A solution of the externality problem using strategic ... - Springer Link

Soc Choice Welfare (1992) 8:73-88

Social Choice Welfare

© Springer-Verlag 1992

A solution of the externality problem using strategic matching* Joel M. Guttman and Adi Schnytzer Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel Received October 26, 1988/Accepted June 6, 1991

Abstract. This paper develops a Coase-like solution of the problem of inducing Pareto optimal behavior in the presence of reciprocal externalities. In place of Coasean direct compensation between the parties to an externality problem, actors strategically match each other's externality-producing activity, and thus induce counterparts to internalize the external benefits or costs of their actions. The analysis suggests a general framework for analyzing social interactions in the presence of reciprocal externalities. As an application of the theory, a solution of the duopoly problem is noted.

Introduction The well-known Coase Theorem states that externality problems may be solved without third-party intervention, if the two parties to the problem can contract between themselves, so that one party compensates the other for either the external benefit conferred or the external cost imposed. The Theorem has had a powerful impact on economic thought, stimulating the development of the property rights and transactions costs literatures. Economic theorists, however, have had difficulty in formalizing the precise mechanism by which the compensation envisaged by Coase is supposed to take place. If the Coasean mechanism is modelled as a cooperative game - which would seem to be the natural framework in which to model the Coasean contracts - then allowing more than two agents to be involved in the externality leads to problems of an empty core and coalitional instability (Aivazian and Callen [1], Aivazian et al. [2]). One way to avoid these difficulties would be to formulate a non-cooperative procedure for arriving at the private compensation that is the essence of the * We are indebted to L. Danziger, D. Samet, participants in faculty seminars at Bar-Ilan and New York Universities and at the University of Maryland, and two anonymous referees for helpful comments and discussions. All remaining errors are our own.

74

J.M. Guttman and A. Schnytzer

Coase Theorem. Such a procedure would allow the parties to an externality situation to provide each other with Coase-like incentives without making direct transfers, by making their own externality-causing activities conditional on the externality levels of their counterparts. Thus the need for bilateral contracts is avoided, and the interaction of the agents can be modelled as a non-cooperative game. In this paper, we develop this approach to understanding and extending the Coase Theorem. The specific mechanism which we propose is a generalization of the "matching" game previously developed to explain voluntary contributions to the provision of public goods (Guttman [8, 9, 11]; Danziger and Schnytzer [5]).i The matching game consists of two stages. In the first stage, actors determine "matching rates", which linearly link their externality-causing actions. Once made, these matching rates are not changed - either because they cannot be changed, i.e., they can be viewed as precommitments, or because actors have sufficiently strong disincentives to change them. In the second stage, non-negative "flat actions" are chosen, which are then matched at the previously determined matching rates. The equilibrium in each stage is a Nash non-cooperative equilibrium. The matching rates are chosen so as to maximize the actor's utility by predicting the equilibrium of the stage determining the flat actions, which is determined by the vector of matching rates. Thus the solution of the overall game is subgame perfect. Some readers may object that if we assume that the agents cannot renege on their matching offers, then we have, in effect, a cooperative game. It must be recognized, however, that in this game, the choices of the matching rates and of the flat actions are made noncooperatively - i.e., both choices are best (individual) responses to the corresponding matching rates and flat actions of the other agents in the game. Thus our model can be viewed as belonging to the Nash Program, which has been described as "an effort to model cooperative behavior as a noncooperative game by modelling all communication and attempts to coordinate behavior as explicit moves in the extensive form game" (McKelvey [16, p 2]). The present model is thus somewhat akin to the work of Rubinstein [17] and others, in which bargaining is modelled in a non-cooperative framework. In order to show the relevance of the model to real-world interaction in the presence of externalities, we still must justify our assumption of a two-stage game with matching rates that are either binding or self-enforcing. This can be done in two ways. The simpler of the two is to posit the existence of legally binding, unilateral contracts, in which these matching commitments are made. A possible objection to such an assumption would run as follows. Instead of assuming a two-stage game in which linear matching functions are chosen, one could simply let actors sign conditional contracts in which the Pareto optimal levels of their externality-causing activities are specified, with the proviso that if any other agent in the relevant society does not sign such a contract, one's own externality-causing activity would revert to its individually optimal (Nash) level. All actors would optimally sign such contracts. It should be noted, however, that such a solution requires all Pareto optimal levels of individual externality-causing activities to be known in advance. Our mechanism, in contrast, allows these Pareto optimal levels to emerge as a Nash equilibrium, much as individual demands for a public good See Guttman [10] and the references cited in Guttman [11, n~.3] for empirical evidence of matching behavior. See also the discussion in [11] of U.S. Federal-state matching grants as an example of matching behavior.

Solution of externality problem

75

are revealed by the Groves-Ledyard [7] mechanism. 2 Individual agents need not know Pareto optimal externality-causing actions before entering into the process. (It should be noted, in passing, that any agent is free to abstain from the matching game by choosing a zero matching rate. Such zero matching rates, however, are not in general individually optimal, as we show.) But since such matching contracts are rarely observed in practice - exceptions are in intergovernmental finance of public investments and in philanthropic fundraising - this would be a normative, rather than descriptive interpretation of our model. A seemingly more descriptive approach would be to postulate that agents have an incentive to maintain reputations for matching at the rates supported by a Nash equilibrium in our model. In a repeated game, such reputations can have considerable value. In order to avoid the Folk Theorem result for infinitely repeated games of infinitely many equilibria (including Pareto optimal equilibria), we note that real-world interactions of the type analyzed here generally occur in finitely repeated relationships. Kreps et al. [15] have shown that if agents are slightly uncertain that they are playing against agents who mechanically use a tit-for-tat (TFT) rule in the finitely repeated Prisoner's Dilemma, then the unique equilibrium of a sufficiently large number of repetitions is cooperation over most of the game. In the game of Kreps et al., a rational player mimics TFT behavior in order to maintain his opponent's doubt that he is a TFT type. This uncertainty of the agents that they are playing against "irrational" TFT players has been explained, using an evolutionary model, by Guttman [12]. Guttman [12] analyzes a population consisting of two types, rational maximizers and mechanical TFT players. The rational actors play sequentially rational strategies against the population mixture of rational and TFT types. Guttman shows that, if there is a cost of making optimizing calculations, and if the number of rounds in Prisoner's Dilemma interactions is not too small, there exists an evolutionarily stable mixture of rational and TFT types, in which the proportion of TFT types is high enough to support cooperative, TFT-like behavior by the rational actors. We do not formally model this reputation-maintaining mechanism here because of the complexity of the matching game, even when the reputation-maintaining mechanism is assumed a priori to operate. Our treating the matching precommitments as if they were binding contracts reflects our view that there remain interesting problems to explore in understanding the incentives to cooperate in the presence of externalities, aside from the issue of enforcement of matching commitments. Thus our work complements the work of Kreps et al. [15] and Guttman [ 12], in which the focus is on the reputational mechanism, and in which a simpler underlying game is assumed (i.e., the binary cooperate-defect choice of the Prisoner's Dilemma). Section 1 of this paper outlines the matching game, and presents a lemma which defines matching rates that induce a Pareto optimal equilibrium in the game. Section 2 proves the existence of an equilibrium characterized by these matching rates, when there are two actors. We believe that this case captures the essential aspects of the matching game as a solution of the externalities problem. 2 Admittedly, one type of explanation of how players arrive at a Nash equilibrium involves their knowing each other's payoff or utility functions (see, e.g., Crawford [4, p 196]), and from this knowledge the Pareto optimal levels of the externality-causing actions can be derived. Nevertheless, if the actors happen to be at such an equilibrium they would have no reason to depart from it, even without such knowledge of other actors' utility functions.

76

J. M. Guttman and A. Schnytzer

Section 3 concludes the paper, and suggests an explanation of observed inefficient outcomes by relaxing the assumption of perfect information of one's counterparts' matching rates.

1. The matching game There are n individuals, each of whom chooses a (non-negative) action x~ which affects the payoffs of the other individuals. Let u~(x 1..... xn) be the payoff of actor i. This payoff is assumed to be measurable in monetary terms, in order to allow interpersonal comparisons - in particular, to allow us to identify a Pareto optimum with a maximum of the sum of the payoffs. By assumption, ~3ui/~ xj ~=0 for all i~=j. As stated in the Introduction, x~ is determined implicitly, as the result of two separate decisions in a two-stage game. The first decision of actor i is to choose a vector of "matching rates" b~.j, which link his xi to the actions of his fellow actors, in a manner to be specified. Actor i's rationale for choosing matching rates is that this provides other actors with an incentive to act in a manner which benefits actor i. For example, if xj positively affects ui, then actor i may wish to choose a positive matching rate, thus giving actor j an incentive to increase xj. Thus the matching rate b~j can be viewed as an indirect subsidy offered by actor i to actor j, akin to the direct compensation envisioned by Coase. These matching rates are simultaneously chosen by the n actors in the first stage of the game, and the equilibrium in this stage is a Nash noncooperative equilibrium. Once the matching rates are chosen, they become common knowledge of the actors. The second decision of actor i, which is made in the second stage of the game, is to choose a non-negative "flat action" a~. This flat action determines (in conjunction with the matching component of x~) the level of xi, according to

xi=ai@ 2 bijaJ •

(1)

j~i

[The matching rate biy is constrained to be at least the value that makes x i zero; for identical ai across actors, this means that bij> - 1 / ( n - 1).] The first term on the right-hand side is actor i's flat action, and the second term is the matching component of his total total action x~. The n actors simultaneously choose their flat actions in the second stage of the game, and the equilibrium is again a Nash equilibrium. Actor i chooses" his optimal matching rates b~j as follows. Given the matching rates of the other actors, any specific choice of a vector of matching rates bij by actor i completes a matrix of matching rates b. For any given matrix b, actor i can predict the Nash equilibrium in the upcoming subgame determining the flat actions, which we denote as Gb. 3 Denote the equilibrium vector of flat actions in Gb as a (h). Given a (b) and the matrix of matching rates b, the individual actions x i are computed according to (1). Once the action vector is predicted, actor i can predict his payoff u i as a function of b - or, given the vectors of 3 In principle, there could be a problem of multiple equilibria here, which would impede the ability of an actor to predict the effect of a choice in his matching rate on the equilibrium in Gb. We deal with this problem in our analysis, particularly in the proof of Proposition 3.


77

matching rates of the other actors - as a function Of bij. Actor i chooses bij so as to maximize ui (b;j). Since the biy are chosen on the assumption that the subgame Gb will be in equilibrium, the equilibrium of the bij - together with the equilibrium of Gb constitute a subgame perfect equilibrium for the overall game. We can now prove the following Lemma, which will be used in the following section.

When bii = Pareto optimal. Lemma.

[~lgi/~ Xj

]/[~Uj /~Xi] , the equilibrium in the matching game is

Proof Let the action vector at equilibrium be )?. By definition of Pareto optimality, 4 ~, ~u~/~xj = 0 ,

(2)

i

for a change in the action xj of any actor j. We assume that the payoff functions ui(x~,..., xn) have the necessary convexity properties to insure (2) is a sufficient condition for Pareto optimality. Now the flat action ~i(b) that maximizes i's payoff given b and the rest of the flat actions, at an interior solution, satisfies aui/aa i (x): 0

i.e., ~, ( ~ u , / ~ j ) (~2j/~a,) (5:) = 0 j=l

which implies

~ui/~-~i~- 2 (~ui/~j)bji ~-0 • j:#i By assumption, bji= [Ouj/Oxi]/[~ui/Oxj]. Then (3) becomes

(3)

2 0uj/~x~=O , j=l

which is precisely the condition for Pareto optimality (2).

Q.E.D.

This Lemma states that if the matching rates bij are the ratios of the marginal payoffs that each of a pair of actors receives from his counterpart's externalitycausing action, then the matching game produces a Pareto optimal equilibrium. In the following section, we show that - if there are two actors - such matching rates are indeed an equilibrium vector of matching rates in the game.

2. Existence of Pareto-optimal equilibrium for n = 2 Suppose there are two individuals, each of whom engages in an activity with a positive external effect on his counterpart. (We show below which of our results

4 Cf. Buchanan and Stubblebine [3].

78


carry over to the case of a reciprocal negative externality.) This positive externality, while reciprocal, need not be symmetric - i.e., the marginal payoff to agent 1 from agent 2's action may be greater or less than the marginal payoff to agent 2 from agent l's action. In order to determine whether the Pareto optimal matching rates defined by the Lemma of the previous section can emerge as an equilibrium of the matching game, we first must get a fuller understanding of the second stage of the game, in which the "flat" actions are chosen. Let a* denote the best response of actor i to the "flat" action of actor j. We first wish to demonstrate the following Proposition 1. Proposition 1. Given the Pareto optimal matching rates defined in the Lemma, and for interior solutions for actor i (i.e., a* > 0), any change by actor j in his flat action will lead actor i to make a compensating change in his optimal flat action that leaves his own overall action x i constant.

Proof This Proposition 1 can be demonstrated by reference to Fig. 1, which depicts an indifference curve for actor 1 in x ~ - x2 space. Since x~ and x2 are both "goods" for actor 1, one might expect the usual, downward sloping indifference curve. In the neighborhood of the Pareto optimum, however, actor 1 is induced by actor 2's matching to engage in his externality causing activity beyond the point at which its private marginal payoff is zero, so that the social marginal payoff at the Pareto optimum (which includes actor 2's positive marginal payoff) will be zero, as required by (2). Thus the marginal payoff to actor 1 from x~ is negative, and his indifference curve is positively sloping, as shown in Fig. 1.

p, /Z

J

,,/""7 / //

/

o'

////I pill ./.#A 0

B

al*

>

Fig. 1. Determination of optimal flat action at Pareto optimal equilibrium

xf

Suppose that actor 2's flat action a 2 is initially zero. Then, by (1), x 1= al, and x2=b2~ a~. Thus the opportunity locus facing actor 1 is the ray OO' in Fig. 1, whose slope is bzl. That is, if actor 1 sets a~ = 0, then x~ = x 2 = 0. As he increases a~, both x~ and x 2 increase linearly. Point A denotes actor l's optimal point, determining a*. Now let actor 2 increase his flat action to an arbitrary level a °. (We assume that a ° is not so large as to drive actor 1 to a corner solution of

79


a i = 0 ; this would occur if A was between O and B' in Fig. 1.) At first glance, this should shift actor l's opportunity locus up to P P ' . But this neglects the fact that actor 1 must match the increase in a 2 at the rate b~2. Note that, given our assumption, that the matching rates are those specified in the Lemma, b12 = 1/b2~. Therefore the increase in x 1 that will be induced by the increase in a 2 from zero to a °, is precisely O B . Thus, even if actor 1 were now to choose a flat action equal to zero, he would still have to engage in x 1 at least at the level O B . Therefore, his opportunity locus does not shift to P P ' , but rather becomes B ' O ' , which is the same as OO' without the segment O B ' . His optimal point (assuming an interior optimum) therefore r e m a i n s a t A. His flat action, which is chosen to bring x I to its individually optimal level, now is ( a * - O B ) , and x~ remains the same as before. This is, of course, precisely what Proposition 1 states. Proposition 1 easily generalizes to the case of a reciprocal, negative externality. The indifference curve in Fig. t becomes concave rather than convex from below, since now x~ has a positive marginal payoff for actor 1 and a negative marginal payoff to actor 2. [At the Pareto optimum, actor 1 is induced to refrain from bringing x 1 up to its privately optimal level, which is where its (private) marginal payoff is zero.] By the Lemma, the matching rates at the Pareto optimum are again positive, implying that the opportunity loci look just as in Fig. 1. Thus the proof carries through to this case. The only difficulty in extending the analysis to the case in which x~ affects actor 2 positively, but x 2 affects actor t negatively, is that the Lemma then implies negative matching rates at the Pareto optimum, and we have restricted the matching rates from being so negative as to make the x i negative. This restriction may well prevent the matching game from achieving a Pareto optimum in this case. Given Proposition 1, we can easily draw best-response functions for the subgame Gb; see Fig. 2. Let x* be the individually optimal level of xl (given b2~). We have, by (1), a* = x ~ - b12 a 2 .

Differentiating totally (given b12) and recalling Proposition 1, da*l = d x * - b12 d a 2 = - b~2 d ( l 2

,

implying that (for interior solutions) da*/da 2 =

-

b12

.

Similarly, d a * / d a 1 = _ b21 = _ l/bl2 .

Thus we obtain two parallel, linear best-response (or reaction) functions in a~ - a 2 space, as in Fig. 2. The heavy lines on the axes represent corner solutions, in which a i is driven to zero by high levels of aj. (It will soon be clear that these two reaction curves coincide at the Pareto optimal equilibrium. They are drawn separately for expositional convenience. )S

5 It should be noted that, if the actors are not at the Pareto optimal equilibrium generated by the matching rates specified in the lemma, the slopes of the reaction curves in Fig. 2 are not ~-b12 and -b2~. The slopes that actually would obtain are discussed in the Appendix.

8O


E

(11

Fig. 2. Best response curves for flat actions at Pareto optimal equilibrium

R F 1 is the reaction function of actor 1. Its intercept on the a~ axis is the flat action needed to induce x* when a 2 = 0. Since, in this case, x* = a 1, this fiat action is x*. The slope of R F 1 is - 1 / b 1 2 (recall that d a * / d a 2 = - b ~ 2 and that a~ is measured on the horizontal axis). Thus the vertical intercept of R F 1 is x~/bi2. If a 2 rises above this level, actor 1 will be at a corner solution with a* = 0. (This occurs, in Fig. 1, when B ' is northeast of A on O O ' . ) The intercept of R F 2 on the a 2 axis is, by similar reasoning, x*. Comparing this with the vertical intercept of R F 1, we find that the condition that R F 2 enclose R F I , as in Fig. 2, is that x* > x * / b t 2 . If this condition obtains, then, in the equilibrium of the stage determining the fiat actions (E in Fig. 2),

a*=0

and

a*=x*

.

(4)

If, on the other hand, x*/b~2 > x * , then, in the equilibrium of this stage, a* 1 -- -

*

X 1

and

x*=0

(5)

Only when x~/bxa = x* will the two reaction functions coincide, and then there will be an infinite number of equilibria in Gb, but all imply the same x~ and x 2 (by Proposition 1), and thus all yield each actor with the same utility. Since the matching rates specified in the L e m m a assume an interior solution for all actors, it follows that at these matching rates, x * / b l 2 = x * , and R F 1 and R F 2 coincide. We can now prove the existence of a Pareto optimal equilibrium for the case of external economies, which is asserted in the following Proposition.


81

Proposition 2. In the case of positive reciprocal externalities, if both x I and x 2 are normal goods for both actors (in the sense specified in the Appendix), then there exists an equilibrium in the overall matching game, in which the matching rates are those given in the Lemma.

Proof The first step in the proof is to identify the direction of change of x* and x* when there is a deviation in either matching rate from the rates given in the lemma. The first-order condition in Gb for actor 1 is aUl/~Xl _c b21 ~U 1/~X2

~_

0 ,

implying (~U 1/~X1)/(~U 1 / ~ X 2 ) = -- b2i .

(6)

We assume that (7)

a2Ul/aX 2 < 0 .

It is also assumed throughout that

~2u,/ax ~,
al

.~a I

Fig. 4. Shifts in best response functions for flat actions: negative externality

a2 A

C \\

C!

A

equilibrium can result (point D in the figure), as well as two corner equilibria, at A and C. By the argument given at the beginning of the proof, equilibrium point A gives the deviating actor (actor 1) the same payoff as before his deviation, since it is on the original (common) reaction line, A A ' . In order to consider equilibrium point D, we must refer to the Appendix, which presents an analysis of the best-response function in Gb of an actor when the matching rates are not those which lead to a Pareto optimal equilibrium. While the analysis there deals with the case of positive externalities, it can be made to apply to the case of negative externalities as well, with appropriate modifications. From Fig. 5 in the Appendix, it is clear that when an actor reduces his matching rate from the level required to support a Pareto optimal equilibrium (and the other actor keeps his matching rate constant at the corresponding level), the deviating actor's payoff increases with an increase in the other actor's flat action. In the case of negative externalities, the indifference curves are concave rather than convex, and the deviating actor's payoff decreases with an increase


85

in his counterpart's flat action. Conversely, if the deviating actor increases his matching rate, the shift in the reaction lines (the opportunity loci) in Fig. 5 are in the opposite direction, and the deviating actor's payoff decreases (for positive externalities) and increases (for negative externalities), with an increase in his counterpart's flat action. This implies that actor l's payoff in Fig. 4c increases as we move up his reaction line from A to B. Therefore D represents a higher payoff level than A, and thus if D were to result from actor l's deviation in his matching rate, his payoff would increase. Turning now to equilibrium point C, if we could show that it represents a higher payoff level for actor 1 than B, This would imply afortiori that actor l's payoff is higher at C than at A. Note that a 2 is higher at C than at B. Since actor 2's action negatively affects actor 1, this represents a cost of going from B to C. For infinitesimal changes, this cost will be u2 da2, where u 2 is actor 1 's (negative) marginal payoff from x2. On the other hand, going from B to C allows actor 1 to increase his action by b~2 da 2, increasing his payoff by the same amount times his marginal payoff from his own action, u~. Thus the change in his payoff is (bl2Ul + u2)da 2. Before actor 1 increased his matching rate, the term in parentheses was [by (6)] zero. Now that his matching rate has increased, this term is positive. Thus actor l's payoff is higher at C than at B, implying that his payoff is higher at C than at A. Thus, if the new equilibrium in Gb were to occur at C, actor l's upward deviation would again be worthwhile. To sum up, actor l's payoff may increase, and cannot decrease, from an upward deviation in his matching rate from the level required to attain a Pareto optimal equilibrium, when actor 2's matching rate stays constant. Thus, when the configuration of the reaction lines following the deviation is as shown in Fig. 4c, the Pareto optimal equilibrium does not exist. Q.E.D. If we examine the effect of a downward deviation in actor l's matching rate, we find that the reaction lines in Fig. 4c are simply reversed, and there is a unique, interior equilibrium. Using the analysis of the Appendix, it can easily be shown that this interior equilibrium leads to a lower payoff to actor 1 than what he received before his deviation. Thus the only deviation that actor 1 would want to make in his matching rate is in an upward direction. An example of a negative reciprocal externality is duopoly. Each firm's output has a negative effect on its rival's profits, and a positive effect on its own profits. Proposition 3 implies that under certain conditions (specifically, that the marginal profit of the firm's own output is relatively insensitive to increases in its output, and that its marginal loss from its rival's output is relatively insensitive to its rival's output), a perfectly collusive outcome could result from the matching game. Under other conditions, and for certain specifications of the form of the demand and cost functions, this outcome may be unattainable. 6 Guttman and Miller [13, Sect. 3], analyzed such a duopoly game, using linear demand and quadratic cost functions. Specifically each firm faces a demand function p=K-(q,+qj)

,

6 For an analysis of "price matching" somewhat similar in spirit to the present model, see Doyle [6]. In Doyle's analysis, however, prices (not quantities) are matched, and the strategy space does not allow for continuously variable rates of matching.

86


and has a cost function Ci=O~iqi 4- fliq2i ,

where the subscripts i and j represent the two firms, and the parameters K, e, and/~ are all positive. The (linear) reaction functions for the two firms in Gb are given in Eq. (6) in [13]. Using that equation, the following coefficients can be calculated to be the vertical coordinates of B and C, and the horizontal coordinates of A and C', in Fig. 4c, assuming identical firms for simplicity: B = ( K - ~ ) / [ ( 1 + e) (2 + 2/~ + 1)] C =(K-cO/[4+2fl+2e]

A =(X-a)/[4+2Pl C' = ( K - c Q / [ 4 + 2 fl + e I ,

where e is firm i's deviation from unity in its matching rate, while firm j keeps its matching rate at unity. It is easily verified that, for positive e, we obtain the outcome depicted in Fig. 4c, implying that the perfectly collusive outcome is unattainable in the specification of [ 13]. Indeed, the simulations reported there all yielded only imperfectly collusive equilibria. In the simulations with identical firms, for example, the equilibrium matching rates were considerably above unity (the level required for a Pareto optimal equilibrium), consistent with the result obtained above regarding the optimality of upward deviations in the matching rates from the levels required for a Pareto optimum. 3. Concluding remarks The above analysis illustrates the usefulness of the matching game as a Coaselike, voluntary solution of the externalities problem. Given the generalization of this game to the n-persori case in the public goods problem as presented by Guttman [11] and by Danziger and Schnytzer [5], we expect that the game can be extended to the n-person case in the externalities problem as well. Such a generalization would thus circumvent the difficulty of extending the Coase Theorem to the n-person case using cooperative game theory, noted in the Introduction. We should stress that, in order to derive empirical implications from our analysis, the theory must first be modified to take account of imperfect information, particularly of other actors' matching rates. As pointed out by Guttman [8-11] and recently demonstrated by Sandler, Sterbenz and Posnett [18], such information costs can easily impede the solution of the externalities problem. In the present analysis, these costs were assumed away. Thus the theory of strategic matching leads to a characterization of Pareto-relevant externalities as externality situations in which information is costly, instead of situations in which property rights are ill-defined. We have made little progress in studying the uniqueness of the Pareto optimal equilibria examined in this paper. In [ 14] we show, however, that for two identical actors with preferences of the form ui = (x, x~) ~ -

xi

,


87

where 0 < e < 1/2, the Pareto optimal equilibrium is unique a m o n g the symmetrical equilibria.

References 1. Aivazian VA, Callen JL (1981) The Coase theorem and the empty core. J Law Econ 24: 275-281 2. Aivazian VA, CaUen JL, Lipnowski I (1987) The Coase theorem and coalitional stability. Economica 54:517-520 3. Buchanan JM, Stubblebine WC (1962) Externality. Economica 29:371-384 4. Crawford VP (1985) Dynamic games and dynamic contract theory. J Conflict Resolution 29:195-224 5. Danziger L, Schnytzer A (1991) Implementing the Lindahl voluntary-exchange mechanism. Eur J Polit Econ 6. Doyle C (1988) Different selling strategies in Bertrand oligopoly. Econ Lett 28:387-390 7. Groves T, Ledyard J (1977) Optimal allocation of public goods: a solution to the free-rider problem. Econometrica 45:783-809 8. Guttman JM (1978) Understanding collective action: matching behavior. Am Econ Assoc Papers Proc 68:251-255 9. Guttman JM (1984) Matching behavior and collective action: theory and experiments. BarIlan University Working paper # 8406 10. Guttman JM (1986) Matching behavior and collective action: some experimental evidence. J Econ Behav Organ 7:171-198 11. Guttman JM (1987) A non-Cournot model of voluntary collective action. Economica 54: 1-19 12. Guttman JM (1990) Rational actors, tit-for-tat types, and the evolution of cooperation. Unpublished, Bar-Ilan University 13. Guttman JM, Miller M (1983) Endogenous conjectural variations in oligopoly. J Econ Behav Organ 4:249-264 14. Guttman JM, Schnytzer A (1988) Strategic interactions and the Coase theorem. Unpublished, Bar-Ilan University 15. Kreps D, Milgrom P, Roberts J, Wilson R (1982) Rational cooperation in the finitely repeated Prisoner's Dilemma. J Econ Theory 27:245-252 16. McKelvey RD (1988) Evolutionary explanations of rationality paradoxes. Unpublished, California Institute of Technology 17. Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97-110 18. Sandler T, Sterbenz FP, Posnett J (1987) Free riding and uncertainty. Eur Econ Rev 31: 1605-1617

Appendix The purpose o f this Appendix is to define the "normality" o f preferences assumed in Propositions 2 and 3, and to show that, at matching rates which do not satisfy the condition b~2= 1/b2~ (a condition which characterizes the matching rates specified in the Lemma), Proposition 1 does not apply. See Fig. 5, which is similar to Fig. 1, with the following difference: the matching rate o f the relevant actor (say actor 1) is less than the reciprocal o f his counterpart's matching rate. As in the p r o o f o f Proposition 1, actor 2 increases his flat contribution f r o m zero to OP, and one would expect (before taking account o f actor l's c o m m i t m e n t to m a t c h the increase in a2 at the rate b~2) that actor l's o p p o r t u n i t y locus would shift up to PP'. Once one takes account o f actor l's matching commitment, his opportunity locus shifts back (i.e., downward), but, because b12 < 1/b2~, the shift is not all the way back to the original


88

opportunity locus OO'. Rather, his opportunity locus shifts only to QQ', the shift being determined by the extent of his matching (in this case, measured by OB). Thus actor l's optimal point shifts from A to A', implying a reduction in his optimal level of x 1 from x* to x*'. I P' X2J

I i i I I I II/

..?'

li i i

0

I

/I

/I /

i/

i /

i

i

A

P

?

•

0

I

B

I

I

I

I

BI xfx~

×1

Fig. 5. Determination of optimal flat action outside Pareto optimal equilibrium

At this point, we define the "normality" of preferences assumed in the text. This normality is characterized by A' being northwest of A. While x* has decreased, the matching component of x 1 has increased from zero to OB. Note that this increase in the matching component of x1 is less than the corresponding increase in Fig. 1 by BB'. Because of the normality of actor l's preferences, (x* - x * ' ) < BB'. This, in turn, implies that the reduction in a~ following the increase in a 2 is less than the corresponding decrease in Fig. 1. Thus we obtain the rotation of RF~ in Fig. 3 a. Note that this result would be obtained in the case of negative externalities as well, it being necessary only to make the indifference curves concave from below. The definition of normality, in this case, is unchanged. The analysis for the case in which hi2 > 1/b2~ is exactly parallel to the above analysis, resulting in the rotation of R F 1 in Fig. 3b.