This appendix contains the detailed analysis of the Rudimentary Asymmetric Deterrence Game with incomplete information introduced in section 3.4 and specified in detail in section A1.1. (See also figure 3.7 and figure A1.1.)

Recall that the eight parameter values satisfy 0 < pB < 1, aDD < aSQ < aDC, and bDD− < bDC < bDD+ < bSQ. Moreover, player B is Hard, i.e. B's utility for outcome DD is bDD+, with probability pB, and B is Soft, i.e. B's utility for outcome DD is bDD–, with probability 1 – pB. Observe that there is one-sided incomplete information in this game – there are two types of B, but only one type of A.

Our primary objective is to identify the Bayesian equilibria of the Rudimentary Asymmetric Deterrence Game. For details on Bayesian equilibria and perfect Bayesian equilibria, see Fudenberg and Tirole (1991). We take A's only strategic variable to be x, and B's strategic variables to be yH and yS, as follows:

This produces the game shown in figure A3.1. Bayesian equilibria will be denoted [x; yH, yS].

First, yH = 1 and yS = 0 at any Bayesian equilibrium. This is easy to verify (see figure A3.1): at node 2, B must choose D (y = 1), yielding outcome DD, or C (y = 0), yielding outcome DC.