Two states, A and B, are involved in a potential military conflict. State Bs strength is common knowledge, but state As is not. A is Strong with probability p = .5 or Weak with probability 1 p = .5. The interaction unfolds as follows: first Nature draws As strength and reveals it to A. Then A decides whether to challenge B or not. B observes the challenge (if any), and then decides whether to fight or back down.

If A does not challenge, the game ends, and each state earns a payoff of 0. If A challenges and B fights, there is a war. The winner of the war earns a payoff of 1 c, and the loser earns 1 c, where 1/2 < c < 1. If A is Strong, it wins the war with probability = .75 and B wins with probability 1 = .25.

If A is Weak, it wins with probability 1 = .25, and B wins with probability .75.

If A challenges and B backs down, A earns 1/2 and B earns 1/2.

(a) What are the information sets of state A? What are As pure strategies? Answer the same questions for state B.

(b) Suppose As strategy is Challenge if Strong, Do not challenge if Weak. What is Bs belief of As strength after being challenged? Given this belief, what is Bs expected utility of Fighting? Of backing down? What is Bs best response?

(c) Is there a perfect Bayesian equilibrium in which A uses the strat- egy specified in the previous part? Hint. Consider whether the Weak type of State A actually prefers not to challenge, given the best response of B to challenge specified above.

(d) Suppose As strategy is Do not challenge if Strong or Weak. Suppose B believes that if A does challenge (which is off the path of play for this strategy), A is strong with probability .5. What is Bs best response to a challenge?

(e) Is there a PBE in which A uses the strategy and B holds the off-path beliefs specified in the previous part?

(f) Suppose As strategy is Challenge if Strong or Weak. Is this a PBE?

(g) Can there be a PBE in which the Weak type of A is more likely to challenge than the Strong type of A?