The Customary International Law Supergame February 23, 2004 The model we present here formalizes certain considerations that remain informal in much of the political science literature, and has other varying features that we describe 2. The N-Person prisoner's dilemma In this section we explain our choice of the n-person prisoner's dilemma as the basic model for the Cil problem, examine the potential strategies that might be played within this game, and explain some of our assumptions. The n-person iterated prisoner's dilemma is really a group of varying models, with a number of varying features including the number of players and their preferences a. Choice of game noted, As we consider the application of game theory to the Cil setting, it should be that there is a wide choice of potential game structures to apply. The basic payoff structure assumed in the prisoners dilemma game captures the essential problem of cooperation in a horizontal social setting with externalities, in which parties have a choice between compliance and defection under circumstances in which they can enrich themselves individually through defection while they could enrich society through compliance. Of course, some CIL contexts might be better modeled using other 577(1992); Joel P. Trachtman, The Domain of wTo Dispute Resolution, 40 HARV. INT'L LJ.333,346-47(1999) For a useful analysis of the "fit" of other games, including"battle of the sexes"and stag hunt, see Swaine, supra note 2. See also Fiona McGillivray Alastair Smith Trust and Cooperation Through Agent-Specific PunishmentS, 54 INT'L ORG. 809, 810 (2000)(noting that the prisoner's dilemma is often used to model international Coo peau y use of the term"externalities, "we mean to be quite inclusive, including both iniary and non-pecuniary externalities: any circumstance in which an action by one state has adverse or beneficial effects on another state At another level of complexity, it would be possible to model the game of formation of a Cil rule separately from enforcement. See James D. Fearon, Bargaining Enforcement, and International Cooperation, 52: 2 INT'L. ORG. 269(1998)(separating the bargaining problem, modeled as a coordination game, from the enforcement problem, modeled as a prisoners dilemma); Stephen D. Krasner, Global Communications and ational Power: Life on the Pareto Frontier, 43 WORLD POL. 336(1991)(arguing that many international issues are better modeled as coordination games). Fearon's two stage approach may be more appropriate to the treaty context than to the custom context Fearon points out that relatively large "shadows of the future "might inhibit bargaining to achieve an initial coordination game agreement, while making the enforcement game more tractable. In the CIl context, there is less natural separation, and there may even be
The Customary International Law Supergame February 23, 2004 10 The model we present here formalizes certain considerations that remain informal in much of the political science literature, and has other varying features that we describe below. 2. The N-Person Prisoner’s Dilemma In this section we explain our choice of the n-person prisoner’s dilemma as the basic model for the CIL problem, examine the potential strategies that might be played within this game, and explain some of our assumptions. The n-person iterated prisoner’s dilemma is really a group of varying models, with a number of varying features, including the number of players and their preferences. a. Choice of Game As we consider the application of game theory to the CIL setting, it should be noted that there is a wide choice of potential game structures to apply. 29 The basic payoff structure assumed in the prisoner’s dilemma game captures the essential problem of cooperation in a horizontal social setting with externalities, 30 in which parties have a choice between compliance and defection under circumstances in which they can enrich themselves individually through defection while they could enrich society through compliance. 31 Of course, some CIL contexts might be better modeled using other 577 (1992); Joel P. Trachtman, The Domain of WTO Dispute Resolution, 40 HARV. INT'L L.J. 333, 346-47 (1999). 29 For a useful analysis of the “fit” of other games, including “battle of the sexes” and “stag hunt,” see Swaine, supra note 2. See also Fiona McGillivray & Alastair Smith, Trust and Cooperation Through Agent-Specific Punishments, 54 INT'L ORG. 809, 810 (2000) (noting that the prisoner’s dilemma is often used to model international cooperation). 30 By use of the term “externalities,” we mean to be quite inclusive, including both pecuniary and non-pecuniary externalities: any circumstance in which an action by one state has adverse or beneficial effects on another state. 31 At another level of complexity, it would be possible to model the game of formation of a CIL rule separately from enforcement. See James D. Fearon, Bargaining, Enforcement, and International Cooperation, 52:2 INT’L. ORG. 269 (1998) (separating the bargaining problem, modeled as a coordination game, from the enforcement problem, modeled as a prisoner’s dilemma); Stephen D. Krasner, Global Communications and National Power: Life on the Pareto Frontier, 43 WORLD POL. 336 (1991) (arguing that many international issues are better modeled as coordination games). Fearon’s two stage approach may be more appropriate to the treaty context than to the custom context. Fearon points out that relatively large “shadows of the future” might inhibit bargaining to achieve an initial coordination game agreement, while making the enforcement game more tractable. In the CIL context, there is less natural separation, and there may even be
The Customary International Law Supergame February 23, 2004 methods, but by using the prisoners dilemma we hope to capture the essence of informal contracting under opposed interests. 32 One of the reasons that we choose this game is because it allows us to contextualize a number of insights and concerns that cannot easily be included in other analytical models. For example, we believe that the n-person prisoner's dilemma can take account of a number of the diverse considerations often referred to together as reputation"or "reputational sanctions. We also believe that the n-person prisoners dilemma must be at the core of a rationalist explanation of the effectiveness of social norms. Finally, we believe that the n-person prisoners dilemma offers parsimony: the factors that it takes into account seem necessary, and there are no factors that seem superflow In a non-cooperative, single-play circumstance, with a standard prisoners dilemma payoff structure, we would expect non-compliance. This is each player's dominant strategy, and a Nash equilibrium. This is because under the payoffs assumed in the prisoners dilemma, each party is better off defecting, no matter what the other party does. Therefore, under the rather restrictive assumptions of the true prisoner's dilemma, the parties each invariably choose the strategy that results in reduced individual welfare, and reduced aggregate welfare, compared to the non-defecting strategy. This is n inefficient outcome. By analogy, states playing the CIl game(assuming prisoner dilemma-type payoffs)in a bilateral single-play setting would fail to form or comply with a Cil rule that increased individual and aggregate welfare. Cooperation is strongly dominated, and the unique Nash equilibrium is for both states to defect. The same is true of a prisoner's dilemma game repeated a finite number of times known in advance to each period. "A subgame perfect equilibrium is a strategy profile that induces a Nash o the players. Now the unique sub-game perfect equilibrium is for each player to defect equilibrium in every subgame interest in this article is not so much in establishing CIL rules, as in enforcing then first mover advantages that would counteract the effect Fearon suggests. Finally, or For a discussion of the use of coordination games to model certain types of nternational contexts, see Barbara Koremenos, Charles Lipson, duncan Nidal Rational Design of International InstitutionS, 55: 4 INT'L ORG. 761, 774(2001); Dur Snidal, Coordination versus Prisoner's Dilemma: Implications for international Cooperation and Regimes, 79: 4 AM. POL. SCL. REV. 923(1985) See Guzman, supra note 2 STEPHEN MARTIN, ADVANCED INDUSTRIAL ECONOMICS 98(1993) a"dominant strategy"is one which, no matter what the other player does, will provide a higher payoff to the acting player. A" Nash equilibrium"is a set of"strategies h that each players strategy is an optimal response to the other players' strategies DREW FUDENBERG JEAN TIROLE, GAME THEORY 11(1991) ld. at 111 37 M.J. OSBORNE, AN INTRODUCTION TO GAME THEORY(Oxford University Press, 2004)
The Customary International Law Supergame February 23, 2004 11 methods, but by using the prisoner’s dilemma we hope to capture the essence of informal contracting under opposed interests. 32 One of the reasons that we choose this game is because it allows us to contextualize a number of insights and concerns that cannot easily be included in other analytical models. For example, we believe that the n-person prisoner’s dilemma can take account of a number of the diverse considerations often referred to together as “reputation” or “reputational sanctions.” 33 We also believe that the n-person prisoner’s dilemma must be at the core of a rationalist explanation of the effectiveness of social norms. Finally, we believe that the n-person prisoner’s dilemma offers parsimony: the factors that it takes into account seem necessary, and there are no factors that seem superfluous. In a non-cooperative, single-play circumstance, with a standard prisoner’s dilemma payoff structure, we would expect non-compliance. 34 This is each player’s dominant strategy, and a Nash equilibrium. 35 This is because under the payoffs assumed in the prisoner’s dilemma, each party is better off defecting, no matter what the other party does. Therefore, under the rather restrictive assumptions of the true prisoner’s dilemma, the parties each invariably choose the strategy that results in reduced individual welfare, and reduced aggregate welfare, compared to the non-defecting strategy. This is an inefficient outcome. By analogy, states playing the CIL game (assuming prisoner’s dilemma-type payoffs) in a bilateral single-play setting would fail to form or comply with a CIL rule that increased individual and aggregate welfare. Cooperation is strongly dominated, and the unique Nash equilibrium is for both states to defect. 36 The same is true of a prisoner’s dilemma game repeated a finite number of times known in advance to the players. Now the unique sub-game perfect equilibrium is for each player to defect in each period. “A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in every subgame.” 37 first mover advantages that would counteract the effect Fearon suggests. Finally, our interest in this article is not so much in establishing CIL rules, as in enforcing them. 32 For a discussion of the use of coordination games to model certain types of international contexts, see Barbara Koremenos, Charles Lipson, & Duncan Snidal, The Rational Design of International Institutions, 55:4 INT’L ORG. 761, 774 (2001); Duncan Snidal, Coordination versus Prisoner’s Dilemma: Implications for International Cooperation and Regimes, 79:4 AM. POL. SCI. REV. 923 (1985). 33 See Guzman, supra note 2. 34 STEPHEN MARTIN, ADVANCED INDUSTRIAL ECONOMICS 98 (1993). 35 A “dominant strategy” is one which, no matter what the other player does, will provide a higher payoff to the acting player. A “Nash equilibrium” is a set of “strategies such that each player’s strategy is an optimal response to the other players’ strategies.” DREW FUDENBERG & JEAN TIROLE, GAME THEORY 11 (1991). 36 Id. at 111. 37 M.J. OSBORNE, AN INTRODUCTION TO GAME THEORY (Oxford University Press, 2004)
The Customary International Law Supergame February 23, 2004 This conclusion is inescapable in theory, given the constraints of the game: by definition, the outcome of the prisoners dilemma game is an inefficient strategic equilibrium. This conclusion presents a normative goal: to modify the real world circumstances so as to produce stable equilibria that are efficient. This is the major role of CIL, and of international law generally respo. Of course, in a world of effective third-party enforcement of agreements,the response to the prisoner's dilemma is clear: the parties enter into a binding agreement to cooperate, thereby modifying the payoff structure and escaping the prisoner's dilemma The prisoners dilemma assumes, however, that its prisoners are held separately and have violated the law, so they cannot negotiate, reach or enforce a binding agreement. In the CIl setting, there is no court of general mandatory jurisdiction nor any publicly-appointed"policeman. While we may draw analogies to the World Court and the U.N., these institutions have substantial differences compared to domestic courts and police. Therefore, we begin by assuming that there is no capacity to make agreements that are binding. This is obviously a simplifying assumption. Our model is meant to show that there are substitutes for formal binding agreements In the cil game, there are five important distinctions from the assumptions of the players can communicate with one another, and can do so more readily today th 5 non-cooperative game theory in general, and the prisoner's dilemma in particular. Fi during the classical period of formation of CIL. Second, the players can enter into treaties that are at least somewhat binding. Third, states play an iterated game with one another with no defined end date, and so can respond at a later time to something done at an earlier time. Updating of information and punishment are possible. Fourth, not only is the narrow game characterized by a particular Cil rule, like the three-mile territorial sea, but it is embedded in a dense fabric of relationships. Fifth, information regarding compliance is often readily accessible; more so today, it would appear, than during the 19century. Each of these distinctions alone may be sufficient to transform the game into something quite different from the prisoners dilemma-while nothing resolves the true prisoner's dilemma, modifications may result in stable and efficient equilibria Indeed, it may be useful to use cooperative game theory to analyze some Cil circumstances As will be illustrated below, one of the more difficult types of multilateral cooperation problem is a commons problem in which, as in the cartel context, incentives to defect increase with the number of other states that comply. where incentives to defect increase with the number of players, we would expect the most severe challenge to cooperation. Not all Cil contexts exhibit this characteristic For an application of cooperative game theory in the international context, see Daniel G. Arce M. Todd Sandler, A Cooperative Game Theory of Non-Contiguous Allies, 3 J PUB. ECoN THEORY 391(2000)(applying cooperative game theory to international security arrangements
The Customary International Law Supergame February 23, 2004 12 This conclusion is inescapable in theory, given the constraints of the game: by definition, the outcome of the prisoner’s dilemma game is an inefficient strategic equilibrium. This conclusion presents a normative goal: to modify the real world circumstances so as to produce stable equilibria that are efficient. This is the major role of CIL, and of international law generally. Of course, in a world of effective third-party enforcement of agreements, the response to the prisoner’s dilemma is clear: the parties enter into a binding agreement to cooperate, thereby modifying the payoff structure and escaping the prisoner’s dilemma. The prisoner’s dilemma assumes, however, that its prisoners are held separately and have violated the law, so they cannot negotiate, reach or enforce a binding agreement. In the CIL setting, there is no court of general mandatory jurisdiction nor any publicly-appointed “policeman.” While we may draw analogies to the World Court and the U.N., these institutions have substantial differences compared to domestic courts and police. Therefore, we begin by assuming that there is no capacity to make agreements that are binding. This is obviously a simplifying assumption. Our model is meant to show that there are substitutes for formal binding agreements. In the CIL game, there are five important distinctions from the assumptions of non-cooperative game theory in general, and the prisoner’s dilemma in particular. First, the players can communicate with one another, and can do so more readily today than during the classical period of formation of CIL. Second, the players can enter into treaties that are at least somewhat binding. Third, states play an iterated game with one another with no defined end date, and so can respond at a later time to something done at an earlier time. Updating of information and punishment are possible. Fourth, not only is the narrow game characterized by a particular CIL rule, like the three-mile territorial sea, but it is embedded in a dense fabric of relationships. Fifth, information regarding compliance is often readily accessible; more so today, it would appear, than during the 19 th century. Each of these distinctions alone may be sufficient to transform the game into something quite different from the prisoner’s dilemma—while nothing resolves the true prisoner’s dilemma, modifications may result in stable and efficient equilibria. Indeed, it may be useful to use cooperative game theory to analyze some CIL circumstances. 38 As will be illustrated below, one of the more difficult types of multilateral cooperation problem is a commons problem in which, as in the cartel context, incentives to defect increase with the number of other states that comply. Where incentives to defect increase with the number of players, we would expect the most severe challenge to cooperation. Not all CIL contexts exhibit this characteristic. 38 For an application of cooperative game theory in the international context, see Daniel G. Arce M. & Todd Sandler, A Cooperative Game Theory of Non-Contiguous Allies, 3 J. PUB. ECON. THEORY 391 (2000) (applying cooperative game theory to international security arrangements)
The Customary International Law Supergame February 23, 200 dilemma-type circumstances, may achieve resolution despite theory. Elinor Ostrom u Observation suggests that even certain commons problems, or other prisoner states that "[a] substantial gap exists between the theoretical prediction that self- nterested individuals will have extreme difficulty in coordinating collective action and the reality that such cooperative behavior is widespread, although far from inevitable Ostrom cites considerable evidence regarding the amount and circumstances of cooperation by individuals in n-person collective action problems. The evidence show that individuals contribute to the resolution of these problems in substantially greater the evidence is obtained in circumstances where the assumptions of the prisoner p huch of amounts than the standard prisoner's dilemma model would suggest. Of course, dilemma are violated-by allowing individuals to communicate, enter into agreements and iterate play. We do not deploy any data in this article. However, we advance a plausible theory of potentially stable and efficient equilibria in the n-player prisoners dilemma, and suggest some possible approaches to empirical testing b. Choice of Strategy within the prisoners dilemma, we must postulate a particular strategy for states to play. There are many choices. We mention only three: tit-for-tat, grim trigger, and penance 1. Bilateral or multilateral retaliation An important aspect of the structure of the game pertains to the ability to impose sanctions in a discriminatory manner States may have trouble discriminating in the application of sanctions for several reasons. First, they may not be able to obtain information regarding the author of the violation. This might occur, for example, with respect to pollution at sea, or terrorist attack. Second, it may be costly for states to respond in a discriminating way. For example, if the sanction involves trade barriers, the sanctioning state must instruct its customs officers to discriminate among goods by origin. Third, and most important, the relevant good being produced by cooperation may be non-excludable. This would occur with the provision of public goods or the protection of international commons To the extent that states are unable to discriminate. their retaliation, if any, must be multilateral, instead of bilateral. This obviously limits the strategy that they are able to play and the relationships that they may enter into. Thus, given that the strategies available to a state are"cooperate"or"defect, there are at least two possibilities that we need to consider in connection with a multilateral game Elinor Ostrom, Collective Action and the evolution of social Norms, 14: 3 J ECoN PERSP.137,138(2000
The Customary International Law Supergame February 23, 2004 13 Observation suggests that even certain commons problems, or other prisoner’s dilemma-type circumstances, may achieve resolution despite theory. Elinor Ostrom states that “[a] substantial gap exists between the theoretical prediction that selfinterested individuals will have extreme difficulty in coordinating collective action and the reality that such cooperative behavior is widespread, although far from inevitable.” 39 Ostrom cites considerable evidence regarding the amount and circumstances of cooperation by individuals in n-person collective action problems. The evidence shows that individuals contribute to the resolution of these problems in substantially greater amounts than the standard prisoner’s dilemma model would suggest. Of course, much of the evidence is obtained in circumstances where the assumptions of the prisoner’s dilemma are violated—by allowing individuals to communicate, enter into agreements and iterate play. We do not deploy any data in this article. However, we advance a plausible theory of potentially stable and efficient equilibria in the n-player prisoner’s dilemma, and suggest some possible approaches to empirical testing. b. Choice of Strategy Within the prisoner’s dilemma, we must postulate a particular strategy for states to play. There are many choices. We mention only three: tit-for-tat, grim trigger, and penance. i. Bilateral or Multilateral Retaliation An important aspect of the structure of the game pertains to the ability to impose sanctions in a discriminatory manner. States may have trouble discriminating in the application of sanctions for several reasons. First, they may not be able to obtain information regarding the author of the violation. This might occur, for example, with respect to pollution at sea, or terrorist attack. Second, it may be costly for states to respond in a discriminating way. For example, if the sanction involves trade barriers, the sanctioning state must instruct its customs officers to discriminate among goods by origin. Third, and most important, the relevant good being produced by cooperation may be non-excludable. This would occur with the provision of public goods or the protection of international commons. To the extent that states are unable to discriminate, their retaliation, if any, must be multilateral, instead of bilateral. This obviously limits the strategy that they are able to play and the relationships that they may enter into. Thus, given that the strategies available to a state are “cooperate” or “defect,” there are at least two possibilities that we need to consider in connection with a multilateral game: 39 Elinor Ostrom, Collective Action and the Evolution of Social Norms, 14:3 J. ECON. PERSP. 137, 138 (2000)
The Customary International Law Supergame February 23, 2004 (1) Defection by state i against state j leads to punishment of i only by state j bilateral defection leads to bilateral punishment;or (ii) Defection by state i against state j leads to punishment of i by all states: bilateral defection leads to multilateral punishment. We focus our analysis on(i)for three reasons. First, it makes the analysis simpler without changing any of the qualitative conclusions. Second, multilateral retaliation ply increases the incentives to comply over bilateral retaliation. Since (i) involves milder punishment of defection than(ii), the conditions that support cooperation with bilateral punishment under(i)will certainly support cooperation with multilater punishment under(ii). In other words, where multilateral punishment is possible, the conditions that we identify below are sufficient but not necessary to support a multilateral rule: any discount factor that supports a multilateral Cil rule with bilateral punishment will also support such a rule with multilateral punishment. Third, there is some force to the argument that (i) is a more plausible scenario than(ii) in most of the situations in which the formulation of Cil is likely to be considered. Articles 42, 48 and 54 of the International Law Commissions Rules on State Responsibility generally prevent retaliation by third states for truly bilateral injury, while there seems no basis for retaliation by an injured state against non-injuring states. These rules seem to limit the formal possibility for multilateral sanctions against truly bilateral defection Thus, in our model, we assume that retaliation is applied bilaterally--that if state defects vis-a-vis state j, only state will respond, and only against state i. If multiple states responded against state i, it would simply make cooperation more likely by increasing the punishment for defection Given (1), we can represent a multilateral prisoner's dilemma game as a set of bilateral games, with a typical bilateral game being as specified in Table l(page 29). As will be seen below, this is not the same as assuming a bilateral game. Rather, it is a multilateral game with bilateral punishment ii. Tit-for-Tat Under"tit-for-tat, states may respond to defection with a single defection. Tit- for-tat is one of the most frequently-discussed strategies in connection with iterated prisoners dilemmas. While tit-for-tat may win evolutionary games, it is not subgame Articles on Responsibility of States for Internationally Wrongful Acts, in Report of the International Law Commission on the Work of Its Fifty-third Session, UN GAOR 56th Sess., Supp. No 10, at 43, UN DoC. A/56/10(2001), available at <www.un.org/law/ilc>,subsequentlynotedbythegeneralAssemblyinA/res/56/83, Responsibility of States for Internationally Wrongful Acts, adopted 12 Dec 2001. These articles do not themselves represent international law, but are an attempt to codify existing custom. See roBert AXelrod, THE EvoLUtIoN OF COOPERATION (1984)
The Customary International Law Supergame February 23, 2004 14 (i) Defection by state i against state j leads to punishment of i only by state j: bilateral defection leads to bilateral punishment; or (ii) Defection by state i against state j leads to punishment of i by all states: bilateral defection leads to multilateral punishment. We focus our analysis on (i) for three reasons. First, it makes the analysis simpler without changing any of the qualitative conclusions. Second, multilateral retaliation simply increases the incentives to comply over bilateral retaliation. Since (i) involves milder punishment of defection than (ii), the conditions that support cooperation with bilateral punishment under (i) will certainly support cooperation with multilateral punishment under (ii). In other words, where multilateral punishment is possible, the conditions that we identify below are sufficient but not necessary to support a multilateral rule: any discount factor that supports a multilateral CIL rule with bilateral punishment will also support such a rule with multilateral punishment. Third, there is some force to the argument that (i) is a more plausible scenario than (ii) in most of the situations in which the formulation of CIL is likely to be considered. Articles 42, 48 and 54 of the International Law Commission’s Rules on State Responsibility 40 generally prevent retaliation by third states for truly bilateral injury, while there seems no basis for retaliation by an injured state against non-injuring states. These rules seem to limit the formal possibility for multilateral sanctions against truly bilateral defection. Thus, in our model, we assume that retaliation is applied bilaterally—that if state i defects vis-à-vis state j, only state j will respond, and only against state i. If multiple states responded against state i, it would simply make cooperation more likely by increasing the punishment for defection. Given (i), we can represent a multilateral prisoner’s dilemma game as a set of bilateral games, with a typical bilateral game being as specified in Table 1 (page 29). As will be seen below, this is not the same as assuming a bilateral game. Rather, it is a multilateral game with bilateral punishment. ii. Tit-for-Tat Under “tit-for-tat,” states may respond to defection with a single defection. Titfor-tat is one of the most frequently-discussed strategies in connection with iterated prisoner’s dilemmas. While tit-for-tat may win evolutionary games, 41 it is not subgame 40 Articles on Responsibility of States for Internationally Wrongful Acts, in Report of the International Law Commission on the Work of Its Fifty-third Session, UN GAOR, 56th Sess., Supp. No. 10, at 43, UN Doc. A/56/10 (2001), available at <www.un.org/law/ilc>, subsequently noted by the General Assembly in A/RES/56/83, Responsibility of States for Internationally Wrongful Acts, adopted 12 Dec. 2001. These articles do not themselves represent international law, but are an attempt to codify existing custom. 41 See ROBERT AXELROD, THE EVOLUTION OF COOPERATION (1984)