Campaigning Revisited:

Two political candidates are scheduled to campaign in two states, in one in period t = 1and in the other in t = 2. In each state each candidate can choose either a positive campaign that promotes his own agenda (P for player 1, p for player 2) or a negative one that attacks his opponent (N for player 1, n for player 2). Residents of the first-period state do not mind negative campaigns, which are generally effective, and payoffs in this state are given by the following matrix:

In the second-period state, residents dislike negative campaigns despite their effectiveness, and the payoffs are given by the following matrix:

a. What are the Nash equilibria of each stage-game? Find all the purestrategy subgame-perfect equilibria with extreme discounting (δ = 0). Be precise in defining history-contingent strategies for both players.

b. Now let δ = 1. Find a subgame-perfect equilibrium for the two-stage game in which the players choose (P,p) in the first stage-game.

c. What is the lowest value of δ for which the subgame-perfect equilibrium you found in (b) survives?

d. Can you find a subgame-perfect equilibrium for this game in which the players play something other than (P, p) or (N, n) in the first stage?