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Game Theory and the Spiral Model

Published online by Cambridge University Press:  13 June 2011

Andrew Kydd
Andrew Kydd
Affiliation:
University of California
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Abstract

When one state engages in a military buildup, other states sometimes take this as a sign that it is more aggressive or expansionist than they previously thought. Some argue that such increases in mutual suspicion can drive arms races and even lead to war. Psychological bias is often invoked to explain this pattern of growing suspicions leading to hostility. This article presents an incomplete information model of an arms race and investigates when escalations should rationally generate increased fears and when, in order to reduce such fears, security seekers can refrain from building. It shows that escalations rationally provoke fear even in the absence of bias and that weak states and states facing high costs of arms racing and war will be especially likely to refrain from building as a way of signaling benign intentions.

Type
Research Article
Information
World Politics , Volume 49 , Issue 3 , April 1997 , pp. 371 - 400
Copyright
Copyright © Trustees of Princeton University 1997

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References

1 Woodward, E. L., Great Britain and the German Navy (Oxford: Clarendon Press, 1935), 168–69, 198Google Scholar.

2 Jervis, , Perception and Misperception in International Politics (Princeton: Princeton University Press, 1976), chap. 3Google Scholar.

3 Glaser, , “Political Consequences of Military Strategy: Expanding and Refining the Spiral and Deterrence Models,” World Politics 44 (July 1992)CrossRefGoogle Scholar.

4 Jervis(fn.2),71.

5 Ibid., chap. 4.

6 Oye, , “Explaining Cooperation under Anarchy: Hypotheses and Strategies,” World Politics 38 (October 1985)CrossRefGoogle Scholar.

7 Many historians note the aggressive motivations of the continental states before 1914, for instance. In the case of Germany, see Fischer, Fritz, Germany's Aims in the First World War (New York: W. W. Norton, 1967), esp. 11–20Google Scholar.

8 The information structure thus contains two levels of uncertainty. There is uncertainty over the utility of the other side, as well as uncertainty over their beliefs about your own utility. This somewhat complex information structure is necessitated by the logic of the spiral model, as will become clear below. The types in this model are comparable to those in Glaser (fn. 3).

9 I assume without loss of generality that l-e>e. If 1—e were less than e, the signal would simply imply the opposite of its value, so that g would indicate a security seeker and s a greedy type. It is further assumed that the draws that produce these messages, as well as those that produce the player's greed, are uncorrelated. The interpretation of the messages as private intelligence gathering is in accord with a lack of correlation. As for the first draws over greed, it is difficult to see why greedy and security- seeking states should rationally infer different probabilities that an opponent is greedy from the same public information, although this kind of divergence may occur for psychological reasons. A final point to emphasize is that these messages are not sent by the other player in any conscious way but simply represent information that the receiving state gathers on its own and that influences its beliefs about the other side's type. Thus they are not “signals” in the sense of consciously manipulated signs. Rather, states signal with their armament choices, as described below.

10 Of course, these beliefs may arise in other ways as well. The intelligence-gathering story is offered as one intuitively plausible way in which such divergent beliefs could arise.

11 I will sometimes refer to beliefs after Nature's move but before the first round of play as “priors” because they precede any move by the players, but it should be remembered that they are posterior to Nature's move.

12 Note that the subscript on the p's is the opposite of that on the γ's. I subscript the p's to reflect the player holding the belief, whereas the γ's are subscripted to reflect the object of the prior probability.

13 This may seem like redundant notation, but it will become convenient later on when these beliefs about the other side's fear are updated and the formulas become much less simple.

14 Further rounds of play would complicate the model without adding greatly to its capacity to address spiral arguments. Note in particular that unlike many models of international relations, such as the repeated Prisoner's Dilemma and related models of arms races, cooperation here is not a function of the shadow of the future; hence the restriction to a finite number of rounds does not fundamentally change the nature of the game, as it does in the repeated PD. See Axelrod, Robert, The Evolution of Cooperation (New York: Basic Books, 1984)Google ScholarPubMed; and Downs, George W. and Rocke, David M., Tacit Bargaining, Arms Races, and Arms Control (Ann Arbor: University of Michigan Press, 1990)CrossRefGoogle Scholar.

15 No building choice is included in the second round because building is only worthwhile if there is a subsequent war, but in this final round of the game there are no subsequent possibilities for attack. Hence no player would build, regardless of type; including the choice would therefore complicate the description of the game to no purpose.

16 Discounting would not affect the results in any significant way and would further complicate an already complex model.

17 A first-strike advantage is necessary for there to be a danger of war between security seekers, a key subject of concern in the spiral model.

18 Fudenberg, Drew and Tirole, Jean, Game Theory (Cambridge: MIT Press, 1991), chap. 8Google Scholar.

19 Differences between family members are primarily off the equilibrium path and in the second round, after learning has taken place. A more complete description of the equilibria in the model is available from the author upon request.

20 Thus, within this upward spiral equilibrium, a downward spiral is also possible, because the ST type signals its type by not building. The next equilibrium will focus on when the SF type can also refrain from building.

21 Note again, the subscript refers to the holder of the belief, not the object of the belief, so a GF type's level of fear is pif and its belief that the other side is fearful is qig.

22 This dependence of spiraling on the distinction between trusting and fearful states is what necessitates the incorporation of the second level of uncertainty in the model (fn. 8).

23 Glaser (fn. 3).

24 Woodward (fn. 1), 11; Ritter, Gerhard, The Sword and the Scepter: The Problem of Militarism in Germany (Miami, Fla.: University of Miami Press, 1970), 145Google Scholar; Kennedy, Paul, Strategy and Diplomacy, 1870–1945 (London: Fontanna Press, 1983), chap. 5Google Scholar; Fischer, Fritz, War of Illusions (New York: W. W. Norton, 1975), chap. 7Google Scholar.

25 In 1907 the kaiser granted an interview with a reporter from the Daily Telegraph, in which he claimed to have given the British the winning strategy in the Boer War and also that the German navy was being built against the Japanese. These preposterous claims only worsened Anglo-German relations and led to a serious political crisis in Germany. See Padfield, Peter, The Great Naval Race (London: Hart-Davis, 1974), 223–24Google Scholar; and Massie, Robert K., Dreadnought: Britain, Germany and the Coming of the Great War (New York: Random House, 1991), chap. 37Google Scholar.

26 Steinberg, Jonathan, “The Copenhagen Complex,” Journal of Contemporary History 1, no. 3 (1966)CrossRefGoogle Scholar.

27 Fischer (fn. 24), 114.

28 Kennedy, Paul, The Rise of the Anglo-German Antagonism, 1860–1914 (London: Ashfield Press, 1980), 443–44. For the showdown within the Liberal Party over naval expenditures during the naval scare of 1909, see also Massie (fn. 25), chap. 33Google Scholar.

29 In a variant of the spiral equilibrium in which GF types attack in the first round, the ratio between the posteriors is not preserved. However, this equilibrium does still witness the polarization of beliefs identified below.

30 For a summary of the debate, see Gaddis, John Lewis, “The Emerging Post-Revisionist Synthesis on the Origins of the Cold War,” Diplomatic History 7 (Summer 1983)CrossRefGoogle Scholar.

31 See the chapters in Breslauer, George W. and Tetlock, Philip E., Learning in U.S. and Soviet Foreign Policy (Boulder, Colo.: Westview Press, 1991)Google Scholar, for arguments along these lines, especially Philip E. Tetlock, “Learning in U.S. and Soviet Foreign Policy: In Search of an Elusive Concept,” and Ernst B. Haas, “Collective Learning: Some Theoretical Speculations.”

32 Of course, the less military importance the concession has, the greater the danger that the aggressive state may try to mimic it, depriving it of its signaling value. If the concession had no military value, then it would simply be cheap talk and the aggressive state would not hesitate to make it too. Thus the signal has to have some middling value of risk in order to be persuasive but not too costly.

33 For a discussion of costly signals in deterrence theory, see Fearon, James, “Threats to Use Force: Costly Signals and Bargaining in International Crises” (Ph.D. diss., University of California, Berkeley, 1993), chap. 3Google Scholar. For a discussion of the role of signals in building trust, see Glaser, Charles, “Realists as Optimists: Cooperation as Self Help,” International Security 19, no. 3 (1995)Google Scholar; Kydd, Andrew, “Signaling and Structural Realism” (Manuscript, University of California, Riverside, 1996)Google Scholar, and Larson, Deborah, “Distrust, Missed Opportunities and Unilateral Initiative in Foreign Policy” (Manuscript, University of California, Los Angles, 1996)Google Scholar. Note that these three authors discuss costly signals in a reassurance context but do not formally demonstrate their possibility, as is done here.