Frank C. Zagare

Limited-move equilibria In 2 × 2 games

Summary and ConclusionThis article introduces the concept of a limited-move equilibrium, an outcome that is stable when players are able to make more than the minimum (i.e., one) but less than the maximum (i.e., four) number of logically possible moves and countermoves in a 2 × 2 game. It defines two types of limitedmove equilibria and their associated decision rules, a Type II equilibrium where the total number of strategy switches permitted in a game is two (Rule II), and a Type III equilibrium where three total moves and countermoves (Rule III) from an initial outcome or status quo point are possible. When games that contain a limited-move equilibrium were compared to games that contain a Nash (Type I) or nonmyopic (Type IV) equilibrium, the 78 distinct 2 × 2 ordinal games identified by Rapoport and Guyer were shown to fall into three mutually exclusive categories:1.Rule-neutral games (29 games — 37%). In this class of games, a single outcome in each game is rendered stable regardless of the number of moves and countermoves assumed to be available to the players.2.Rule-dependent games (34 games — 44%). This is the largest of the three categories. Within this class are the games in which all decision rules associated with a stable outcome are associated with the same outcome. The congruence of decision rules and stable outcomes range from the case in which all decision rules except one (i.e., Rule IV) induce the same equilibrium outcome to the case in which a game has an equilibrium only when one of the four decision rules is assumed.3.Rule-variable games (15 games — 19%). In this category, different decision rules imply different stable outcomes. In rule-variable games, then, the impact of the operative decision rule is potentially the most significant, especially since players in these games are not indifferent towards which of the various outcomes emerge in a stable state. Consequently, an important dimension of conflict in these games might very well concern adjudication of the rules governing the ability of the players to make moves and countermoves. Perhaps the most interesting observation that can be gleaned from this analysis is that at least one outcome in each of the 78 distinct 2 × 2 games is stable under one of the four postulated decision rules. Consequently, there is an equilibrium of some type in every strict ordinal 2 × 2 game. In nine of these games — six rule-dependent games, and three rule-variable games — only limited-move equilibria exist. Moreover, there are four games without a Type IV equilibrium that have a limited-move equilibrium different from a unique Type I equilibrium.In general, limitations on the ability of players to make moves and countermoves tend to foster stability rather than destroy it in 2 × 2 games. Of the 78 games of this type, 69(88%) contain one or more Type I and Type II equilibria, 57(73%) have Type III equilibria, while only 37(47%) have outcomes that exhibit Type IV or nonmyopic stability. Environmental limitations on the number of moves and countermoves players can make, then, may help explain much of the stability observed in ongoing, real life games that do not contain Type IV equilibria.In this context, it is worth pointing out that one of the underlying premises of this work has been that the concept of a limited-move equilibrium constitutes a useful and necessary adjunct to the concept of a nonmyopic equilibrium and of the dynamic conception of a game it implies, especially for empirical studies where ideosyncratic features of real life games might enter into the calculus of a player considering the long-term consequences of a strategy change. For while it is probably true that the assumption, defining a Nash equilibrium, that players are able to make only a single unilateral deviation from an outcome, is overly restrictive and not representative of the fluid nature of many actual conflict-of-interest situations, it is also probably true that the assumption, defining a nonmyopic equilibrium, that players can make an unlimited number of moves and countermoves, is frequently not satisfied in many empirical settings. Consequently, the two intermediate equilibrium concepts identified in this essay should provide a firmer theoretical base for, and enhance the explanatory power of, applications of simple game-theoretic models to a world characterized by a wide variety of actual decision rules.

Rationality and Deterrence

This essay explores the relationship between deterrence theory and two distinct notions of rationality associated with it. The author argues that the concept of procedural irrationality, which is the basis of many critiques of contemporary deterrence theory, is not necessarily inconsistent with the instrumental definition of rationality implicit in many models of deterrence. He also argues that some recent rational choice models of deterrence are, nonetheless, deficient in their application of the rationality postulate. Finally, he provides an outline of a possible resolution of the paradox of deterrence.

Credibility, Uncertainty, and Deterrence

In this paper the connection between deterrence stability and threat credibility is examined conceptually and theoretically. We formulate as a model of bilateral deterrence a game of incomplete information in which each player is uncertain about its opponents preferences should it unilaterally alter the status quo. Uncertainty about the preferences of ones opponent leads to uncertainty about the opponents willingness to retaliate. By identifying the credibility of each players retaliatory threat with the probability that a player prefers retaliation to capitulation, we maintain consistency with both the traditional strategic literature, where credibility usually means believability, and with game theory, where credibility is usually synonymous with sequential rationality (i.e., subgame perfect equilibrium). We analyze formally the strategic implications of this conception of credibility and thus explore the critical role played by uncertainty in deterrence. By explicitly modeling uncertainty, we are able to understand the role of threats in contributing to, or detracting from, the robustness of a deterrence relationship.

RECONCILING RATIONALITY WITH DETERRENCE A RE-EXAMINATION OF THE LOGICAL FOUNDATIONS OF DETERRENCE THEORY

This article argues that classical (or rational) deterrence theory is logically inconsistent, empirically inaccurate and prescriptively deficient. In its stead it offers an alternative theoretical framework – perfect deterrence theory – that makes consistent use of the rationality postulate and is in accord with the empirical literature of deterrence. Perfect deterrence theory’s axiomatic base, its empirical expectations and its most significant policy prescriptions are highlighted and contrasted with those of classical deterrence theory. The theory’s implications for current policy debates about a national missile defense system, arms control, US policy toward China and Russia, and inter-state negotiations in general, are discussed.

Deception in simple voting games

Abstract The calculations of sophisticated voters who successively eliminate undesirable strategies are analyzed in three-person voting games in which one voter with complete information can, as a deceiver, induce the other two voters with incomplete information to vote in such a way as to ensure a better outcome than the deceiver could ensure in a game of complete information. Deception which is “tacit,” wherein a deceiver votes consistently with his announced preference scale, is distinguished from deception which is “revealed,” wherein a deceivers action deviates from his announced preference scale. Among the conclusions drawn from the study is that revealed deception is generally a more potent tool than tacit deception in securing a more-preferred outcome, and deception opportunities are greater the more disagreement there is among the nondeceivers.

Classical deterrence theory: A critical assessment

This essay summarizes and critically evaluates the conceptual underpinnings of Classical (or Rational) Deterrence Theory. The paradigms origins are traced to Realist political thought and balance ...

The long‐term stability of deterrence

In this article, we construct a model of deterrence that specifically integrates both systemic and decision‐making variables. After contrasting its underlying structure with more standard views of the deterrence relationship, we demonstrate the logical consistency of the power transition model with the expected utility framework. The model we develop combines and extends the insights of each of these two approaches, thereby permitting us to develop a theory of the necessary and sufficient conditions of major power war and conflict initiation. In other words, for the first time, we are able to specify, precisely, the theoretical consequences of variations in power dynamics, evaluations of the status quo, salience, and attitudes toward risk. As theories of the necessary conditions for international conflict, neither the power transition theory nor the expected utility model provide this information. Thus, by combining and then extending these two frameworks, we merely refine propositions implicit in each of...

All Mortis, No Rigor

Like Caesar’s view of Gaul, Stephen Walt’s evaluation of the recent rational choice literature in strategic studies is divided into three parts.1 But all the king’s horses and all the king’s men could not put his article back together again: the analysis of the second section does not follow from the arst, and the conclusions of the third cannot be drawn from the second. In the end, Walt’s discussion provides a clear illustration of why formal models are so valuable: they provide the strongest possible protection against improper argumentation. Walt’s arst section is a reasoned and balanced discussion of the underlying premises of rational choice theory and the rationale for formal modeling. In fact, Walt’s summary of the foundations of this methodological technique is refreshing. Unlike many other efforts to evaluate the contributions of game theory to international affairs, it is no caricature.2 Also uplifting is the absence of vitriol that turned one recent exchange between scholars into an intellectual food aght.3 Walt begins by noting the usefulness of mathematical models in ensuring logical consistency, one of three criteria he lists as important for evaluating theories and bodies of literature. Insightfully, he recognizes that the formal literature is not monolithic, that there are important differences among those who use game theory to analyze international politics. As well, Walt’s discussion demonstrates a sophisticated understanding of the rationality postulate. Although he does not discuss the issue explicitly, he does not fall into the

Nonmyopic Equilibria and the Middle East Crisis of 1967

Although game-theoretic models are frequently used as a normative/analytical device by international relations theorists, seldom has this powerful tool been used in analyzing the actual interactions of international politics or for retrodicting-historical events.’ One reason why theorists have eschewed game-theoretic models in their empirical work is that most real-life games are ongoing, while game-theoretic models are preceived to be essentially static (Snyder and Diesing, 1977, p. 416), a fact that even the co-founders of game theory thought necessary to point out (von Neumann and Morgenstern, 1953, pp. 44-45). Partly responsible for the static nature of game-theoretic models, especially of nonzero-sum theory, has been the dominance in gametheoretic analyses of an equilibrium notion developed by Nash (195 1). According to Nash, an outcome is an equilibrium if no single player in a game benefits by unilaterally switching to another strategy. Implicit in Nash’s equilibrium concept is the notion that players, in deciding whether or not to change strategies, make only an estimate of the immediate advantages and disadvantages of a unilateral move, and do not take into account the possible response of another player. In this sense, Nash’s equilibrium concept is static. It is also myopic since it assumes that players do not (or can not) calculate the long-term consequences of moves and countermoves. Clearly, the underlying assumptions of Nash’s equilibrium concept distort the dynamic nature of international politics, wherein strategy choices are usually made sequentially, and where probes and strategems are frequently used-especially in crisis situations (Young, 1968, p. 218)-that permit responses to countermoves and counterprobes. This is not to suggest that there have been no attempts to develop dynamic game models.* Unfortunately, though, these attempts have

Assessing Competing Defense Postures: The Strategic Implications of “Flexible Response”

A two-stage Asymmetric Escalation Game is developed to explore the connection between stage credibility and deterrence stability. There are two players in the model: Challenger and Defender. Challenger may initiate or not. If Challenger initiates, Defender may do nothing, respond in kind, or escalate; Challenger may then escalate or counterescalate, and so on. Each player is uncertain about the others intentions at the final stage of the game. Escalation represents a choice that both players believe is qualitatively different from other available responses. Thus the model applies to any situation in which Defender may respond by crossing a threshold, thereby inducing a (psychologically) distinct level of conflict. The Perfect Bayesian Equilibria are identified and interpreted, and inferences are drawn about the viability of limited war options and various competing flexible response deployment policies. In general, the model reveals that substrategic deployments add little to overall deterrence stability. Under certain relatively rare conditions, a policy called no-first-use in the super power context offers Defender advantages that might conceivably warrant the deployment stance associated with it. But a war fighting deployment never benefits Defender. Within the confines of the model, therefore, limited or substrategic wars are possible but unlikely.