the states. However, the current situation is such that each state could opt for self-sufficiency, or they could form coali-

116 MICHAEL SHEEHAN and K. C. KOCIKU

tions where low-cost sites would be used wherever found and costs and benefits apportioned in some acceptable fashion.

Gately has estimated characteristic function values by computing optimal investment scenarios using a large mixed-integer programming algorithm and local data. He gives a number of solutions, along the lines of the rules we have presented above. The conclusion reached is that cooperative action would bring substantial benefits to all. Unfortunately, the states involved seem to have decided on a policy of self-sufficiency out of dislike for one another.?

Another article focusing on the Indian subcontinent deals with the optimum development of the Ganges and its tributaries for flood control, hydroelectric generation, and irrigati0n.f The Ganges flows out of eastern India (West Bengal) into Bangladesh where it reaches a confluence with two other large rivers, the Bramaputra and Ghora Urta. The highs and lows of these rivers currently present Bangladesh with yearly disasters, while at the same time they are the key to the future prosperity of the region.

The problem arises out of nature’s penchant for extremes (or man’s tendency to live where he shouldn’t). During monsoon (June-October) no crops grow but rice, due to a general flooding of the rivers, which also causes a good deal of direct damage and loss of life each year. On the other hand, during the dry season it is difficult to grow anything because of a more or less general condition of drought and lack of irrigation.

The development strategy which ought to be followed is a familiar one. Both India and Bangladesh can agree to cooperate to build a jointly optimal pattern of upstream storage facilities (because of the geography of the region this would put most facilities in India) and then share the costs in a mutually acceptable fashion. Alternatively, they could opt for individual action, which, while not overly damaging for India, would be much more costly for Bangladesh. The best alternative. for flood control alone would be the construction of 4700 miles of high- cost embankments, which accomplish nothing in the way of power production or storage of flood waters for the dry season.

In his study, Rogers uses linear programming a la Maass,§ with local data to estimate characteristic functions for six strategy variants. He then suggests the outlines of an agreement between the two nations, based on a particular pattern of water works of various sorts along with side-payments from Bangladesh to India to offset costs. However, as in our other examples, Rogers is not sanguine in his expectations.

Our third example comes from Japan.q In the area south of Tokyo along the harbor there are several cities

tAn article which points up this apparently universal lack of cooperation in the development of Indian water and power

resources is Cautam,[26]. $Rogers[27]. BMaass et a/.([28], chapt. 12 and 13). BSuzuki and Nakayama4[8]. liWe have not presented this rule above, as the mathematics

are advanced and complicated and the amount of the improve- ment over our present group of rules is uncertain.

ttDay[9]. 1BThe logic also applies in the reverse, but because of the

proportions, much less forcefully.

(Yokohama, Kanagawa, Kawasaki) which are anticipat- ing water supply problems in the next several decades. In the same neighborhood are several agricultural asso- ciations which have prior appropiation rights to a significant portion of the area’s natural water supply. However, there are two rivers, the Sakawa and the Sagarni, which are prime candidates for control and further exploitation. Solutions to the cities’ water supply problem will take one of the following forms: cities can, singly or in coalition, build a dam on either river, or buy water from an agricultural association, or attempt some combination of the two.

Suzuki and Nakayama use mathematical programming to compute characteristic function values but find that the common sense conditions are not met for the “solution” produced. This leads them to the point of their article, which is the presentation of another rule, the “nucleolus,” similar to the kernel, which gives a sort of “second best” solution.” However, it is not clear to US, in the absence of any mitigating insitiutional limitations on participant mobility, why the nucleolus solution would not be abandoned by a subset of the players. Yet, this criticism notwithstanding, the example is an interesting one; and it is to be lamented that the real world development and the solution of the problem were not further developed by the authors.

Our last example deals with the growing water pollution problem in the Great Lakes.tt The situation as it now stands has the population of the area at about 35 m; 27 m of these are American and 8m are Canadians. Day estimates that by the year 2000 the area will have a population of 57m, with 42m of these on the U.S. side. Thus, both the ratio of Americans to Canadians and the total numbers of each are changing. Problems arise in this situation because all these people, as well as the industries of the two nations, want to dump their wastes into one or another of the Lakes. If the present pollution control standards for the composition of these wastes are main- tained, the added volume will produce a significant and increasing degradation in the quality of the waters of the Lakes.

What needs to be decided collectively, because these are international waters, is how the standards for wastes to be dumped into the Lakes ought to be tightened over time and who should pay the additional costs of the more expensive facilities. However, the situation is complicated by the presence of a “free rider” problem.

Let us assume that the Canadains currently contribute water pollution in proportion to their population in the Lake area, i.e. 25%. If the Canadaians can assume that the U.S. is committed to cleaning up the U.S. share of the pollution and if the Canadainas find that they are more or less satisfied with the lakes being only one- quarter as polluted as before (i.e. the cost to Canada of cleaning up the Canadian wastes is greater than the extra benefits to Canada of going from a one-quarter polluted lake to an unpolluted lake), then the Canadians will leave the cleanup to the Americans.§§ This results in a Prisoner’s Dilemma type of situation.

Assume, using hypothetical figures, that the cost to either country of putting in new treatment facilities and thereby reducing its own pollution, is $200m. Let’s also estimate that the benefits to each “player” of low levels of lake pollution is $300m over the life (say 30 years) of the facility. On the other hand, if one side pollutes and the other does not (i.e. has built a new treatment plant), a middle ground is reached where each

Game theory analyses applied to water resource problems 117

receives only $150m in benefits over the horizon. Finally, if both parties agree to joint action the total costs of a joint facility will be $2OOm, with benefits equalling $6OOm. Diagrammatically we have the situation given in Table 7. As presented in the table, a joint action solution which will be stable can be found within the common sense constraints. This stability arises because if either player withdraws from the joint solution each player then will receive only the rewards from the next best solution, which is the no treatment/no treatment strategy which gives each player zero rewards.

Now let us change the values in the table to reflect the fact that, given the relatively small Canadian population, Canadian lack of participation would not have nearly as serious an impact as would U.S. non-participation. This situation is shown in Table 8. With the net gains from joint action divided equally to make the case as strong as possible,t the Canadians would have an incentive to pull out of the joint action solution as the U.S. would probably elect “treatment” in any case. In such a situation the Canadians would increase their reward from $115m to $140m. Once the U.S. realized this it would return to “joint action” by offering the Canadians at least $140m from the $230m aggregate net gains.

However the Great Lakes situation develops, it will be an interesting one to follow, though Day seems to think, as in the other examples we have investigated, that in the

Table 7. Canadian-US pollution problem with joint action Canadians

“mu COSk3 2B 2OON, * ZOOM ZOOM. 0 *B L!i

NA

3; Benefits

300% 300H 150!4, 150N

*Note: 200H, 2oou Americans, Canadians

Table 8. Canadian-US. pollution problem with no Canadian participation

Canadians

Individual N0 TWatlUC!llt Treatment

3: Joint Action

58 300M. loon 300% 0 NA

5; 400M, 14011 310H, 140M

end zero-gain strategies will prevail, just because man is contrary and the situation complex.

It is our feeling that the use of game theory in such applied contexts will help negotiators see the benefits of cooperative solutions in many situations of potential conflict.

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game theory analyses applied to water resource problems

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