Consider the following "war of attrition" game. Interaction between players 1 and 2 takes place over discrete periods of time, starting in period 1. In each period, players choose between "stop" (S) and "continue" (C) and they receive payoffs given by the following stage-game matrix:
The length of the game depends on the players' behavior. Specifically, if one or both players select S in a period, then the game ends at the end of this period. Otherwise, the game continues into the next period. Suppose the players discount payoffs between periods according to discount factor
d. Assume x
(a) Show that this game has a subgame perfect equilibrium in which player 1 chooses S and player 2 chooses C in the first period. Note that in such an equilibrium, the game ends at the end of period 1.
(b) Assume x = 0. Compute the symmetric equilibrium of this game. (Hint: In each period, the players randomize between C and S. Let a denote the probability that each player selects S in a given period.)
(c) Write an expression for the symmetric equilibrium value of a for the case in which x is not equal to 0.