¹ Among more recent works, see Alexandroff, Alan and Rosecrance, Richard, “Deterrence in 1939,” World Politics 29 (April 1977), 404CrossRefGoogle Scholar–24; de Mesquita, Bruce Bueno, The War Trap (New Haven: Yale University Press, 1981Google Scholar); George, Alexander and Smoke, Richard, Deterrence in American Foreign Policy (New York: Columbia University Press, 1974Google Scholar); Jervis, Robert, Perception and Misperception in International Politics (Princeton: Princeton University Press, 1976Google Scholar); Mearsheimer, John, Conventional Deterrence (Ithaca, NY: Cornell University Press, 1983Google Scholar); and Snyder, Glenn and Diesing, Paul, Conflict Among Nations (Princeton: Princeton University Press, 1977Google Scholar).

² My comments are not intended as criticism of an ambitious and insightful enterprise. The authors' descriptive use of game theory is appropriate, given an ultimate concern with different models of decision making and bargaining rather than with game theory, which is used only in "the limited role … [of] depicting the structure of a crisis" Snyder and Diesing (fn. i), 87. Nevertheless, if any analytical game theory approach were attributed to the work, a game theorist could reasonably object that it had been emasculated (cf. p. 182).

³ Snidal, Duncan, “The Limits of Hegemonic Stability Theory,” International Organization 39 (Autumn 1985), 579–614CrossRefGoogle Scholar.

⁴ The articles in this volume presuppose a basic familiarity with game-theoretic approaches. A good introduction is Hamburger, Henry, Games as Models of Social Phenomena (San Francisco: W.H. Freeman, 1979Google Scholar); Shubik, Martin, Game Theory in the Social Sciences: Concepts and Solutions (Cambridge: MIT Press, 1983Google Scholar) provides a more advanced treatment. Two accessible “classics” are Luce, R. Duncan and Raiffa, Howard, Games and Decisions (New York: Wiley, 1957Google Scholar), and Schelling, Thomas, Strategy of Conflict (Cambridge: Harvard University Press, 1960Google Scholar).

⁵ Landau, Martin, “On the Use of Metaphor in Political Analysis,” Social Research 28 (Autumn 1961), 331Google Scholar–53.

⁶ The Hobbesian (domestic) state of nature is contrasted with the current international situation in Beitz, Charles, Political Theory and International Relations (Princeton: Princeton University Press, 1979Google Scholar). Decentralized international cooperation is dealt with from a strategic-actor perspective in Keohane, Robert O., After Hegemony (Princeton: Princeton University Press, 1984Google Scholar), and from a more purely game-theoretic perspective in Snidal, Duncan, “Interdependence, Regimes and International Cooperation” (Ph.D. diss., Yale University, 1981Google Scholar). See also the related articles in Krasner, Stephen, ed., International Regimes (Ithaca, NY: Cornell University Press, 1983Google Scholar).

⁷ Metaphors are distinguished by the looseness of their correspondence rules, but not necessarily by the absence of mathematical sophistication. For an example of a mathematically sophisticated (though ultimately nonfruitful) metaphor, recall Paul Smoker's use of the harmonic motion of springs as a metaphor for arms rivalry, in “The Arms Race as an Open and Closed System,” Peace Research Society (International) Papers 7 (1967), 41–62Google Scholar. For a critical discussion, see Busch, Peter, “Mathematical Models of Arms Races,” an appendix to Bruce Russett, What Price Vigilance? (New Haven: Yale University Press, 1970Google Scholar).

⁸ The oligopoly analogy is widespread. The correspondences presented here can be found in Waltz, Kenneth, Theory of International Politics (Reading, MA: Addison-Wesley, 1979Google Scholar), esp. chap. 7. Other examples of microeconomic analogies are Gilpin, Robert, War and Change in World Politics (New York: Cambridge University Press, 1981CrossRefGoogle Scholar) and Keohane, Robert O., “The Demand for International Regimes,” International Organization 36 (Spring 1982), 325CrossRefGoogle Scholar–55'

⁹ On the “economic approach” (of rational, maximizing behavior) versus economics as analogy, see Barry, Brian, Sociologists, Economists and Democracy (Chicago: University Chicago Press, 1978Google Scholar).

¹⁰ I do not want to convey too pristine a view of how models are developed. Models are always constructed with an eye toward some of their inferences (what assumptions are needed to produce a certain conclusion). Moreover, the ceteris paribus clause is often invoked to deal with important correspondences that are not contained in the model. However, the logical structure of the model forces our theoretical assumptions and conclusions to be consistent and leads to other wholly unanticipated inferences. The ceteris paribus clause should not become a refuge from incorporating further considerations into the model; it should be a stimulus for its progressive refinement.

¹¹ Models are not primarily distinguished from analogies and metaphors by mathematical sophistication. (See note 7.) For example, physical models and analogue machines (including computer simulations viewed as physical machine representations) are models that are not in explicit mathematical form. Mathematics is simply a particular way of expressing a model. It is useful in forcing us to tighten up correspondences, in exposing weaknesses in a model or metaphor, and in providing a powerful means of pursuing deductive implications.

¹² Again, the distinction between a theory and its model is not always clear. Mary Hesse argues that “almost any model or interpretation carries some surplus meaning. If, however, a model is used in a way that exploits this surplus meaning in prediction and explanation, we shall call it a theoretical model.” See Hesse, “Models and Analogy in Science” in Edwards, Paul, ed., The Encyclopedia of Philosophy (New York: Macmillan and Free Press, 1967), 354Google Scholar–59.

¹³ For example, waves provide a model for both water motion and light; similarly, the coupled differential equations which can be interpreted as a model of an arms race in international relations may have a very different meaning in thermodynamics. Indeed, it is impossible to speak of a model of something—as opposed to a purely logical and empirically uninterpreted model—without theory to guide us on the correspondence rules. Any empirical model must be embedded in a theory. However, the theoretical richness is often tightly circumscribed in the model. A good example is the use of the “as if assumption to establish a correspondence at a purely observational level without plumbing the deeper implication of the observed behavior. (See note 17.)

¹⁴ Keohane, Robert O., “Theory of World Politics: Structural Realism and Beyond,” in Finifter, Ada, ed., Political Science: The State of the Discipline (Washington, DC: American Political Science Association, 1983Google Scholar).

¹⁵ Here we can agree that “perhaps every science must start with metaphor and end with algebra; and perhaps without the metaphor there would never have been any algebra”—although by the argument of this section, the word “model” should be substituted for “algebra.” See Black, Max, Models and Metaphors (Ithaca, NY: Cornell University Press, 1962), 242Google Scholar.

¹⁶ Axelrod, Robert, The Evolution of Cooperation (New York: Basic Books, 1984Google Scholar), chap. 5.

¹⁷ This position agrees with Milton Friedman's well-known “as if argument on one level, but differs from it on another. Friedman's argument is that it does not matter if the actors being modeled actually make (strategic) calculations as long as they act “as if they did. For him, the proof of the pudding is the accuracy of the predictions that result from the assumption. But if we are to understand and explain behavior in addition to predicting it, his argument will be insufficient. To understand state behavior in international politics, and to avoid post hoc reconstruction of behavior as “rational,” we must pay attention to the nature and limits of state rationality. See Friedman, , “The Methodology of Positive Economics,” in Hahn, Frank and Hollis, Martin, eds., Philosophy and Economic Theory (New York: Oxford University Press, 1979Google Scholar), and Blaug, Mark, The Methodology of Economics (New York: Cambridge University Press, 1985Google Scholar).

¹⁸ Strains of this narrow interpretation of rationality are apparent even among the best proponents of Realism. See, for example, Waltz (fn. 8), 70. Sophisticated versions of Realism—and certainly those that have incorporated game-theoretic notions—have employed an understanding of strategic rationality. For a clear-headed discussion of narrow rationality, see Bueno de Mesquita (fn. 1), 29–33.

¹⁹ See Elster, Jon, Ulysses and the Sirens: Studies in Rationality and Irrationality (New York: Cambridge University Press, 1979Google Scholar).

²⁰ See Keohane (fn. 6), and Snidal (fn. 6).

²¹ Allan, Pierre, Crisis Bargaining and the Arms Race (Cambridge, MA: Ballinger, 1983), 5–6Google Scholar, and Hahn and Hollis (fn. 17), 15.

²² Armatya Sen, “Rational Fools: A Critique of the Behavioral Foundations of Economic Theory,” in Hahn and Hollis (fn. 17), 92; emphasis in original.

²³ Olson, Mancur and Zeckhauser, Richard, “An Economic Theory of Alliances,” Review of Economics and Statistics 48 (No. 3, 1966), 226Google Scholar–79; Snidal (fn. 6), chaps. 5–6.

²⁴ Keohane (fn. 6), chap. 7, and Allan (fn. 21), chap. 4. There are limits to rational theory, however: it cannot always incorporate contending approaches except in a trivializing way. Even some issues that are internal to the theory—especially problems of preference aggregation in determining a “national interest”—are far from resolved.

²⁵ Russell Hardin demonstrates how to narrow the range of relevant games for deterrence problems in “Unilateral Versus Mutual Disarmament,” Philosophy and Public Affairs 12 (Summer 1983), 236Google Scholar–54.

²⁶ Examples of these deductions include those discussed above. The success of rational-actor approaches in other areas of political science is due to precisely this sort of approach (for example, the assumption that candidates maximize votes leads to conclusions about their behavior). For a discussion of this (and a critique of the trivializing use of revealed preferences by imputing utility to “citizen's duty” to explain voting), see Barry (fn. 9), chap. 2.

²⁷ Ashley, Richard, “The Poverty of Neo-Realism,” International Organization 38 (Spring 1984), 236Google Scholar–86.

²⁸ Ordinal payoffs correspond to “first,” “second,” and so forth. Interval measurement requires meaningful “units” (e.g., degrees of temperature or units of payoff) for the “distance” between outcomes (e.g., change in the temperature or in a state's payoff). Cardinal measurement requires a meaningful “zero” (e.g., absolute zero in temperature scales) and is largely irrelevant for game theory. Other levels of measurement may fall between these categories (e.g., partial orderings may give us interval-level comparisons between some outcomes, but no direct comparison between others).

²⁹ For example, see Jervis, Robert, “Cooperation under the Security Dilemma,” World Politics 30 (January 1978), 167–214CrossRefGoogle Scholar, at 174. Nevertheless, ordinal payoffs can carry our analysis very far, and econmists once (erroneously) even believed they were sufficient for virtually all purposes. An interesting account of how measurement is integrally related to the questions we are investigating is Cooter, Robert and Rappoport, Peter, “Were the Ordinalists Wrong About Welfare Economics?,” Journal of Economic Literature 22 (June 1984), 507Google Scholar–30. For a thoughtful but more technical discussion of these issues, see Shubik (fn. 4), chaps. 4 and 5.

³⁰ For examples of the impact of vulnerability, see Jervis (fn. 29), 171–73.

³¹ Complications of inconsistent time preferences are ignored here. See Elster (fn. 19), chap. 2.

³² Cooperation will make sense in anticipation of, and in response to, cooperation by the other party. An early formulation of the impact of iterated Prisoners' Dilemma games is Shubik, Martin, “Game Theory and the Paradox of the Prisoner's Dilemma,” Journal of Conflict Resolution 14 (June 1970), 181CrossRefGoogle Scholar–93. The result is worked out formally in Taylor, Michael, Anarchy and Cooperation (New York: Wiley, 1976Google Scholar) and extended via tournament techniques in Axelrod (fn. 16).

³³ The coordination aspects of the two-person Prisoners' Dilemma supergame can be seen in the matrix of supergame strategies in Taylor (fn. 32), 39. For a discussion of the differences between iterated Prisoners' Dilemma and iterated Coordination, see Snidal, Duncan, “Coordination Versus Prisoners' Dilemma: Implications for International Cooperation and Regimes,” American Political Science Review (forthcoming, December 1985CrossRefGoogle Scholar).

³⁴ A further dynamic adjustment process in terms of the evolution of strategies, whereby more successful strategies in one period are more likely (for reasons of survival, imitation, or learning) to occur in subsequent periods, is added by Axelrod (fn. 16). Wagner, R. Harrison, “Theory of Games and International Cooperation,” American Political Science Review 77 (June 1983), 330CrossRefGoogle Scholar–46, provides a useful critique of the attempt to embody dynamic assumptions in a single 2x 2 game through the use of sequential games that deny “players the opportunity to cheat (by assuming that they will cooperate conditionally) … “(pp. 332–33). However, he commits a similar error in assuming conditional behavior within the extensive game (p. 344). Iterated game analysis keeps the decision period much cleaner and less subject to artificial insertion of “conditional” cooperation that is based, in effect, on the ability either to predict the future or to recover from an adversary's behavior before payoffs accrue.

³⁵ The formal results for cooperation through time require either that the game continue forever or that there be uncertainty about its termination date. Luce and Raiffa (fn. 4) show that cooperation will not be rational if the termination date is known. Russell Hardin argues in Collective Action (Baltimore: Johns Hopkins University Press, 1982Google Scholar) that since this s implausible, futurei play will provide incentives to cooperate.

³⁶ This is a case where the same model is a model for two different theories (e.g., evolutionary and rational). Axelrod's discussion (fn. 16) recognizes the alternative interpretation of his model in terms of learning and adaptation, especially in his chapter on cooperation in trench warfare. Nowhere does he provide a vulgar evolutionary view of politics. Nevertheless, it is important to emphasize the very different interpretations of his model of iterated play under rational as opposed to evolutionary theory. See Elster (fn. 19), chap. 1, and Keohane (fn. 14).

³⁷ When should a situation be treated as an N-person game? In some cases the answer is obvious because it is technically impossible for the actions of many states to be insulated from one another. For example, in conservation of fish stocks in a “commons,” all states fishing that commons will be relevant. The “N-ness” of the problem will depend on the exact nature of the commons. Iffishspecies are nonmigratory and the commons is territorially divided, then only territorial states (perhaps as few as one) need be involved. But if species are migratory and/or the commons is not divided, then any state that fishes those species may be a relevant actor. This determination may be complicated if states act strategically and misrepresent the extent of their interest in the issue, or if the number of participants itself is not exogenous to the regime. For example, in the construction of economic regimes (e.g., trading blocs), deciding the scope of membership may hinge on the expected impact on the regime. An overview of the technical game theory material is available in Rapoport, Anatol, N-Person Game Theory (Ann Arbor: University of Michigan Press, 1970Google Scholar) and in Shubik (fn. 4).

³⁸ Hardin (fn. 35); Schelling, Thomas, Micromotives and Macrobehavior (Ne w York: Norton, 1978Google Scholar).

³⁹ Snidal (fn. 3).

⁴⁰ Axelrod's analysis has every actor playing the sam e strategy (e.g., Tit-for-Tat) against every other actor ; but that may mea n behaving differently vis-a-vis different states on any particular move (according to how they behaved on the previous turn). However, the linked nature of the 2 × 2 games is centra l to his analysis since the evolutionary survival of actors depends o n comparisons of how each fares (on average) against all the others. Axelrod (fn. 16), chap. 3.

⁴¹ The hardest issues to analyze will be nonsymmetric ones involving intermediate numbers of states with limited capacities to discriminate their actions. That category, of course, covers much of the ground in international politics.

⁴² This problem is not unique to N-person games; it also crops up in two-person games (e.g., in the differences between outcomes predicted between maximax versus minimax strategies). While solution concepts sometimes converge in N-person games, they often do not, and the complexity of the strategic structure makes it harder to compare or choose among them than in two-person games. See Rapoport (fn. 37), and Shubik (fn. 4).

⁴³ Two relevant works to build upon are Kaplan, Morton, System and Process in International Politics (New York: Wiley, 1957Google Scholar) and Bueno de Mesquita (fn. 1).