- Two states decide in sequence whether to initiate a military conflict. Each has private information about its own strength. The strength of state i is High with probability p, and Low with probability 1−p. The strength of each state is drawn by nature and is independent of the strength of the other state. First nature draws types and revealssi to state i. Then state 1 decides to initiate a conflict or not. If state 1 does not initiate, the game ends and there is no war. If state 1 initiates, state 2 can either fight back or stand down. If state 2 fights back, there is a war; if state 2 stands down, there is no war. If the states have unequal strength, the stronger state wins a war with probability 1. If they have equal strength, they each win with probability .5. The cost of war is c to each state. Payoffs to each state are 1 − c for winning a war; −c for losing a war; and .5 if there is no war.
- (a) What are the types and pure strategies of state 1?
- (b) What are the types and pure strategies of state 2?
- (c) Suppose c = 2/3. Is there a PBE in which war occurs?
- (d) Suppose c = 1/3. Suppose state 1 initiates conflict for any strength. What is the expected utility of each type of state 2 from fighting back? From standing down?
- (e) For c=1/3 again,is it aPBE(for any p in[0,1])for state 1to initiate conflict for any strength?
- (f) For c = 1/3 again, suppose state 1 initiates conflict if and only if its strength is High. What is the expected utility of each type of state 2 from fighting back? From standing down?
- (g) For c=1/3 again,is it a PBE(for any p in[0,1])for state 1 to initiate conflict if and only if its strength is High?

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