Game theory and the spiral model

World Politics

Volumn 49, Issue 3, 1997, Pages 371-400

  Kydd, Andrew ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000457224     PISSN: 00438871     EISSN: None     Source Type: Journal    
DOI: 10.1353/wp.1997.0012     Document Type: Article

References (39)

1. E. L. Woodward, Great Britain and the German Navy (Oxford: Clarendon Press, 1935), 168-69, 198.
2. Jervis, Perception and Misperception in International Politics (Princeton: Princeton University Press, 1976), chap. 3.
3. Glaser, "Political Consequences of Military Strategy: Expanding and Refining the Spiral and Deterrence Models," World Politics 44 (July 1992).
4. Jervis (fn. 2), 71.
5. Ibid., chap. 4.
6. 6 Oye, "Explaining Cooperation under Anarchy: Hypotheses and Strategies," World Politics 38 (October 1985).
7. Many historians note the aggressive motivations of the continental states before 1914, for instance. In the case of Germany, see Fritz Fischer, Germany's Aims in the First World War (New York W. W. Norton, 1967), esp. 11-20.
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 1-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 Robert Axelrod, The Evolution of Cooperation (New York: Basic Books, 1984); and George W. Downs and David M. Rocke, Tacit Bargaining, Arms Races, and Arms Control (Ann Arbor: University of Michigan Press, 1990).
15. 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 Robert Axelrod, The Evolution of Cooperation (New York: Basic Books, 1984); and George W. Downs and David M. Rocke, Tacit Bargaining, Arms Races, and Arms Control (Ann Arbor: University of Michigan Press, 1990).
16. 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.
17. 16 Discounting would not affect the results in any significant way and would further complicate an already complex model.
18. 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.
19. Drew Fudenberg and Jean Tirole, Game Theory (Cambridge: MIT Press, 1991), chap. 8.
20. 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.
21. 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.
22. 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.
23. 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).
24. Glaser (fn. 3).
25. 24 Woodward (fn. 1), 11; Gerhard Ritter, The Sword and the Scepter: The Problem of Militarism in Germany (Miami, Fla.: University of Miami Press, 1970), 145; Paul Kennedy, Strategy and Diplomacy, 1870-1945 (London: Fontanna Press, 1983), chap. 5; Fritz Fischer, War of Illusions (New York: W. W. Norton, 1975), chap. 7.
26. 24 Woodward (fn. 1), 11; Gerhard Ritter, The Sword and the Scepter: The Problem of Militarism in Germany (Miami, Fla.: University of Miami Press, 1970), 145; Paul Kennedy, Strategy and Diplomacy, 1870-1945 (London: Fontanna Press, 1983), chap. 5; Fritz Fischer, War of Illusions (New York: W. W. Norton, 1975), chap. 7.
27. 24 Woodward (fn. 1), 11; Gerhard Ritter, The Sword and the Scepter: The Problem of Militarism in Germany (Miami, Fla.: University of Miami Press, 1970), 145; Paul Kennedy, Strategy and Diplomacy, 1870-1945 (London: Fontanna Press, 1983), chap. 5; Fritz Fischer, War of Illusions (New York: W. W. Norton, 1975), chap. 7.
28. 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 Peter Padfield, The Great Naval Race (London: Hart-Davis, 1974), 223-24; and Robert K. Massie, Dreadnought: Britain, Germany and the Coming of the Great War (New York Random House, 1991), chap. 37.
29. 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 Peter Padfield, The Great Naval Race (London: Hart-Davis, 1974), 223-24; and Robert K. Massie, Dreadnought: Britain, Germany and the Coming of the Great War (New York Random House, 1991), chap. 37.
30. Jonathan Steinberg, "The Copenhagen Complex," Journal of Contemporary History 1, no. 3 (1966).
31. Fischer (fn. 24), 114.
32. Paul Kennedy, 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. 33.
33. 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.
34. For a summary of the debate, see John Lewis Gaddis, "The Emerging Post-Revisionist Synthesis on the Origins of the Cold War," Diplomatic History 7 (Summer 1983).
35. See the chapters in George W. Breslauer and Philip E. Tetlock, Learning in U.S. and Soviet Foreign Policy (Boulder, Colo.: Westview Press, 1991), 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."
36. 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.
37. For a discussion of costly signals in deterrence theory, see James Fearon, "Threats to Use Force: Costly Signals and Bargaining in International Crises" (Ph.D. diss., University of California, Berkeley, 1993), chap. 3. For a discussion of the role of signals in building trust, see Charles Glaser, "Realists as Optimists: Cooperation as Self Help," International Security 19, no. 3 (1995); Andrew Kydd, "Signaling and Structural Realism" (Manuscript, University of California, Riverside, 1996), and Deborah Larson, "Distrust, Missed Opportunities and Unilateral Initiative in Foreign Policy" (Manuscript, University of California, Los Angles, 1996). Note that these three authors discuss costly signals in a reassurance context but do not formally demonstrate their possibility, as is done here.
38. For a discussion of costly signals in deterrence theory, see James Fearon, "Threats to Use Force: Costly Signals and Bargaining in International Crises" (Ph.D. diss., University of California, Berkeley, 1993), chap. 3. For a discussion of the role of signals in building trust, see Charles Glaser, "Realists as Optimists: Cooperation as Self Help," International Security 19, no. 3 (1995); Andrew Kydd, "Signaling and Structural Realism" (Manuscript, University of California, Riverside, 1996), and Deborah Larson, "Distrust, Missed Opportunities and Unilateral Initiative in Foreign Policy" (Manuscript, University of California, Los Angles, 1996). Note that these three authors discuss costly signals in a reassurance context but do not formally demonstrate their possibility, as is done here.
39. For a discussion of costly signals in deterrence theory, see James Fearon, "Threats to Use Force: Costly Signals and Bargaining in International Crises" (Ph.D. diss., University of California, Berkeley, 1993), chap. 3. For a discussion of the role of signals in building trust, see Charles Glaser, "Realists as Optimists: Cooperation as Self Help," International Security 19, no. 3 (1995); Andrew Kydd, "Signaling and Structural Realism" (Manuscript, University of California, Riverside, 1996), and Deborah Larson, "Distrust, Missed Opportunities and Unilateral Initiative in Foreign Policy" (Manuscript, University of California, Los Angles, 1996). Note that these three authors discuss costly signals in a reassurance context but do not formally demonstrate their possibility, as is done here.