the cost of the game: a taxonomy of social interactions

Ž .European Journal of Law and Economics, 9:2; 99]114 2000Q 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

The Cost of the Game: A Taxonomy ofSocial InteractionsFRANCESCO PARISI [email protected] Mason Uni ersity School of Law, 3401 North Fairfax Dri e, Arlington, VA 22201

Abstract

This paper considers the role of reciprocity rules in various strategic settings. After distinguishing fourtypical categories of social interaction, the paper examines three forms of reciprocity constraints. An

Ž .ideal rule of perfect incentive alignment ‘‘structural reciprocity’’ serves as a benchmark for the analysisof a ‘‘golden rule’’ of reciprocity, characterized by a mechanical linking of one player’s strategy tothat of the other player; and a ‘‘silver rule’’ of stochastic reciprocity, characterized by a probabilisticsymmetry in the relationship between the players.

Keywords: Customary law, social norms, reciprocity, game theory

This paper explores the effect of reciprocity-type constraints for the emergence ofefficient spontaneous equilibria, resisting the idea that all stable social practicesare efficient, given the preferences of the group’s members. In Part I, I distinguishfour typical categories of social interaction for the purpose of delimiting thesituations that pose problems for efficient spontaneous outcomes. In doing so, Imap the relationship between the payoffs of two players in a two-dimensional spacefor an immediate visual understanding of the outcomes in terms of aggregatewelfare. In Part II, I distinguish three general forms of reciprocity, namely,structural reciprocity, induced reciprocity and stochastic reciprocity, and considerthe effect of such constraints on the problematic cases of spontaneous order.

The conclusions reached shed some light on the understanding of real examplesof customary law, such as the medieval law merchant and the evolution ofinternational customary law, as well as explaining the role of legislatively inducedreciprocity constraints as aids for the emergence of spontaneous law.

I. A taxonomy of social interactions

Game theory appraises a distinctive subset of the general problems of optimalchoice, characterized by the interdependence of individual choices,1 and distin-guishes broad groups of strategic interaction under the general labels of conflictand coordination.

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The game-theoretic categories are useful to appraise the wide array of socialinteractions. I shall proceed considering four typical categories of strategic interac-tion.

1. The Pure Common-Interest Case. The first class of games consists of positive-sum games characterized by a dominant strategy which yields a Pareto optimaloutcome. The optimal outcome is obtained in a stable Nash equilibrium. SchellingŽ . 21980 describes this category as ‘‘Pure Common Interest Games,’’ while otherscholars have used the equally intuitive label of ‘‘Self-Enforcing Contracts.’’3 Givena choice between entering into an exchange or not, the payoff structure of thegame guarantees that if both parties find the exchange strategy beneficial, ex-change will obtain in equilibrium.

Amounting to what is known in the economic literature as the notion of ‘‘perfectincentive alignment,’’ the game’s payoff structure excludes the possibility of shirk-ing and opportunism. One could think of this ideal environment as the result ofoptimal contract enforcement mechanisms, institutional safeguards, relationshipsinvolving trust and reputation, or any other device which renders adversarialpossibilities non advantageous or inaccessible to the players. In this category ofgames, there is no temptation to defect unilaterally because there are safeguardsthat eliminate all the payoff advantages of unilateral defection.4

Generally speaking, solutions to this class of games are not troublesome, sinceall players gain by cooperating. Perfect incentive alignment guarantees that thespontaneous equilibrium of the game will occur at an optimizing point.5 Whetherthe incentive-alignment is endogenous or exogenously determined by existing lawsor norms, there is no need for additional intervention in either situation.

2. Battle of the Sexes Games. This well-known class of games encompassespositive-sum games characterized by mixed conflict-coordination motives. Althoughthe coordination problem is likely to be solved by allowing sequential decisionmaking or pre-commitment strategies, whenever the payoff structure is asymmetricthere is no guarantee that an equilibrium will be achieved where the aggregatepayoffs are maximized. Generally speaking, it appears that all games characterizedby multiple Nash equilibria with non-constant sums generate a mixed set ofincentives for the participating players, with the potential for costly miscoordina-tion.

Laws and norms may be unnecessary with respect to this group of cases, in thatŽseveral solutions are available to induce coordination e.g., sequential moves,

.signaling, pre-commitment strategies, etc. . The conflictual component of thisgroup of cases leaves, instead, room for the emergence of norms or rules of lawwhich may avoid the efficiency loss produced by a suboptimal conflictual outcome.For example, a fairness norm may induce a ‘‘middle-way’’ solution that maximizesaggregate utility under conditions of decreasing returns. Similarly, repeated inter-action under conditions of role-reversibility may induce convergence toward anoptimal outcome, given sufficiently small discount rates. Finally, keeping thecomplete information assumption in mind, courts and legislators may induce the

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Žoptimal equilibrium by assigning rights e.g., under the form of a first-mover.advantage, or the like .

3. The Prisoners’ Dilemma Games. The third group of games, of which thePrisoners’ Dilemma represents the best known example, encompasses positive-sumgames characterized by a dominant strategy which yields a suboptimal outcome forboth players. In these games, the privately rational strategies undertaken by theparties yields an outcome that is Pareto inferior to alternative non-equilibriumoutcomes. Cooperation problems arise because some or all participants are facedwith dominant defection strategies.

As it is known, absent contract enforcement mechanisms, even Pareto improvingexchanges in the marketplace take the form of a conflict game similar to the wellknown Prisoners’ Dilemma. Exchange will not occur in equilibrium, so that thesurplus value of the potential exchange remains unexploited.6 This group ofstrategic situations is thus plagued by the presence of opportunistic behaviorbecause of the potential accessibility of the off-diagonal non-cooperative outcomes.7

As known, in these games optimizing equilibria may be promoted through enforce-Žable contracts, social norms or formal law e.g., imposing social sanctions, reputa-

.tional costs, or legal penalties for unilateral defection .In Part II of the present paper, it will be shown that reciprocity constraints of the

‘‘golden’’ type will be able to generate cooperative outcomes under generallyplausible conditions.

4. The Inessential Games. The last category encompasses two groups of games:Ž . Ž .1 zero-sum games, and 2 positive-sum games where all obtainable Nash equilib-ria have a constant aggregate payoff. These two groups have an important common

Ž .feature: all the relevant outcomes i.e., those obtainable in equilibrium yieldconstant total payoffs. The winners gain an amount equal to the losers’ loss.8 I callthis category ‘‘Inessential Games,’’ a label which political scientists sometimes useto describe these kinds of conflict situations.9

While pure conflict situations arise in the absence of gains from reciprocalŽ .cooperation or coordination i.e., in pure constant-sum games , the class of

Inessential Games is a composite category which encompasses other games thatyield constant payoffs in the relevant sets. Although a constant sum characterizesthe relationship between all Nash equilibria, no single outcome is mutually pre-ferred by the two players.

As it is known, the outcomes obtained in a zero-sum game are on the sameindifference curve of a social welfare function. The same can be said of thebroader class of Inessential Games, as defined above. However, this conclusion isof limited relief to policy analysts, for truly inessential games are empirically rare.

Furthermore, prior to classifying a game as inessential, the incentives and socialcosts generated by the game should be considered.10 For example, litigationbetween two private parties is usually modeled as a zero-sum game. Yet differentprocedural rules may affect the extent of resource dissipation in the parties’ racefor the appropriation of the rents from litigation. Likewise, compensation rules intorts may appear as addressing a zero-sum problem, yet the choice of alternative

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rules is crucial for the parties’ ex ante incentives. Thus, what may appear anessential game on the basis of the private payoffs from the game, it may constitutea negative sum game, once the social costs are taken into consideration.

The cost of the game

As discussed in the previous paragraphs, individual rational self-interest does notguarantee that a Nash equilibrium obtains at a point of aggregate optimization.Legal scholars have long recognized the significance of strategic behavior whenconsidering how legal rules affect individuals. In the principal applications of gametheory to the law, legal rules have been analyzed as corrective devices for gameinefficiencies. Law promotes what individuals cannot spontaneously achieve, orhelps them avoid the traps of strategic social interaction. The logic of thisargument, however, can be reversed in order to use game theory to explain thespontaneous emergence of law. When appraising spontaneous sources of law usinggame theory, one should consider the incentive structure of the originating envi-ronment as well as the possible role of strategic behavior in affecting the equilib-rium outcome.

Figure 1 shows the possible areas of equilibrium in terms of the relationshipbetween the players’ payoffs. The relationship between the payoffs of two players ismapped in a two-dimensional space, allowing for an immediate visual evaluation of

Ž .the outcomes in terms of Pareto and Kaldor-Hicks efficiency Figure 1 .In Figure 1, the axes represent the respective payoffs for the two players. In

order to avoid unnecessary complexity, I assume a linear relationship betweenpayoff and utility, so that the axis also represent the level of utility to the individualplayer.11 While such a classification obviously omits many important details, itprovides a viable benchmark for the purpose of the present analysis.

Ž .It is worth noting that customs and patterns of behavior that fall in area 1 , ifobtainable through a Nash strategy of the players, will emerge even in the absence

Ž .of side-payments or relational bonds. Conversely, outcomes obtained in areas 2Ž .and 3 create asymmetric incentives for the two players: one player receives

negative payoffs, while the other receives positive payoffs. The criterion of Kaldor-Hicks efficiency further allows to draw a qualitative distinction between the

Ž . Ž .outcomes obtained in area 2 and those obtained in area 3 . Indeed the zero-sumŽ .line separates the two sets, such that all the points found in area 2 represent a

Ž . Ž .social improvement over the status quo origin , while all points found in area 3yield a social welfare loss.

For the purpose of this paper, it should be noted that even though points in areaŽ .2 are socially preferable to the status quo, those outcomes will be achieved inequilibrium only if the players have access to enforcement mechanisms of somesort. Likewise, given an exit option, contract or relationship will be necessary to


Figure 1. The cost of the game.

ensure that the gainers properly compensate the losers. Only under such circum-stances, all parties will be better off by participating in the game and concurring inthe formation of a welfare-enhancing practice.

In the following section, I will consider the effect of reciprocity mechanisms onthe ability of self-interested players to achieve a socially desirable outcome. Theanalysis will shed some light on the understanding of historical illustrationsregarding the emergence of customary law, where close-knit relationships, familybonds and other forms of structural or stochastic reciprocity allowed parties tocapture some of the welfare benefits of the asymmetric payoffs from the game.

II. Reciprocity constraints and the rules of social interaction

According to the theory of norms, there are two formative elements of sponta-Ž .neous social rules: 1 the practice should emerge out of the spontaneous and

Ž .uncoerced behavior of various members of a group, and 2 the parties involvedmust subjectively believe in the obligatory or necessary nature of the emerging

Ž . 12practice opinio iuris . To an economist, the first element is essentially a ratherstandard assumption of rational choice, while the second element may be viewed as

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a belief of social obligation, emerging in response to game inefficiencies, in supportof rules of behavior that avoid aggregate losses from strategic behavior.13

While these elements represent two necessary conditions for the emergence ofcustomary rules, they do not provide much information on the remaining suffi-ciency conditions for the emergence of efficient custom. In this section, I examinesome additional structural conditions for the emergence of spontaneous socialrules, whenever the strategic conditions of the game are such that socially optimaloutcomes are not achievable through Nash strategies.

The stylized settings considered below shed light on the more problematic casesof conflicting incentives with inconsistent individual preferences over alternativeoutcomes. I suggest that in conditions of stochastic or induced symmetry, sponta-neous processes of law formation may be successful even in the presence oforiginally misaligned individual interests of the Prisoners’ Dilemma type and of theasymmetric Battle of the Sexes type.

( )A. The ideal setting: structural reciprocity perfect incenti e alignment

Perfect incentive alignment occurs when the parties’ rankings of preferencesconverge toward a mutually desirable outcome. As known, this condition impliesthat neither party has an incentive to defect unilaterally, nor has a reason to feardefection by the other party. Structurally symmetrical situations are relativelyeasily characterized by stable relationships of mutual cooperation. Because strate-gies that maximize individual expected payoffs also maximize group welfare, no onehas any reason to challenge the emerging equilibrium.

Ž . Ž .The perfect alignment of interests can be i endogenous, or ii exogenouslyinduced. The former case implies that the players naturally find themselves in sucha heavenly relationship. The latter case implies that the parties have been able toovercome an}otherwise present}conflict of interests through exogenous devices.Such is the case of an exchange supported by a perfect contract enforcementmechanism. Paradoxically, therefore, there is no need for law or norms in anenvironment already characterized by perfect incentive alignment, as contracts andsocial arrangements are self-enforcing.

The perfect alignment of individual interests, however, rarely occurs in real lifesituations. In the absence of proper enforcement mechanisms, even a Paretoimproving exchange opportunity creates a temptation for shirking and ex postopportunism. When shirking and post-contractual opportunism becomes a domi-nant strategy for one or both players, the exploitation of opportunities for mutualexchange becomes difficult or unobtainable.14 The absence or inadequacy ofcontract enforcement mechanisms leads to the search and eventual emergence ofalternative safeguards.

The following discussion focuses on conditions of reciprocity and role-reversibil-ity as alternatives to the above described structural harmony.


( )B. The golden rule: induced reciprocity symmetry constraint

Golden rules of reciprocity can be thought as the result of a successful binding ofeach player’s strategy to that of his opponent. Automatic reciprocity of thegolden-type creates a symmetry constraint on the players’ strategies, producing animportant change in the traditional results.

Unlike the atomistic world of non-strategic economics, suboptimal equilibriamay emerge in a game theoretic setting because players are only allowed to choosestrategies, and cannot single-handedly determine outcomes. Because of the poten-tial accessibility of off-diagonal non-cooperative outcomes, players are tempted todefect from optimal strategies, generating outcomes that are Pareto inferior forall.15

By eliminating the accessibility of the asymmetric outcomes of the game, goldenrules of reciprocity induce the parties to choose strategies after taking into accountthe effect of a reciprocal choice of the other player. In other words, golden-typereciprocity renders the reward for unilateral defection unobtainable. As a result,no rational player will employ defection strategies in the hope of obtaining higherpayoffs from unilateral defection, nor will rational players be induced to selectdefection strategies as a merely defensive tactic. Automatic reciprocity mechanismsthus guarantee the destabilization of mutual defection strategies and the shifttoward optimizing cooperation.16

Figure 2 depicts the equilibrium obtained in the absence of a reciprocityconstraint. Two players are faced with a cooperation problem. Strategies I, II, and

ŽIII represent three levels of cooperation with strategy I being the highest level and

Figure 2. A cooperation problem without constraint.

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.strategy III being the lowest . Even though mutual cooperation at level I generatesthe highest aggregate payoff, strategy III dominates in equilibrium, as it is shownby the Nash arrows for the two players.

This equilibrium should be contrasted with the outcome induced by a reciprocityconstraint as illustrated in Figure 3.

Figure 3 shows the effect of a reciprocity constraint on the equilibrium obtainedin Figure 2. By eliminating the accessibility of asymmetric outcomes, golden-typereciprocity compels the parties to take into account the effect of the opponent’sreciprocal choice, when selecting their optimal strategy. In this way, the dominanceof strategy III obtained in Figure 2 is transformed in a dominance of strategy I,with optimal levels of cooperation for the two players.

Interestingly, where custom is recognized as a primary source of law, mecha-nisms of automatic reciprocity are generally regarded as meta-rules of the system.

ŽThe so-called law of nations the system that governs the relationship between.states serves as an example. The process of voluntary recognition of rules by

sovereign states implies that, absent a commonly accepted standard of conduct,lawless freedom applies. Lawless environments are characterized by structuralreciprocity. In its original form, custom emerges in the absence of an establishedlegal system or commonly recognized rule of law.

Rules of reciprocity are among the oldest principles of customary law. In theabsence of law, reciprocity implies that parties can do back to others what has beendone to them, subject to the limits of their reciprocal strengths. Ancient customs ofretaliation, based on conceptions of symmetry and punitive balance, provide anintriguing illustration of the principle of reciprocity at work. While practices of

Figure 3. A cooperation problem with a reciprocity constraint.


literal retaliation are formally discouraged, the principle of reciprocity is recog-nized as a crucial pillar for the process of international law formation.17

Often, situations of post-contractual behavior capable of modifying states’ obli-gations arise in the law and practice of international relations.18 The internationallaw process provides states with numerous occasions for opportunistic behavior,including hold-out strategies and free riding. Left unconstrained, states’ unilateral

Ž .defection strategies would dominate in equilibrium Figure 2 .To cope with this reality, basic norms of reciprocity have emerged as interna-

Ž .tional customary law. Art. 21 1 b of the 1969 Vienna Convention articulates anestablished custom of reciprocity, creating a mirror-image mechanism in the caseof unilateral reservations: ‘‘Legal Effects of Reservations and of Objections toReservations: A reservation established with regard to another party . . . modifiesthose provisions to the same extent for that other party in its relations with thereserving state.’’ The effects of this automatic reciprocity mechanism are similar toa tit-for-tat strategy without the need for active retaliation by states: whenever atreaty is modified unilaterally in favor of one state, the result will be as if all theother states had introduced an identical reservation against the reserving state. Byimposing a symmetry constraint on the parties’ choices, this rule offers a possiblesolution to Prisoners’ Dilemma problems.

As an additional illustration, one can think of a voluntary contribution gamebetween several people for the realization of a common project. If the benefitsfrom the realized project are non-excludable, the various players are faced with ausual public good problem, with dominant free-riding strategies. As an application

Žof a golden-type reciprocity constraint, imagine that the parties agree and credibly.commit to a ‘‘lower denominator’’ matching grant solution. Every player agrees to

contribute to the common cause an amount equal to the least of the other players’contributions. The players are now faced with a choice of strategy subject to a

Žcredible reciprocity constraint. By unilaterally defecting i.e., offering a contribu-. 19tion of zero , each player can precipitate the failure of the joint enterprise. No

Ž .player can obtain the unilateral defection temptation payoff, because by withhold-ing or reducing the amount of his contribution, each player causes a mirror-imagerevocation or reduction of the other players’ contributions. Thus, although no

Žplayer can unilaterally determine the success of the joint enterprise unanimous.participation is necessary , each individual player can determine its failure. Under

most circumstances, this arrangement is capable of preventing unilateral defectionand free-riding strategies from dominating in equilibrium. The ability of eachplayer to lower the minimum denominator in the group adds robustness to thecooperative solution, in spite of the apparent fragility occasioned by the unilateralveto effect of the reciprocity rule.

As an important consideration, one should note that, while the principle ofreciprocity solves conflict situations characterized by a Prisoners’ Dilemma struc-

Ž .ture in both symmetric and asymmetric cases , golden-type reciprocity is on itsown incapable of correcting other strategic problems. When a conflict occurs along

Žthe diagonal possibilities of the game such that the obtainable equilibria are

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.already characterized by symmetric strategies , a reciprocity constraint will notalter the dynamic of the game.20 Reciprocity constraints are effective only whereincentives for unilateral defection are present. This reciprocity will be ineffectivefor Battle of the Sexes games and unnecessary in the Inessential Game and PerfectIncentive Alignment cases.21

( )C. The sil er rule: stochastic reciprocity role re¨ersibility

Silver-type reciprocity requires a stochastic symmetry in the relationship betweentwo or more players. This stochastic symmetry could result from a randomdistribution of asymmetric payoffs in a iterated game or, more commonly, it couldresult from an alternation of roles among the various players.

Unlike golden-type reciprocity, silver-type reciprocity further necessitates apre-commitment of each player to a meta-strategy for the entire duration of theplay. That is to say, a player cannot adopt different strategies at different nodes of

Ž .an iterated game, but must choose a strategy be it pure or mixed that he willunconditionally follow for the entire duration of the supergame. Under suchconditions, the optimal meta-strategy will be that which maximizes the presentvalue of the expected flow of payoffs from the supergame. In situations ofsilver-type reciprocity with a relatively high probability of future interaction and arelatively low time preference, cooperative strategies will be more likely to domi-nate in equilibrium.

An important consideration in most analyses of spontaneous cooperation is thediscount factor, which serves two purposes. First, it acts as a function of theplayers’ time preference. Faced with very high time preference, players in a repeatgame become less likely to give up part of their present payoff for an expectedincrease in the payoffs from future interactions. As time preference increases, thepresent discounted value of future payoffs decreases. Where time preference isinfinite, payoffs from future interactions have zero value. Second, the discountfactor is a function of the probability of future interactions. When the probabilityof future interactions is low, players are less likely to give up part of their presentpayoff for an expected increase in the payoffs from future iterations. As theprobability of future interactions increases, so does the present expected value ofcooperation.22

One should further note that, unlike golden reciprocity, silver reciprocity doesnot in itself correct repeated Prisoners’ Dilemma problems. Indeed, even if theplayers found themselves in such an iterated game, silver-type reciprocity wouldnot alter the usual results obtained in the Chainstore paradox23 and the Folktheorem.24 Conversely, silver-type reciprocity is capable of correcting other strate-gic problems that the golden-type rule cannot effectively address, such as theasymmetric Battle of the Sexes game.


The history of customary law provides illustrations of silver-type reciprocity atwork. During the formative period of the medieval mercantile law, travelingmerchants acted in the dual capacity of buyers and sellers in order to exploit pricedifferentials between various medieval markets. In their commercial dealings theycommitted themselves to following a set of rules and customs that would govern alltheir future business transactions. This privately created body of law, known as the

Ž .law merchant lex mercatoria , applied indiscriminately to commercial contractsentered into by professional merchants, without possibility of a dual standard. Ifthose rules were biased in their favor of merchants in their role as sellers, theywould have the opposite effect when the same merchants acted as buyers, and

Ž .vice-versa. In this setting, Fuller 1969 observed that frequent role changesfostered the emergence of mutually recognized and accepted duties in a society ofeconomic traders: ‘‘By definition the members of such a society enter direct andvoluntary relationships of exchange. . . . Finally economic traders frequently ex-change roles, now selling now buying. The duties that arise out of their exchangesare therefore reversible, not only in theory but in practice.’’25

In this mercantile context, role reversibility changed an otherwise structurallyŽ .asymmetrical situation i.e., buyer v. seller into one that was stochastically sym-

metrical,26 eliminating the temptation to articulate one-sided rules. The lawmerchant therefore illustrates a successful system of spontaneous and decentral-ized law.

The emergence of cooperative strategies for the supergame requires, as indi-Žcated above, an effective pre-commitment of the players to a single strategy pure

.or mixed to be followed for the entire duration of the game. This will ensure thatthe ex ante choice of a given rule will not be followed by an ex post breach of therule, once the roles are reversed.

In real life, where relational rules are violated following role reversal, norms ofsocial reprobation play a collateral yet crucial role in sanctioning case-by-caseopportunism. Likewise, a merchant who invokes a particular rule when buying yetrefuses to abide by the same rule when selling would be regarded as violating abasic norm of business conduct, and would suffer reputational costs within thebusiness community.27 Conditions of role reversibility, coupled with norms thatgenerate disincentives for the adoption of opportunistic double standards,28 aretherefore likely to generate optimal rules via spontaneous processes.

III. Conclusions: the nature of reciprocity

Reciprocity constraints of the ‘‘golden’’ type serve an important function whenapplied to simultaneous or sequential Prisoners’ Dilemma games. At the origin ofthe problematic results of the Prisoners’ Dilemma lies the fact that players cannotchoose outcomes, but rather only select strategies. Outcomes are beyond thecontrol of any individual player, generated instead by the combination of selectedstrategies.

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Reciprocity constraints of the golden type eliminate this problematic feature ofthe game by preventing asymmetric combinations of strategies. Under a goldenrule of reciprocity, players know that by selecting a strategy they are actuallydetermining the outcome of the game. The incentives for unilateral defection areeliminated because the reciprocity constraint would transform a situation ofunilateral defection into one of mutual defection with reciprocal losses for bothplayers.

Regarding the ‘‘silver’’ rule of reciprocity, one should think that, following thesame optimization logic they employ for all economic choices, individuals chooseamong alternative rules of behavior. They are unlikely to reveal strategic prefer-ences when individual interests are not aligned. Traditionally, such strategies areviewed as hindrances to the spontaneous emergence of cooperation. I argue,however, that strategic preference revelation has no effect in situations of rolereversibility or stochastic symmetry.29 Because the expected costs and benefits ofalternative rules are the same for all members of the group, each member has theincentive to agree to a set of rules that benefits the entire group, thus maximizinghis expected share of the wealth.30

In long-term human interactions, reciprocity and close-knittedness provide indi-viduals with an optimal set of incentives for choosing socially desirable strategies.Recent analyses indicate that individual incentives are more likely to be alignedwhere the horizons of individual maximization are extended to include the payoffsfrom future interactions and there is a concern for the well-being of close memberswithin the group. The presence of reciprocity constraints allows for a far moreoptimistic prediction of spontaneous order. This insight is consistent with theprediction of evolutionary models and the findings of experimental studies of socialinteraction, where low discount rates for future payoffs and the close-knittednessof the group are found to be positively correlated with the emergence of optimalsocial norms. These conclusions are qualitatively similar to those discussed abovefor the case of role reversibility. Repeat game situations with role reversibility,reciprocity constraints, and structural integration facilitate the emergence andrecognition of rules of custom. In consideration of reciprocal constraints under-taken by the other members of the community, individuals who frequently ex-change roles in their social interactions have incentives to constrain their behaviorto conform to socially optimal norms of conduct. An even stronger logic explainsthe emergence of cooperation in situations of induced reciprocity. In both cases,the non-idealistic and self-interested behavior of human actors is yet capable ofgenerating optimal norms.

Notes

1. The assumption of interdependence of individual choices indicates that the environment contem-plated in this paper is different from the neo-classical literature’s atomistic world of perfectcompetition, which assumes the independence of individual decisions.


Ž .2. Thomas C. Schelling, The Strategy of Conflict, 2nd ed., p. 88 1980 .Ž .3. The term can be found in Klein, ‘‘Self-Enforcing Contracts,’’ UCLA Working Papers undated .

4. These conditions are consistent with the neo-classical economic model, in which the dynamic of freeexchange implicitly assumes the existence of contract enforcement mechanisms, and confines therole of strategies to pre-contractual profit maximizing efforts, excluding post-contractual oppor-tunism and unilateral defection.

5. Similarly, pure coordination problems are characterized by the perfect coincidence of the players’interests and by the additional feature of multiple equilibria. The convergence of individual andcollective interests fosters an optimal outcome on the basis of a mere coordination of self-interestedstrategies. It has been argued, however, that the solution to coordination problems may be delayedif it relies exclusively on decentralized processes of legal and social order. The multiplicity of Nashequilibria in a coordination game creates difficulties for decentralized solutions. For example, ifeveryone in a country needs to coordinate on a basic set of traffic conventions, such as driving onthe same side of the road, the emergence of spontaneous]but heterogeneous]clusters of trafficcustoms would consolidate local equilibria that do not possess the features of universality requiredin a modern society. Ironically, however, the most universal traffic rules are those for waternavigation, which emerged through spontaneous rulemaking processes. For an interesting historical

Ž .background, see John H. Wigmore. The Maritime Legal System 1928 ; Nicholas J. Healy and DavidŽ .J. Sharpe, Admiralty Cases and Materials, 2nd ed. St. Paul, MN: West Publishing, 1986 ; IanŽ .Brownlie, Principles of Public International Law, 4th ed. Oxford: Clarendon Press, 1990 ; Thomas J.

Ž .Schoenbaum, Admiralty and Maritime Law St. Paul, MN: West Publishing, 1987 .Ž6. As noted in the recent literature see among others, Muris, 1981; Williamson, 1983; Parisi, 1995,

.etc. , the main problem faced by traders in a lawless environment is that of opportunism.7. For analysis of the role of conventions in the correction of Prisoners’ Dilemma situations, see

Ž .Andrew Schotter, Economic Theory of Social Institutions, Cambridge, 1981 ; D. Lewis, Con¨ention:Ž .A Philosophical Study, Harvard, 1969 ; Harvey Leibenstein, The Prisoners’ Dilemma in the Invisible

Ž .Hand: An Analysis of Intrafirm Productivity. AEA Papers and Proceedings. 72, 92]97 1982 .8. Additionally, any change in outcome will make at least one party worse off, since there is no surplus

to allow the winners to effectively compensate the losers.9. See, e.g., the interesting study of Sue Crawford and Elinor Ostrom, A Grammar of Institutions.

Ž .American Political Science Re¨iew. 89, 582]600 1995 .10. The study of the incentives generated by the rules of the game, in other words, further limits the

applicative scope of this class of games. Most allocational rules have relevant effects on theindividual incentives of future players.

11. Most of the results would still hold using ordinal preferences, but this would prevent the use of theKaldor-Hicks criterion in analyzing the outcome.

Ž .12. See, more extensively, Parisi 1996b .13. The existence of a belief of social obligation is often a valuable element in the legal test of the

qualitative condition of legal necessity and desirability. As an illustration, consider the two classicalstrategies of cooperation and defection in response to the usual Prisoners’ Dilemma payoffstructure.

Ž .14. See Kronman 1985 .Ž .15. For an insightful discussion, see Buchanan 1975 .

Ž .16. For a similar argument relying on tit-for-tat strategies in iterated games, see Axelrod 1981 andŽ .1984 .

17. Many structural similarities exist between the emergence of legal rules in ancient law and theformative process of international law, including the lack of a supreme legislative body. The rules

Ž .governing international relations are left to the spontaneous interaction of states custom or toŽ .contracts expressly entered into treaties .

18. Having signed a treaty, states may introduce unilateral reservations at the time of ratification.Standards of compliance in treaty implementation also rely heavily on the subsequent practice of

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states. The post-contractual behavior of states can shape and modify the content of an alreadyfinalized agreement, or even abrogate a treaty.

19. Nor would it be sensible for the defending player to believe that a new arrangement could bereached among the remaining players. Subsequent hold-ups by other players in the absence ofreciprocity would undermine the obtainability of the cooperative outcome.

Ž20. Positive-sum games are solvable through enforceable contracts e.g., treaties with well-defined.protocols, etc. , while zero-sum games are likely to remain conflictual throughout.

21. Both cases result in identical equilibria, the existence of a reciprocity constraint notwithstanding,indicating the relevant constraint is not binding.

22. As well-known in the literature, the discount factor’s role in evolutionary models is critical.Environments promoting a high probability of future interaction and low time preference are moreconducive to cooperative outcomes. In the case of a one-shot game, on the other hand, theprobability of future interaction is zero, so that the expected value of future payoffs is also zero.

Ž .23. Selten 1978 at pp. 127]159.Ž .24. See, among others, Fudenberg and Maskin 1986 at pp. 533]554.

Ž .25. Fuller 1969, p. 24Ž .26. For a study of role reversibility among medieval merchants, see Greif 1989 .

27. The group’s ability to impose a sanction depends on the recordation of an individual’s past behavior.ŽReputation serves as a source of collective knowledge regarding past actions. Benson 1992, pp.

.5]7 explores the role of reputation in situations of repeated market interaction.28. Internalization of the norm is a source of spontaneous compliance. For example, individuals

internalize obligations when they disapprove of and sanction other individuals’ deviations from therule, or when they suffer direct utility loss from the violation of the norm. A legal custom willtherefore successfully evolve in a community when the ex ante individual incentives are aligned with

Ž .the collective public interest. Cooter 1994, p. 224 calls this proposition the ‘‘alignment theorem.’’Ž .29. See Parisi 1996b .

30. See Id.

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