In the chapter, we argue that the payoff to the United States is 210 when (either type) Soviets defy the U.S. threat; these payoffs are illustrated in Figure 14.3. Suppose now that this payoff is in fact 212 rather than 210.
(a) Incorporate this change in payoff into a game tree similar to the one in Figure 14.4.
(b) Using the payoffs from your game tree in part (a), find the effectiveness condition for this version of the U.S.–USSR brinkmanship game.
(c) Using the payoffs from part (a), find the acceptability condition for this game.
(d) Draw a diagram similar to that in Figure 14.5, illustrating the effectiveness and acceptability conditions found in parts (b) and (c).
(e) For what values of p, the probability that the Soviets are hard-line, is the pure threat (q - 1) acceptable? For what values of p is the pure threat unacceptable but brinkmanship still possible?
(f) If Kennedy was correct in believing that p lay between 13 and 12, does your analysis of this version of the game suggest that an effective and acceptable probabilistic threat existed? Use this example to explain how a game theorist’s assumptions about player payoffs can have a major effect on the predictions that arise from the theoretical model.