¹ The essays presented here focus on nation-states as primary actors in world politics, treat national preferences as sovereign, and assume that any ultimate escape from international anarchy is unlikely. Our focus is on non-altruistic cooperation among states dwelling in international anarchy.

² In this essay, I use elementary game theory in a purely instrumental fashion. First, although some references to the formal literature are provided, the text does not furnish formal proofs on the existence or location of equilibrium points in different categories of games. As Thomas Schelling notes, the equilibrium solutions identified by formal game theorists may stabilize convergent expectations among mathematicians, but unless equilibria can also be reached through “alternative less sophisticated routes,” such solutions may have little influence on international outcomes. See Schelling, , The Strategy of Conflict (London: Oxford University Press, 1963), 113Google Scholar–14. Accordingly, the contributors search for “alternative less sophisticated routes” to reach mutually beneficial equilibrium points and for simple strategies to restructure situations to create mutually beneficial equilibrium points.

³ For an extended discussion of the uses and abuses of game theory in the empirical study of international politics, see Duncan Snidal, “The Game Theory of International Politics,” in this collection.

⁴ For the definitive classification of ordinally defined games, see Anatol Rapoport and Melvin Guyer, “A Taxonomy of 2 × 2 Games,” General Systems 11 (1966), 203Google Scholar–14. For an extended reinterpretation of crisis bargaining in light of payoff structures, see Snyder, Glenn H. and Diesing, Paul, Conflict Among Nations: Bargaining, Decisionmafying, and System Structure in International Crises (Princeton: Princeton University Press, 1977Google Scholar).

⁵ For examples, see Jervis, Robert, “Cooperation under the Security Dilemma,” World Politics 30 (January 1978), 167–214CrossRefGoogle Scholar; Williamson, Oliver E., “Credible Commitments: Using Hostages to Support Exchange,” American Economic Review (September 1983), 519Google Scholar–40; Ruggie, John Gerard, “International Regimes, Transactions, and Change: Embedded Liberalism in the Postwar Economic Order,” in Krasner, Stephen D., ed., International Regimes (Ithaca, N.Y.: Cornell University Press, 1983Google Scholar).

⁶ For orthodox game-theoretic analyses of the importance of iteration, see Luce, R. Duncan and Raiffa, Howard, Games and Decisions (New York: Wiley, 1957Google Scholar), Appendix 8, and Kreps, David M., Milgram, Paul, Roberts, John, and Wilson, Robert, “Rational Cooperation in Finitely-Repeated Prisoner's Dilemma,” Journal of Economic Theory 27 (August 1982, 245CrossRefGoogle Scholar–52. For the results of laboratory experiments, see Radlow, Robert, “An Experimental Study of Cooperation in the Prisoner's Dilemma Game,” Journal of Conflict Resolution 9 (June 1965), 221CrossRefGoogle Scholar–27. On the importance of indefinite iteration to the emergence of cooperation in business transactions, see Telsor, Robert, “A Theory of Self-Enforcing Agreements,” Journal of Business 53 (January 1985), 27–44CrossRefGoogle Scholar.

⁷ On how iterated Prisoners' Dilemmas environments literally select for Tit-for-Tat strategies, see Axelrod, Robert, The Evolution of Cooperation (New York: Basic Books, 1984Google Scholar). For a formal statement on the effects of reciprocity on equilibrium outcomes in iterated games, see Drew Fudenberg and Eric Maskin, “The Folk Theorem in Repeated Games with Discounting and with Incomplete Information,” Econometrica, forthcoming.

⁸ On enhancing iterativeness through decomposition of payoffs over time, see Schelling (fn. 2), 43–46, and Axelrod (fn. 7), 126–32.

⁹ Ibid., 139–41.

¹⁰ See Shubik, Martin, Games for Society, Business and War: Towards A Theory of Gaming (New York: Elsevier, 1975Google Scholar). For a formal statement on the importance of the number of players to cooperation in iterated games, see Fudenberg and Maskin (fn. 7).

¹¹ See Keohane, Robert O., After Hegemony: Cooperation and Discord in the World Political Economy (Princeton: Princeton University Press, 1984Google Scholar), and Krasner (fn. 5).

¹² See Conybeare, John A. C., “International Organization and the Theory of Property Rights,” International Organization 34 (Summer 1985), 307CrossRefGoogle Scholar–34, and Oye, Kenneth A., “Belief Systems, Bargaining, and Breakdown: International Political Economy 1929–1936,” Ph.D. diss. (Harvard University, 1983Google Scholar), chap. 3.

¹³ See Jervis (fn. 5); Ruggie (fn. 5); McKeown, Timothy J., “Firms and Tariff Regime I Change: Explaining the Demand for Protection,” World Politics 36 (January 1984), 215CrossRefGoogle Scholar–33. I On the effects of ambiguity of preferences on the prospects of cooperation, see the concluding sections of Jervis (fn. 5).

¹⁴ For an extended discussion of the distinction between cooperation and harmony, see Keohane (fn. 11), 51–55.

¹⁵ Wagner, , “The Theory of Games and the Problem of International Cooperation,” American Political Science Review 70 (June 1983), 330CrossRefGoogle Scholar–46.

¹⁶ Kenneth Waltz borrowed Rousseau's parable of the staghunt to illustrate the infeasibility of realizing mutual interests under international anarchy. Rousseau used the staghunt to illustrate the possibility of cooperation during his first period of primative social interdependence. He argued that individuals could cooperate on “mutual undertakings” to realize “present and perceptible interest” through “some kind of free association that obligated no one and lasted only so long as the passing need that formed it.” This essay returns to Rousseau's use of the staghunt. See Waltz, , Man, the State, and War (New York: Columbia University Press, 1959Google Scholar), and Jean Jacques Rousseau: The First and Second Discourses, trans. , Roger D. and Masters, Judith R. (New York: St. Martins, 1964), 165Google Scholar–67.

¹⁷ The illustrative preference orderings strike most mature observers as perverse: the drivers need not place themselves in the game.

¹⁸ Haas, , Williams, , and Babai, , Scientists and World Order: The Uses of Technical Knowledge in International Organizations (Berkeley: University of California Press, 1977Google Scholar).

¹⁹ Williamson (fn. 5).

²⁰ See Nitze, Paul, “Assuring Strategic Stability in an Era of Detente,” Foreign Affairs 54 (January 1976), 207CrossRefGoogle Scholar–32, for the seminal article in this tradition. Nitze's recommendations hinge on acceptance of the precepts of what has come to be known as nuclear utilization theory. Jervis's recommendations depend on acceptance of the precepts of mutual assured destruction (fn. 5).

²¹ See Newhouse, John, Cold Dawn: The Story of SALT I (New York: Holt, Rinehart & Winston, 1973Google Scholar).

²² See Haas, “Words Can Hurt You; Or Who Said What to Whom About Regimes,” in Krasner (fn. 5).

²³ This section is derived largely from Axelrod (fn. 7), and Telsor (fn. 6).

²⁴ See Rapoport, Anatol and Chammah, Albert, Prisoner's Dilemma (Ann Arbor: University of Michigan Press, 1965CrossRefGoogle ScholarPubMed), and subsequent essays in Journal of Conflict Resolution.

²⁵ One common objection to this line of argument centers on the irrationality of cooperation if a sequence of Prisoners' Dilemmas has a known last element. On the known last play,; the immediate gain from squealing cannot be offset by expectations of future cooperation. On the next-to-last play, the immediate gain from squealing is not offset by expectations % of future cooperation, since both actors know that cooperation is irrational on the last move. : And so on back toward the initial move. This line of analysis collapses iterated Prisoners' : Dilemma into single-play Prisoners' Dilemma. To analysts of international relations, the importance of this objection is limited. In international relations, no experimenter decrees 1 that a series of Prisoners' Dilemmas shall end on the 10th move or at noon. Although any series of transactions will terminate sooner or later, governments do not generally know when the last play will occur. On all rounds of play, the actors' decisions are conditioned by the possibility of future interaction. For a formal analysis of how uncertain time horizons can lead to a resolution of the Prisoners' Dilemma, see Luce and Raiffa (fn. 6), Appendix 8. Discount parameters such as Axelrod's ”w” may capture the effects of uncertainty. Possible future payoffs may be discounted both because the value placed on future benefits is lower than present benefits and because the stream of future benefits may be interrupted if the structure of the game changes.

²⁶ This conclusion rests on the assumption that dyadic interactions are moderately independent. For an argument on how defection can provide a benefit (external to a dyadic interaction) by discouraging the entry of other actors, see Shibley Telhami, “Cooperation or Coercion: Tit for Tat and the Realities of International Politics,” unpub. (Swarthmore College, January 1985). Note also that cooperation can also encourage (mutually beneficial) entry of other actors.

²⁷ On iterated Chicken, see Snyder and Diesing (fn. 4), 43–44.

²⁸ See George, Alexander L., Managing U.S.-Soviet Rivalry: Problems of Crisis Prevention (Boulder, CO: Westview, 1983Google Scholar).

²⁹ Axelrod shows that in iterated Prisoners' Dilemma, where actors can reliably distinguish between cooperation and defection by others and respond in kind, Tit-for-Tat performs better than do alternative strategies. When recognition and control are perfect, iterated environments strongly favor the emergence of cooperation.

³⁰ Schelling (fn. 2), 43–46, and Axelrod (fn. 7), 126–32.

³¹ For analyses of issue linkage, see Tollison, Robert D. and Willett, Thomas D., “An Economic Theory of Mutually Advantageous Issue Linkages in International Negotiations,” International Organization 33 (Autumn 1979) 425–49CrossRefGoogle Scholar; Oye (fn. 12), chap. 3, “Bargaining: The Logic of Contingent Action”; and Axelrod and Keohane in the concluding essay of this symposium.

³² See Olson, Mancur Jr., The Logic of Collective Action: Public Goods and the Theory of Groups (Cambridge: Harvard University Press, 1965Google Scholar), and Olson, Mancur and Zeckhauser, Richard, “An Economic Theory of Alliances,” Review of Economics and Statistics 48 (August 1966), 266CrossRefGoogle Scholar–79. For a recent elegant summary and extension of the large literature on dilemmas of collective action, see Hardin, Russell, Collective Action (Baltimore: Johns Hopkins University Press, 1982Google Scholar).

³³ See Waltz, Kenneth N., “The Stability of a Bipolar World,” Daedalus 93 (Summer 1964Google Scholar), and Rosecrance, Richard N., “Bipolarity, Multipolarity, and the Future,” Journal of Conflict Resolution (September 1966), 314CrossRefGoogle Scholar–27.

³⁴ On hegemony, see Gilpin, Robert, U.S. Power and the Multinational Corporation (New York: Basic Books, 1975), 258CrossRefGoogle Scholar–59. On duopoly, see McKeown, Timothy, “Hegemonic Stability Theory and 19th-century Tariff Levels in Europe,” International Organization 37 (Winter 1983), 73–91CrossRefGoogle Scholar. On regimes and cooperation, see Keohane (fn. 11), and Krasner (fn. 5). On two-person games and N-person public-goods problems, see Kindleberger, Charles, “Dominance and Leadership in the International Economy: Exploitation, Public Goods, and Free Rides,” International Studies Quarterly 25 (June 1981), 242CrossRefGoogle Scholar–54.³⁵See Keohane (fn. 11), chap. 6, for extensions of these points.

³⁶ See Dunn, Lewis A., Controlling the Bomb (New Haven: Yale University Press, 1982Google Scholar).

³⁷ In addition to providing a partial solution to the problems of large numbers, regimes may affect the order and intensity of actor preferences as norms are internalized, and may heighten the iterativeness of situations as interaction becomes more frequent.

³⁸ For a full analysis of intra-alliance cooperation on East-West trade, see Mastanduno, Michael, “Strategies of Economic Containment: U.S. Trade Relations with the Soviet [Union,” World Politics 37 (July 1985), 503CrossRefGoogle Scholar–31, and Crawford, Beverly and Lenway, Stephanie, [”Decision Modes and International Regime Change: Western Collaboration on East-West Trade, World Politics 37 (April 1985), 375–402CrossRefGoogle Scholar.

³⁹ For a pure libertarian argument on private exchange as an alternative to public man-iagement, see Conybeare (fn. 12).

⁴⁰ See George, Alexander and Smoke, Richard, Deterrence in American Foreign Policy (New York: Columbia University Press, 1974Google Scholar) for a seminal example of how an austere theoretical framework and detailed historical cases can promote both development of theory and historical understanding.