Two states decide in sequence whether to initiate a military conflict. Each has private information about its own strength.

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 reveals si 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.

1. (a) What are the types and pure strategies of state 1?
2. (b) What are the types and pure strategies of state 2?
3. (c) Suppose c = 2/3. Is there a PBE in which war occurs?
4. (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?
5. (e) Forc=1/3 again,isitaPBE(foranypin[0,1])forstate1to initiate conflict for any strength?
6. (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?
7. (g) For c=1/3 again,is it a PBE (for any pin[0,1])for state1 to initiate conflict if and only if its strength is High?