1 See, e.g., Chinen, Mark A., Afterword, 23 Mich. J. Int’l L. 201 (2001)Google Scholar; Chinen, Mark A., Game Theory and Customary International Law: A Response to Professors Goldsmith and Posner, 23 Mich. J. Int’l L. 143 (2001)Google Scholar; Fon, Vincy & Parisi, Francesco, Customary Law and Articulation Theories: An Economic Analysis (George, Mason L. & Econ. Res. Paper No. 02–24, 2002), at <http://papers.ssrn.com/sol3/papers.cfm?abstract_id=335220>Google Scholar Goldsmith, Jack L. & Posner, Eric A., Further Thoughts on Customary International Law, 23 Mich. J. Int’l L. 191 (2001)Google Scholar; Goldsmith, Jack L. & Posner, Eric A., A Theory of Customary International Law, 66 U.Chi. L. Rev. 1113 (1999)CrossRefGoogle Scholar [hereinafter Goldsmith & Posner 1999]; Guzman, Andrew T., A Compliance-Based Theory of International Law, 90 Cal. L. Rev. 1823 (2002)CrossRefGoogle Scholar; Parisi, Francesco, The Formation of Customary Law (George, Mason L. & Econ. Res. Paper No. 01–06, 2001), at <http://papers.ssrn.com/sol3/papers.cfmPabstract_icN262032>Google Scholar; Swaine, Edward T., Rational Custom, 52 Duke L.J. 559 (2002)CrossRefGoogle Scholar; Pierre-Hugues, Verdier, Cooperative States: International Relations, State Responsibility and the Problem of Custom, 42 Va. J. Int’l L. 839 (2002)Google Scholar.

2 The leading article here is Goldsmith & Posner 1999, supra note 1. This work is now incorporated in Goldsmith, Jack L. & Posner, Eric A., The Limits of International Law (2005)Google Scholar. For a self-described “traditionalist’s” response, see Vagts, Detlev F., International Relations Looks at Customary International Law: A Traditionalist’s Defence, 15 Eur. J. Int’l L. 1031 (2004)CrossRefGoogle Scholar.

3 See Vagts, supra note 2 (critiquing the use of examples in Goldsmith & Posner 1999).

4 By “self-interest” we mean merely to refer to maximization of preferences, which can, among other things, be other-regarding or altruistic.

5 In this type of setting, we use the term “multilateral” to refer to contexts in which more than two parties are involved. A more technical term would be “n-person.” Although “multilateral” is the general term, we sometimes use the term “plurilateral” to refer to a group of states that form a subset of a broader group.

6 The “prisoner’s dilemma” is a form of game commonly used in social science analysis. In this article, it serves as a vehicle for providing a formalized analogy to the CIL context. The structure and assumptions that characterize the prisoner’s dilemma are described in parts II and III below.

7 See the cautions expressed in Goldsmith & Posner 1999, supra note 1, and the broader treatment in Snidal, Duncan, The Game Theory of International Politics, in Cooperation Under Anarchy (Oye, Kenneth A. ed., 1986)Google Scholar.

8 While it may be argued that the game theory that we use, initially developed in the context of analysis of individual behavior, cannot be applied to state behavior, it should be noted that game theory has been applied to the behavior of firms, as well as to that of states. For an argument regarding the adaptation of these types of models to international law, see Dunoff, Jeffrey L. & Trachtman, Joel P., Economic Analysis of International Law, 24 Yale J. Int’l L. 1 (1999)Google Scholar.

9 “Commons problems” are circumstances where persons share a particular resource, where one person’s use of the resource may reduce the amount of the resource available to others, and where conservation of the resource may increase the amount of the resource available to all. A fishery may, in this context, present a commons problem. “Public goods”—for example, the light of a lighthouse—are goods that are available to all and that can be used by one person without diminishing its availability to others. “Network goods” involves circumstances where wider use of the same good makes that good more valuable to all. Standards often have this characteristic.

10 An “equilibrium” is a strategic setting in which no player has an incentive to change its strategy given that the other players do not change their strategies. It is assumed that each player correctly perceives the strategic constraints under which it operates, and acts rationally in response to those constraints. A “stable equilibrium” is simply an equilibrium in which the players’ strategies remain stable—no player has an incentive to deviate from the equilibrium. By “efficiency,” we refer to Pareto efficiency, meaning that no player may be made better off without some player being made worse off. No equilibrium is necessarily efficient. In fact, the equilibrium outcome of the prisoner’s dilemma is generally inefficient.

11 The Court’s Statute is available at <http://www.icj-cij.org>.

12 Restatement (Third) of the Foreign Relations Law of the United States §102 (1987).

13 Goldsmith & Posner 1999, supra note 1.

14 Mendelson, Maurice H., The Formation of Customary International Law, 272 Recueil Des Cours 155, 268–93 (1998)Google Scholar.

15 Goldsmith & Posner 1999, supra note 1, at 1130.

16 Ellickson, Robert C., Order Without Law: How Neighbors Settle Disputes (1991)Google Scholar.

17 One might argue, however, that the general legal system serves as a background framework that supports the farmers’ social norms. For example, die rules against violence impose constraints on the types of sanctions that farmers can impose on one another for noncompliance with a social norm.

18 For a recent work synthesizing and extending some of the results of the social norms literature, see Posner, Eric A., Law and Social Norms (2000)Google Scholar. See also McAdams, Richard H., Signaling Discount Rates: Law, Norms, and Economic Methodology, 110 Yale L.J. 625 (2001)CrossRefGoogle Scholar.

19 For an example of this type of speculation comparing domestic custom to international custom, see Mendelson, supra note 14, at 165-68. See also Anthony A., D’Amato, The Concept of Custom in International Law (1971)Google Scholar.

20 See, e.g., Keohane, Robert O., After Hegemony: Cooperation and Discord in the World Political Economy (1984)Google Scholar; Keohane, Robert O., International Institutions and State Power: Essays in International Reiations Theory (1989)Google Scholar.

21 See, e.g., Krasner, Stephen D., International Regimes (1983)Google Scholar.

22 See, e.g., Haggard, Stephan & Simmons, Beth A., Theories of International Regimes, 41 Int’l Org. 491 (1987)CrossRefGoogle Scholar.

23 Snidal, Duncan, Political Economy and International Institutions, 16 Int’l Rev. L. & Econ. 121, 124 (1996)Google Scholar.

24 While there is no state at the global level, there is an international legal and organizational order, which is quite a bit more fragmented than most nation-states.

25 We add this qualification because one might argue that the CIL and conventional international law framework, as it exists, is comparable to a municipal state or, at least, to its constitution.

26 Ellickson, Robert C., The Evolution of Social Norms: A Perspective from the Legal Academy, in Social Norms 35 (Hechter, Michael & Karl-Dieter, Opp eds., 2001)Google Scholar. Note that Ellickson assumes multilateral, as opposed to bilateral, retaliation.

27 See Harold, Hongju Koh, Why Do Nations Obey International Law? 106 Yale L.J. 2599 (1997)Google Scholar (reviewing Chayes, Abram & Antonia, Handler Chayes, The New Sovereignty: Compliance with International Regulatory Agreements (1995)Google Scholar). This internalization may be desirable under certain circumstances and may be developed as a tool for enforcing CIL. That is, states may persuade one another to use their domestic legal systems as a means of bringing about compliance with particular international legal obligations. See also Trachtman, Joel P. & Moremen, Philip M., Costs and Benefits of Private Participation in WTO Dispute Settlement: Whose Right Is It Anyway ? 44 Harv. Int’l L.J. 221 (2003)Google Scholar.

28 See Cooter, Robert, Expressive Law and Economics, 27 J. Legal stud. 585 (1998)CrossRefGoogle Scholar.

29 Cooter, Robert, Do Good Laws Make Good Citizens? An Economic Analysis of Internalized Norms, 86 Va. L. Rev. 1577 (2000)CrossRefGoogle Scholar; Cooter, Robert, Models of Morality in Law and Economics: Self-Control and Self-Improvement for the “Bad Man” of Holmes, 78 B.U. L. Rev. 903, 911–912 (1998)Google Scholar; Elster, Jon, The Cement of Society: A Study of Social Order (1989)CrossRefGoogle Scholar. Basu refers to these as “preference-changing norms.” See Basu, Kaushik, Social Norms and the Law, 3 The New Palgrave Dictionary of Economics and the Law 477 (1998)Google Scholar.

30 See Mahoney, Paul G. & Chris, William Sanchirico, Norms, Repeated Games, and the Role of Law, 91 Cal. L. Rev. 1281, 1284 (2003)Google Scholar. Mahoney and Sanchirico explain the state of the social norms literature with respect to the multilateral prisoner’s dilemma. They explain that the objection to these models is that third-party enforcement is not individually rational; the players lack incentives to retaliate.

31 Id. at 1284 n.12 (citing works by EHickson, Katz, McAdams, and Posner).

32 There is disagreement between institutionalists and “realists,” who claim that states’ interests in international relations are characterized by a search for gains relative to other states, rather than absolute gains. Realists reject the possibility of cooperation where it results in relative gains to a competitor. See Busch, Marc L. & Reinhardt, Eric R., Nice Strategies in a World of Relative Gains: The Problem of Cooperation Under Anarchy, 37 J. Conflict Resol. 427 (1993)CrossRefGoogle Scholar; Powell, Robert, Absolute and Relative Gains in International Relations Theory, 85 Am. Pol. Sci. Rev. 1303 (1991)CrossRefGoogle Scholar; Snidal, Duncan, Relative Gains and the Pattern of International Cooperation, 85 Am. Pol. Sci. Rev. 701 (1991)CrossRefGoogle Scholar.

33 See the special Summer 2000 issue of International Organization devoted to the phenomenon of “legalization,” 54 Int’l Org. 385 (2000)Google Scholar.

34 See Abbott, Kenneth W. & Snidal, Duncan, Hard and Soft Law in International Governance, 54 Int’l Org. 421 (2000)CrossRefGoogle Scholar; Setear, John K., Treaties, Custom, Iteration, and Public Choice (2004), at <http://ssrn.com/abstract=492604>>Google Scholar (arguing that custom is more attractive to executive branches). To the extent that CIL is less detailed—less specific— than treaty norms, it is amenable to a rules-versus-standards type of analysis. See Kaplow, Louis, Rules Versus Standards: An Economic Analysis, 42 Duke L.J. 557, 577 (1992)Google Scholar; Trachtman, Joel P., The Domain of WTO Dispute Resolution, 40 Harv. Int’l L.J. 333, 346–47 (1999)Google Scholar.

35 For a useful analysis of the “fit” of other games, including “battle of the sexes” and “stag hunt,” see Swaine, supra note 1. See also McGillivray, Fiona & Smith, Alastair, Trust and Cooperation Through Agent-Specific Punishments, 54 Int’l Org. 809, 810 (2000)Google Scholar (noting that the prisoner’s dilemma is often used to model international cooperation).

36 By use of the term “externalities,” we mean to be quite inclusive, including both pecuniary and nonpecuniary externalities—that is, any circumstance in which an action by one state has adverse or beneficial effects on another state.

37 At another level of complexity, it would be possible to model the game of forming a CIL rule as separate from enforcement. See Fearon, James D., Bargaining, Enforcement, and International Cooperation, 52 Int’l Org. 269 (1998)CrossRefGoogle Scholar (separating the bargaining problem, modeled as a coordination game, from the enforcement problem, modeled as a prisoner’s dilemma); Krasner, Stephen D., Global Communications and National Power: Life on the Pareto Frontier, 43 World Pol. 336 (1991)CrossRefGoogle Scholar (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. He points out that relatively large “shadows of the future” might inhibit bargaining to achieve an initial agreement in a coordination game, while making the enforcement game more tractable. In the CIL context, there is less natural separation, and there may even be first-mover advantages that would counteract the effect that Fearon suggests. Finally, we would expect that states would integrate the lawmaking game with the enforcement game, making the initial legislative action less a coordination game and more a prisoner’s dilemma. Whether states did so would substantially depend on the extent to which they knew how any particular rule would affect their interests.

38 For a discussion of using coordination games to model certain types of international contexts, see Koremenos, Barbara, Lipson, Charles, & Snidal, Duncan, The Rational Design of International Institutions, 55 Int’l Org. 761, 774 (2001)Google Scholar, and Snidal, Duncan, Coordination Versus Prisoners’ Dilemma: Implications for International Cooperation and Regimes, 79 Am. Pol. Sci. Rev. 923 (1985)CrossRefGoogle Scholar.

39 See Guzman, supra note 1.

40 Martin, Stephen, Advanced Industrial Economics 98 (1993)Google Scholar.

41 In the classic prisoner’s dilemma, the prosecutor presents each of two prisoners an opportunity to confess. If neither of them confesses, both will be convicted of lesser offenses and receive comparatively light sentences of, say, three years. If either of them confesses and provides evidence against the other, while the other does not confess, the one who confesses is allowed to go free, whereas the other is given a much heavier sentence of, say, ten years. If both of them confess, they each serve seven years. The prisoners cannot communicate with, or bind, one another. From the prisoners’ perspective, the best collective outcome—a total of six years’ imprisonment—results when neither prisoner confesses. The dilemma, however, is that when each prisoner looks only at his own utility, he finds that, no matter what the other prisoner does, he is better off confessing. Each reasons as follows. If the other prisoner confesses, I have two options. If I do not confess, I serve ten years. But if I also confess, I serve only seven years. Therefore, I confess. Likewise, I have two options if the other prisoner does not confess. If I also do not confess, I serve three years. But if I confess, I serve no time at all. Therefore, I confess. The outcome is that each of the prisoners will always confess, and each will receive a longer sentence than if neither confessed. In this case, bilateral defection leads to an inefficient outcome. And since, under the game’s payoff structure, each party is better off defecting, no matter what the other party does, the outcome is a “Nash equilibrium”—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).

42 This is a “Nash equilibrium.” Id.

43 This is a subgame-perfect equilibrium. “A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in every subgame.” Osborne, M. J., An Introduction to Game Theory 166 (2004)Google Scholar.

44 See supra note 10 for definitions of stable equilibria and efficiency.

45 For an application of cooperative game theory in the international context, see Daniel G., Arce M. & Sandler, Todd, A Cooperative Game Theory of Noncontiguous Allies, 3 J. Pub. Econ. Theory 391 (2000)Google Scholar (applying cooperative game theory to international security arrangements).

46 Ostrom, Elinor, Collective Action and the Evolution of Social Norms, 14 J. Econ. Persp. 137, 138 (2000)Google Scholar.

47 See Axelrod, Robert, The Evolution of Cooperation (1984)Google ScholarPubMed. Evolutionary games are designed to compare the success of different strategies when played by a population of individuals in repeated play against other strategies.

48 In the language of game theory, it is not “subgame perfect”; that is, at every stage of a repeated game, no player will have an incentive to deviate from the equilibrium strategy, even when others do. See Osborne, supra note 43, at 444–46.

49 Id. at 444.

50 See supra note 41.

51 Goldsmith and Posner appear to recognize the possibility for stable and efficient equilibria under certain circumstances where states play the grim-trigger strategy. Nevertheless, using as their example the overfishing of the commons, they suggest that the grim trigger would not be used because it would be collectively irrational. Goldsmith & Posner 1999, supra note 1, at 1129–30.

52 Barrett, Scott, A Theory of Full International Cooperation, 11 J. Theoretical Pol. 519 (1999)CrossRefGoogle Scholar; Barrett, Scott, Environment and Statecraft: The Strategy of Environmental Treaty-Making (2003)CrossRefGoogle Scholar.

53 See, e.g. Farrell, Joseph & Maskin, Eric, Renegotiation in Repeated Games, 1 Games & Econ. Behav. 327 (1989)CrossRefGoogle Scholar; Fudenberg & Tirole, supra note 41, at 174.

54 Indeed, this argument is not uncommon in international law discourse. See, e.g., Sullivan, Scott M., Changing the Premise of International Legal Remedies: The Unfounded Adoption of Assurances and Guarantees of Non-repetition, 7 U.C.L.A. J. Int’l L. & Foreign Aff. 265 (2002–2003)Google Scholar.

55 While a precise definition of “renegotiation-proof” has not yet been agreed upon in game theory literature, the treatment by Farrell and Maskin is worth considering. They define a “weakly renegotiation-proof “equilibrium for an infinitely repeated game to be a subgame-perfect equilibrium strategy profile that is not Pareto-dominated by any other subgame-perfect strategy profile. Using this definition, the grim-trigger strategy profile described above is not weakly negotiation-proof, since after defection the payoffs to cooperation Pareto-dominate those of punishment. See Farrell & Maskin, supra note 53.

56 See Fudenberg & Tirole, supra note 41, at 179–82. “Penance” is sometimes referred to as “getting even.” See Myerson, Roger B., Game Theory: Analysis of Conflict 326–27 (1991)Google Scholar.

57 That is, it is both subgame perfect and weakly renegotiation-proof. See Fudenberg & Tirole, supra note 41, at 180 (citing Farrell & Maskin, supra note 53); Eric, van Damme, Renegotiation-Proof Equilibria in Repeated Prisoners’ Dilemma, 47 J. Econ. Theory 206 (1989)Google Scholar.

58 Fudenberg & Tirole, supra note 41, at 180. See also the “defect for deviate” strategy proposed by Mahoney and Sanchirico, supra note 30, at 1296.

59 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> [hereinafter Articles on State Responsibility]. Rather than expressing international law per se, these articles are an attempt to codify existing custom. See Chorzow Factory Case (Pol. v. Ger.), 1928 PCIJ (ser. A) Nos. 7, 9, 17, 19 (concerning obligation to make reparations).

60 See Bederman, David J., Counterintuiting Countermeasures, 96 AJIL 817 (2002)CrossRefGoogle Scholar.

61 Fudenberg & Tirole, supra note 41, at 112.

62 Myerson, supra note 56, at 371.

63 The first law of welfare economics suggests that under perfect competition, the outcome of market transactions is Pareto optimal. Kreps, David M., A Course in Microeconomic Theory 200 (1990)Google Scholar.

64 The Coase Theorem states that where transaction costs are zero, parties will bargain (costlessly) to achieve an efficient allocation. See Coase, Ronald H., The Problem of Social Cost, 3 J.L. & Econ. 1 (1960)Google Scholar.

65 Milgrom, Paul & Roberts, John, Economics, Organization and Management 24 (1992)Google Scholar.

66 A Pareto improvement is one that makes at least one person better off without making anyone worse off.

67 See supra note 9 and accompanying text.

68 While our model deals with games in which information is available to all, Parisi develops the Harsanyian concept of stochastic symmetry and role reversibility: the longer the shadow of the future, the less any one state can be certain of the way in which it will be affected by a particular rule. See Parisi, supra note 1. “Articulations that are made prior to unveiling of conflicting contingencies can be analogized to rules chosen under a Harsanyian veil of uncertainty.” Id. at 19. See also Keohane, Robert O., The Demand for International Regimes, 36 Int’l Org. 325 (1982)CrossRefGoogle Scholar. Harsanyi’s classic article, “Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility,” was published in the 1955 Journal of Political Economy. His concept of stochastic symmetry refers to the idea that parties may enter into an agreement where they are unsure—under a veil of uncertainty—as to precisely how the agreement will affect their particular interests. They can see that the agreement is an overall improvement, and are assumed to be willing to take a chance as to how that welfare improvement is distributed.

69 See Martin, Lisa L., The Rational State Choice of Multilateralism, in Multilateralism Matters: The theory and Praxis of an Institutional form 91, 99 (John, Gerard Ruggie ed., 1993)Google Scholar (suggesting techniques, including delegation as in the Security Council, for reducing the effective number of players).

70 Schachter, Oscar, New Custom: Power, Opinio Juris, and Contrary Practice, in Theory of International Law at the Threshold of the 21 st Century: Essays in Honour of Krzysztof Skubiszewski 531, 536–37 (Makarczyked, Jerzy., 1996)Google Scholar; see also Mendelson, supra note 14, at 194, 215, 225 (in the past, “civilized” states were considered sufficient, and the applicable group of states did not need to be geographically constrained).

71 See Asylum (Colom. v. Peru), 1950 ICJ Rep. 266 (Nov. 20); Right of Passage over Indian Territory (Port. v. India), 1960 ICJ Rep. 6 (Apr. 12). Cf. Shaw, Malcolm N., International Law 72–73 (4th ed. 1997)Google Scholar; Swaine, Edward T., The Local Law of Global Antitrust, 43 Wm. & Mary L. Rev. 627, 706–25 (2001)Google Scholar. ICJ judgments are available at <http://www.icj-cij.org>. See supra note 5 for the definition of “plurilateral.”

72 Goldsmith & Posner 1999, supra note 1, at 1132.

73 Id.

74 Olson, Mancur Jr., The Logic of Collective Action: Public Goods and The Theory of Groups 2 (1965)Google Scholar; see also Oye, Kenneth A., Explaining Cooperation Under Anarchy: Hypotheses and Strategies, in Cooperation Under Anarchy, supra note 7, at 1 Google Scholar.

75 Olson, supra note 74, at 48.

76 Hardin, Russell, Collective Action 43 (1982)Google Scholar.

77 See Lupia, Arthur & Sin, Gisela, Which Public Goods Are Endangered?: How Evolving Communication Technologies Affect the Logic of Collective Action, 117 Public Choice 315 (2003)CrossRefGoogle Scholar; Mitchell, Ronald B., Sources of Transparency: Information Systems in International Regimes, 42 Int’l Stud. Q. 109 (1998)CrossRefGoogle Scholar.

78 Oye, supra note 74, at 18–19.

79 Articles on State Responsibility, supra note 59.

80 See Bowles, Samuel & Gintis, Herbert, The Moral Economy of Communities: Structured Populations and the Evolution of Pro-social Norms, 19 Evolution & Hum. Behav. 3, 11–14 (1997)Google Scholar.

81 Kandori, Michihiro, Social Norms and Community Enforcement, 59 Rev. Econ. Stud. 63 (1992)CrossRefGoogle Scholar.

82 Id.

83 Milgrom, Paul R., North, Douglass C., & Weingast, Barry R., The Role of Institutions in the Revival of Trade: The Law Merchant, Private Judges and the Champagne Fairs, 2 Econ. & Pol. 1, 3 (1990)Google Scholar. Note that these authors are suggesting that information may compensate for infrequency of bilateral interaction.

84 Id. at 8.

85 Id. at 15.

86 Of course, to the extent that players may exchange nature payoffs for present payoffs, the patience variable becomes less important or less diverse; where efficient capital markets allow for the exchange of future payoffs for current payoffs, we would expect the patience variable to be constant across players.

87 A “discount factor” is a mathematical factor structured to reflect the degree of patience of a player. It represents the present value today of a payoff in a future period. A discount factor of 1 means that future payoffs are valued equally to present payoffs. A discount factor of .75 means that future payoffs are valued at 75% of present payoffs. A high discount factor indicates patience, while a low discount factor indicates impatience. Discount factors should be contrasted with “discount rates” applied to future payoffs in order to reduce them to a present value, where the discount factor equals 1/(1 + r), with r representing the discount rate. Discount factors are less than 1 (assuming a positive discount rate), and a low discount rate corresponds to a high discount factor. For example, a discount rate of 10% would result in a discount factor of approximately 91%. We discuss below the circumstances under which a state might be thought to have a high or low discount factor.

88 See Myerson, supra note 56, at 308–69.

89 Fudenberg & Tirole, supra note 41, at 111.

90 Id.

91 Id. at 150 (referring to the Folk Theorem).

92 Fudenberg, Drew & Maskin, Eric, The Folk Theorem in Repeated Games with Discounting or with Incomplete Information, 54 Econometrica 533 (1986)CrossRefGoogle Scholar.

93 See Keohane, supra note 20, at 91; Haas, Ernst, Why Collaborate? Issue Linkage and International Regimes 32 World Pol. 357 (1980)CrossRefGoogle Scholar; McGinnis, Michael D., Issue Linkage and the Evolution of International Cooperation, 30 J. Conflict Resol. 141 (1986)CrossRefGoogle Scholar; Tollison, Robert D. & Willett, Andrew D., An Economic Theory of Mutually Advantageous Issue Linkage in International Negotiations, 33 Int’l Org. 425 (1979)CrossRefGoogle Scholar. McGinnis shows formally that in a prisoner’s dilemma “multisupergame,” players may adopt strategies that create linkages across time and games, providing opportunities for cooperation, whereas cooperation would not be possible for isolated games.

94 See Spagnolo, Giancarlo, Issue Linkage, Credible Delegation, and Policy Cooperation (Center for Econ. Pol’y Res., Discussion Paper No. 2778, 2001), at <http://ssrn.com/abstract=269364>Google Scholar.

95 See, e.g., Douglas Bernheim, B. & Whinston, Michal D., Multimarket Contact and Collusive Behavior, 21 Rand J. Econ. 1 (1990)Google Scholar; Edwards, Corwin D., Conglomerate Bigness as a Source of Power, in Business Concentration and Price Policy 331 (1955)Google Scholar; Matsushima, Hitoshi, Multimarket Contact, Imperfect Monitoring, and Implicit Collusion, 98 J. Econ. Theory 158 (2001)CrossRefGoogle Scholar; Spagnolo, Giancarlo, On Interdependent Supergames: Multimarket Contact, Concavity and Collusion, 89 J. Econ. Theory 127 (1999)CrossRefGoogle Scholar.

96 Spagnolo, supra note 95, at 128. Spagnolo shows that when players have a concave objective function in respect of profits—meaning that they value extra payoffs less than some basic level of payoffs, which induces wealth effects and therefore generates economies of scale in engaging in cooperation—multimarket contact can enhance cooperation. Aconcave objective function in this context arises from an interest in smoother payoffs—and less interest in unusually high payoffs—in any given period. There are reasons why a political “manager” might also have a concave objective function. See Mansfield, Edward D., Milner, Helen V., & Peter Rosendorff, B., Why Democracies Cooperate More: Electoral Control and International Trade Agreements, 56 Int’l Org. 477 (2002)CrossRefGoogle Scholar (modeling trade agreements as signaling and commitment devices in the domestic, democratic context). In the international relations context, in which executives play a two-level game, a concave objective function may arise from accountability via elections or, in the trade context, from a desire to avoid disruption to particular constituencies. What we have here is an example of Corden’s “conservative social welfare function.” Corden, W. M., Trade Policy and Economic Welfare 107 (1974)Google Scholar.

97 Spagnolo, supra note 95, at 133.

98 See Guzman, supra note 1, at 1869–70; Snidal, supra note 38, at 939.

99 See supra note 93.

100 Matsushima, supra note 95, at 164–65.

101 See Koremenos et al., supra note 38, at 764–65.

102 See, e.g., Haas, Ernst B., Beyond The Nation-State: Functionalism and International Organization (1968)Google Scholar; Jeppe, Tranholm-Mikkelsen, Neofunctionalism: Obstinate or Obsolete? 20 Millenium: J. Int’l Stud. 1 (1991)Google Scholar. Neofunctionalism hypothesized that increasing cooperation would accelerate, as cooperation would create the conditions for its own extension.

103 Downs, George W. & Jones, Michael A., Reputation, Compliance, and International Law, 31 J. Legal Stud. 95, 101 (2002)Google Scholar (suggesting that reputation varies by field of activity, and that this segmentation reduces the effects of reputation).

104 See Anne-Marie, Slaughter, The Real New World Order, 76 Foreign Aff. 183 (1997)Google Scholar.

105 Milgrom et al., supra note 83, at 1.

106 See Trachtman, Joel P., Economic Analysis of Prescriptive Jurisdiction, 42 Va. J. Int’l L. 1 (2001)Google Scholar.

107 While it is possible that contractual waivers of immunity might be used, they would not cover extracontractual responsibility and might entail costly negotiations. In any event, these examples are intended as sketches of possible, rather than definitive, analyses.

108 Trail Smelter Case (U.S. v. Can.), 3 R.I.A.A. 1905 (1938 & 1941). The principle of sic utere tuo was included as Principle 21 of the Stockholm Declaration, UN Conference on the Human Environment, Stockholm Declaration, June 16, 1972, UN Doc. A/CONF.48/14, 11 ILM 1416 (1972). See also Principle 2 of the 1992 Rio Declaration on Environment and Development, June 14, 1992, UN Doc. A/CONF.151/5/Rev.l, 31 ILM 874 (1992).

109 See also International Liability for Injurious Consequences Arising out of Acts Not Prohibited by International Law, in Report of the International Law Commission on the Work of Its Fifty-fifth Session, UN GAOR, 58th Sess., Supp. No. 10, at 103, UN Doc. A/58/10 (2003), available at <http://www.un.org/law/ilc/>.

110 Pigou suggested that efficiency is enhanced by legal rules internalizing all externalities; Coase showed that the efficiency of internalization depends on the transaction-cost profile of the circumstance without internalization compared to that of the circumstance with internalization. See Pigou, Arthur C., The Economics of Welfare §11.11.5 (1932), available at <http://www.econlib.org/library/NPDBooks/Pigou/pgEW13.html>>Google Scholar.

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112 Zacher, Mark W., The Territorial Integrity Norm: International Boundaries and the Use of Force, 55 Int’l Org. 215 (2001)CrossRefGoogle Scholar. This norm does not prevent states from exercising influence over policy within other states.

113 Grim trigger is not necessarily tougher than penance. While in some cases grim trigger will provide a greater incentive to cooperate than penance, the opposite will be true if the gain from defection when the other state cooperates is less than the gain from defection when the other state defects. We show this mathematically in the appendix.

114 The International Court of Justice has recognized that even the protection of diplomats has an important multilateral dimension. See United States Diplomatic and Consular Staff in Tehran (U.S. v. Iran), 1980 ICJ Rep. 3, para. 92 (May 24). In that case, the ICJ made the following statement:

115 See, e.g., Pahre, Robert, Multilateral Cooperation in an Iterated Prisoner’s Dilemma, 38 J. Conflict Resol., 326 (1994)CrossRefGoogle Scholar; Snidal, supra note 38, at 929.

116 For a discussion of global public goods, see Providing Global Public Goods: Managing Globalization (Inge Kaul et al. eds., 2003).

117 Other areas of international law (for example, relating to sovereignty and diplomatic protection) may exhibit network effects, which may arise simply from efficiencies due to consistency of arrangements. For a relevant analysis in the corporate law field, see Klausner, Michael, Corporations, Corporate Law, and Networks of Contracts, 81 Va. L. Rev. 757 (1995)CrossRefGoogle Scholar.

118 The classic reference on this last point is Bernheim & Whinston, supra note 95.

119 Chinen, Game Theory and Customary International Law, supra note 1, at 154.

120 See Komesar, Neil K., Imperfect Alternatives (1994)Google Scholar; Hervé, Moulin, Cooperative Microeconomics: A Game-Theoretic Introduction (1995)Google Scholar.

121 “Consensus” is here understood as an absence of express objection.

122 See Trachtman, supra note 34.

123 Goldsmith & Posner 1999, supra note 1, at 1 131–33.

124 D’Amato, supra note 19, at 66.

125 It is worth noting the relationship of this proposition to the first formulation of Kant’s categorical imperative: “[A]ct only on that maxim through which you can at the same time will that it should become a universal law.” Immanuel Kant, Groundwork of the Metaphysics of Morals 31 (Mary Gregor trans. & ed., Cambridge Univ. Press 1998) (1785).

126 See Military and Paramilitary Activities in and Against Nicaragua (Nicar. v. U.S.), 1986 ICJ Rep. 14, para. 207 (June 27) (“Reliance by a State on a novel right, or an unprecedented exception to the principle, might if shared in principle by other States, tend towards a modification of customary international law.”).

127 “[A]t the initial stage of the development of the custom, it is sufficient that the States concerned regard the practice as what the Court. . . referred to as ‘potentially norm-creating’, as conforming to a rule which either already exists or is a useful and desirable rule which should exist.” Thirlway, Hugh, The Law and Procedure of the International Court of Justice 1960–1989 (Part Two), 1990 Brit. Y.B. Int’l L. 1, 43 Google Scholar (citing North Sea Continental Shelf (FRG/Den.; FRG/Neth.), 1969 ICJ Rep. 3, 42 (Feb. 20)). See McDougal, Myres etal., Studies in World Public Order 773–74 (1960)Google Scholar; Mendelson, supra note 14; Swaine, supra note 1, at 615 (suggesting that states that initiate a custom might be analogized to offerors in a contractual setting, and that this conditional obligation is consistent with opinio juris); see also D’Amato, supra note 19, at 73–102. And what of the circumstances in which initial acts are unreciprocated? We might understand these acts as transaction costs—that is, as the costs of identifying feasible transactions or appropriate partners. This analysis is not very different from that of gift-giving practices in many village and other social circumstances. See Posner, supra note 18, at 49–67.

128 See Parisi, supra note 1, at 18 (describing “articulation theories” of CIL in these terms). Parisi suggests special deference to rules chosen prior to conflict: “Articulations that are made prior to the unveiling of conflicting contingencies can be analogized to rules chosen under a Harsanyian veil of uncertainty.” Id. at 19; see supra note 68. This condition is referred to as “stochastic uncertainty.” One might ask, however, why there would be an incentive to produce new rules in the absence of conflict. Perhaps the answer is that the formation of custom is a process occurring over time, and states may participate before they know what their individual positions and interests will be. One might also argue that the veil of uncertainty continues to operate even after conflict arises, at least to the extent that states may not know their position with respect to future conflicts.

129 Comm. on Formation of Customary (Gen.) Int’l Law, Int’l Law Ass’n, Final Report: Statement of Principles Applicable to the Formation of General Customary International Law § 1 (i) (2000) (citing Mendelson, supra note 14, at 399), at <http://www.ila-hq.org/pdf/CustomaryLaw.pdf>.

130 What of states that have no relevant intent, or that object to the formation of a legal rule? We will not deal with the doctrinal problem of the persistent objector. But it seems plausible that states would accept a rule of consensus legislation in international custom: a rule may be formed that binds states that do not object. For an analysis of consensus-based decision making within the GATT/WTO system, see Steinberg, Richard H., In the Shadow of Law or Power? Consensus-Based Bargaining and Outcomes in the GATT/WTO, 56 Int’l Org. 339 (2002)CrossRefGoogle Scholar.

131 The ICJ seems to ignore the text in favor of a reverse reading, seeking to find evidence of international custom in a general practice accepted as law. Rosalyn Higgins, Problems & Process: International Law and How We Use It 18–19 (1994). Of course, there are other formulations. See, e.g., Brownlie, Ian, Principles of Public International Law 4–11 (5th ed. 1998)Google Scholar.

132 Goldsmith & Posner 1999, supra note 1, at 1115.

133 See Keohane, Robert O., Reciprocity in International Relations, 40 Int’l Org. 1 (1986)Google Scholar.

134 Recall that there is an important literature on social norms that considers the potential for preference modification, holding out the possibility of compliance because of modified preferences. See the sources cited supra note 29. It may not be necessary to choose between static and dynamic preferences: the latter may work together with the former and with self-interest. Changes in preferences may effect changes in rules, and changes in rules may have an impact on preferences.

135 But see Swaine, supra note 1 (raising concerns regarding Goldsmith and Posner’s perspective on CIL doctrine).

136 Cf. Guzman, supra note 1, at 1875 (asserting that Goldsmith and Posner’s claim that CIL does not affect state behavior goes beyond what the evidence suggests).

137 For some criticism of Goldsmith and Posner’s description of CIL doctrine, see Vagts, supra note 2.

138 See D’Amato, supra note 19, at 97 (stating that “an ‘illegal’ act by a state contains the seeds of a new legality”).

139 Goldsmith & Posner 1999, supra note 1, at 1157.

140 See, e.g., Shaw, supra note 71, at 69.

141 See Anne-Marie, Slaughter, International Law in a World of Liberal States, 6 Eur. J. Int’l L. 503 (1995)Google Scholar.

142 See Slaughter, supra note 104.

143 This article does not by any means challenge the theory that law can affect behavior by modifying preferences. It merely presents a theory that does not depend on modifying preferences.

144 See Swaine, supra note 1, at 618 (“states do not, in fact, interact solely with respect to one rule or the other, and it is also possible to understand their interaction with respect both to an individual rule and to the system of customary international law”).

145 This assumption keeps the notation reasonably simple and is not unreasonable in the context of international law. The formal analysis is unchanged if we assume that defection against g induces punishment by some group h, which may include g plus other states. Most analyses of the n-person prisoner’s dilemma assume, in fact, that h is the full set of cooperating states other than i.

146 For the sake of notational simplicity, we assume in this part of the discussion that payoffs for state i are determined by the number of cooperating, punishing, and defecting states, but not by their identities. We return to this assumption below.

147 Fudenberg & Tirole, supra note 41, at 180; see also the “defect-for-deviate” strategy proposed by Mahoney and Sanchirico, supra note 30.

148 We could also assume with grim trigger that states defect for a finite number of periods t without affecting the qualitative conclusions.

149 Again, standard analysis indicates that we can always choose the first period in which a state defects as the beginning of the game.

150 We are grateful to an anonymous referee for identifying this possibility. Maggi identifies the World Trade Organization as such an arbiter of trade agreements. Maggi, Giovanni, The Role of Multilateral Institutions in International Trade Cooperation, 89 Am. Econ. Rev. 190 (1999)CrossRefGoogle Scholar. We note, however, that the WTO does not coordinate multilateral retaliation in response to violations of WTO law, but permits only states that have used its dispute settlement facilities to retaliate.

151 There is the related question: what is the optimal deviation for i ? In other words, what is the optimal choice of g? This question, while interesting, will not be considered here since it takes us beyond the specific goals of this article.

152 The classic reference on this phenomenon is Bernheim & Whinston, supra note 95. See also Lohmann, Susanne, Linkage Politics, 41 J. Conflict Resol. 38 (1997)CrossRefGoogle Scholar, for an c the context of a bilateral prisoners’ dilemma.

153 Goldsmith & Posner 1999, supra note 1, at 1125 tbl. 2. Goldsmith & Posner’s table 2, relating to the protection of coastal fishing boats, appears as follows:

154 The more general analysis is available from the authors on request.

155 See supra note 153.

156 This case is the one analyzed by Lohmann, supra note 152.