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The Theory of Games and the Balance of Power

Published online by Cambridge University Press:  13 June 2011

R. Harrison Wagner
R. Harrison Wagner
Affiliation:
University of Texas
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Abstract

The theory of games is used to investigate several controversial issues in the literature on the balance of power. A simple model of an international system is presented as an n-person noncooperative game in extensive form, and the stability of both constant-sum and nonconstant-sum systems is examined. It is shown not only that constant-sum systems with any number of actors from two through five can be stable, but also that stability is actually promoted by conflict of interest. Contrary to much of the literature, however, there is a well-defined sense in which the most stable system is one with three actors. In each type of system, there is at least one distribution of power that leads not only to system stability but also to peace. Some of these peaceful distributions are more stable than others, and these more stable distributions are shown to be characterized by inequality rather than by equality of power. It is possible to distinguish between a bipolar and a multipolar type of stable distribution, the properties of each of which resemble, to some degree, assertions made about them in the literature. Finally, contrary to much of the recent literature on international cooperation, an increase in the ability of states to make binding agreements may actually diminish the stability of international systems.

Type
Research Article
Information
World Politics , Volume 38 , Issue 4 , July 1986 , pp. 546 - 576
Copyright
Copyright © Trustees of Princeton University 1986

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References

1 Waltz, , Theory of International Politics (Reading, MA: Addison-Wesley, 1979), 117Google Scholar.

2 Donnadieu, Leonce, Essai sur la theorie de I'equilibre (Paris: A. Rousseau, 1900)Google Scholar.

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6 Evidence for this assertion is presented in Wagner, R. Harrison, “The Theory of Games and the Balance of Power,” Annual Meetings of the American Political Science Association, Washington, DC, September 1984 (subsequently revised)Google Scholar.

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8 von Gentz, Friedrich, Fragments Upon the Balance of Power in Europe (London: M. Peltier, 1806), 56–70Google Scholar; Donnadieu (fn. 2), 4.

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10 This distinction can also refer to the alliance structure that characterizes an international system.

11 Burns, Arthur Lee, “From Balance to Deterrence: A Theoretical Analysis,” World Politics 9 (July 1957) 494–529CrossRefGoogle Scholar.

12 In game theory, “perfect information” means that each player, when choosing, knows what choices have been made by other players; thus the players do not choose independently. This condition should not be confused with the standard assumption of “complete information,” which means that all features of the game tree are common knowledge among the players.

13 For a discussion of this and related issues, see Kreps, David and Wilson, Robert, “Sequential Equilibria,” Econometrica 50 (July 1982), 863–94CrossRefGoogle Scholar. I would like to acknowledge Robert Powell's help in defining the endpoints of this game and the players' payoff functions.

14 It is important not to confuse the various game-theoretic concepts associated with the word “equilibrium” with the use of that term by balance-of-power theorists. The question we are examining is, in fact, whether any equilibria in a suitably defined noncooperative game are consistent with instability (or a lack of “equilibrium”) in the international system that the game is designed to represent.

15 Burns (fn. 11). See also Kaplan, Morton, Burns, Arthur, and Quandt, Richard, “Theoretical Analysis of the Balance of Power,” Behavioral Science 5 (July 1960), 240–52CrossRefGoogle Scholar.

16 Both Robert Powell and Emerson Niou have independently pointed out to me that, if the victim is allowed to make a preemptive transfer of just enough resources to the unopposed attacker to give it R/2 units, it will lose fewer resources, and will thus prefer to make this transfer. Moreover, if the opposed attacker would like to acquire resources peacefully, then it, too, will prefer this outcome, and this, rather than the outcome described in the text, is the only equilibrium. (For a development of this idea, see Emerson M. S. Niou and Peter C. Ordeshook, “A Theory of the Balance of Power,” Journal of Conflict Resolution, forthcoming.) But what is necessary for the existence of this alternative equilibrium is not simply voluntary transfers, but transfers to which the opposed attacker has no opportunity to respond. Otherwise the reasoning in the text applies, and both the victim and the opposed attacker will prefer to join against the unopposed attacker if it tries to take advantage of the victim's offer. While the rules of the game do not allow voluntary transfers, therefore, the reasoning in the text is consistent with any means of transferring resources from the victim to the unopposed ally that allows the opposed ally to make a counter-offer before the transfer is completed.

17 The quantity of resources with which the victim is left is thus determined by the relation between the rate at which unopposed attackers can absorb resources from their victims and the time required for states to retarget their resources. That is the (somewhat artificial) implication of the particular assumptions made earlier. The specific form of the conclusion is less important than the general point: that the inability of states to prevent their allies from taking advantage of the division of their victims is an important factor in preserving the independence of the victims. The conflict between the U.S. and the U.S.S.R. that arose out of the question of the division of Germany is perhaps a relevant example. See Wagner, R. Harrison. “The Decision to Divide Germany and the Origins of the Cold War,” International Studies Quarterly 24 (June 1980), 155–90CrossRefGoogle Scholar.

18 For the vertical bar in these distributions, read “such that.”

19 Gilpin, Robert, War and Change in World Politics (Cambridge and New York: Cambridge University Press, 1981)CrossRefGoogle Scholar.

20 Waltz (fn.1), 164.

21 See, for example, Jervis, Robert, “Cooperation under the Security Dilemma,” World Politics 30 (January 1978), 167–214CrossRefGoogle Scholar.

22 The problem with Riker's argument (fn. 3), therefore, is neither the zero-sum as sumption, nor an assumption that balance-of-power games are not repeated, but the assumption implicit in the theory of cooperative games that coalition agreements are enforceable.

23 In order to avoid misunderstandings, let me emphasize what should be obvious: that the game analyzed in this article is quite different from either a single-play or a repeated Prisoners' Dilemma game, and therefore the literature on international cooperation based on the analysis of Prisoners' Dilemma games is not relevant to the issues examined here. See also Wagner, R. Harrison, “The Theory of Games and the Problem of International Cooperation,” American Political Science Review 77 (June 1983), 330–46.CrossRefGoogle Scholar.