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Rational Deterrence in an Imperfect World

Published online by Cambridge University Press:  13 June 2011

Barry Nalebuff
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Extract

This paper considers the role of reputation and signaling in establishing deterrence. The cost-benefit calculations of rational deterrence are extended to allow for incomplete or imperfect information. The author uses requirements of a sequential equilibrium (and its refinements) to impose consistency restrictions on how strategic players signal a reputation for strength. This provides a way to interpret potentially misleading reputations and offers a resolution to the reputation paradox of Jervis.

Type
Research Article
Information
World Politics , Volume 43 , Issue 3 , April 1991 , pp. 313 - 335
Copyright
Copyright © Trustees of Princeton University 1991

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References

1 In truth, the calculus of deterrence might better have been called algebra.

2 Quoted in Ikle, F., How Nations Negotiate (New York: Harper and Row, 1964), 67.Google Scholar

3 Jervis, , The Logic of Images in International Relations (Princeton: Princeton University Press, 1970).Google Scholar

4 Sadly, we must report that the Mayflower furniture store recently had its first sale, a going-out-of-business sale.

5 But after an interpreter was found, the crew did not appear to be in immediate danger. The Cambodian captors were apparently persuaded that the Mayagüez was not a spy boat.

6 New YorK Times, Week in Review, May 18, 1975.

7 Jervis, , “The Symbolic Nature of Nuclear Politics” (The Edmund James Lecture, Department of Political Science, University of Illinois at Urbana-Champaign, 1985).Google Scholar

8 Jervis, , “Deterrence and Perception,” in Miller, Steven, ed., Strategy and Nuclear Deterrence (Princeton: Princeton University Press, 1984), 66–67.Google Scholar

9 Although we have found a consistent solution, is it of any use? Here Jervis has some doubts as to the ability of game theorists to say much more about the “black box” of mixed strategies:

Economists have not been able to model the behavior of oligopolists nearly as determin-istically as they have that of the wheat farmer facing a market he cannot influence. … [I]n many situations game theory prescribes a mixed or randomized solution. Of course, game theory yields great insights into how actors try to out-think and out-bluff each other, but the competitive and variable sum nature of the situation means that scholars cannot produce deductions on the model: “In situation X the actor always should (or will) do Y.” We can show that the actors' calculations are consistent with deterrence, but this requires looking into the “black-box” of decision-making. (Jervis, , “Rational Deterrence: Theory and Evidence,” World Politics 41 [January 1989], 183–207)CrossRefGoogle Scholar

In fact, it is possible to characterize the precise equilibrium proportion of mixing. Or to put it more poetically, there is a method to the madness of a random strategy; see Dixit, Avinash and Nalebuff, Barry, Thinking Strategically: A Competitive Edge in Business, Politics, and Everyday Life (New York: W. W. Norton, 1991).Google Scholar To verify whether or not actors follow their prescribed mixed-strategy rules does not require a large number of independent observations from a repeated game. Instead the theory is tested by evaluating how well it predicts cross-sectionally, looking across a variety of different conflict situations.

10 But there is nothing that prevents x and c from being negative, in which case the observed value would represent a net cost and the unobserved parameter would represent a net benefit.

11 Here, the assumption that intervention costs are uniformly distributed is made solely for analytic convenience. The prior belief should be based on the history up to this point.

12 Note that we do not require that c i ≤ c n It is possible that intervening hurts one's reputation. Such an example is presented below.

13 Kreps, David and Wilson, Robert, “Sequential Equilibria,” Econometrica 52 (July 1982), 863–95.CrossRefGoogle Scholar

14 How we decide the action for the c* type is irrelevant.

15 If c* = o then no one should intervene, whereas if c* = I, then everyone should intervene. Both endpoint cases predict that one action is never taken and hence falls under the weaker consistency condition 2(b).

16 This simple relationship is an artifact of the uniform distribution of intervention costs.

17 This type of well-behaved sequential equilibrium exists provided o < c* = x + a/2 < I. Otherwise, the effect of reputation may be so large or the value of intervening so great that we expect everyone to act, c* & I. Alternatively, the value of reputation may be so small (or negative) or the price of intervention so costly that all types choose not to intervene and c* ≤ o.

18 Jervis (fn. 7).

19 Otherwise, there are no zero-probability events. The earlier analysis applies and there is a unique solution.

20 Note that if a = 1/2 and x > 1/4, then the model predicts that a country with c = 0 would strictly prefer to intervene. Hence the reversed equilibrium is ruled out.

21 Pearce, David, “Rationalizable Strategic Behavior and the Problem of Perfection,” Econometrica 52 (July 1984), 1029–50CrossRefGoogle Scholar; Bernheim, B. Douglas, “Rationalizable Strategic Behavior,” Econometrica 52 (July 1984), 1007–28.CrossRefGoogle Scholar

22 Banks, Jeffrey and Sobel, Joel, “Equilibrium Selection in Signaling Games,” Econometrica 55 (May 1987), 647–62.CrossRefGoogle Scholar

23 Grossman, Sanford and Perry, Motty N., “Perfect Sequential Equilibria,” Journal of Economic Theory 39 (June 1986), 97–119.CrossRefGoogle Scholar

24 Note that this domination is over other strategies, not over an opponent.

25 Banks and Sobel (fn. 22).

26 Otherwise, intervention must be expected.

27 Grossman and Perry (fn. 23).

28 Farrell, Joseph, “Credible Neologisms in Games of Communication” (Mimeo, University of California, Berkeley, 1984).Google Scholar

29 Perfect sequential equilibrium is not a panacea. It is much harder to show that something satisfies PSE than to show that it does not. In fact, one of its failings is that it is possible that no equilibrium will satisfy this test. In our example this problem does not arise as the separating equilibrium satisfies PSE.

30 As this intuition suggests, it is not the case that PSE eliminates all no-intervention equilibrium. If x is negative and a is small, the no-intervention equilibrium satisfies PSE.