Consider the following generalization of the Wittman model.

Generalizing the Wittman Model

Consider the following generalization of the Wittman model. There are two parties A and B. Party A is office seeking, that is, its payoff is v > 0 if it wins the election and 0 otherwise. Party B is policy seeking and has ideal policy position 1. Thus, in the event that policy x ∈ R is chosen, party B’s payoff is −|x−1|. Suppose that the voters’ ideal policies are distributed continuously over the real line, with a unique median x_m

(1) Let π (x_A,x_B) denote the probability that party A wins the election. Write down party B’s expected payoff for any arbitrary x_A, x_B.

(2) Argue that both parties choosing x_m is a Nash equilibrium.

(3) Is this the unique Nash equilibrium? If you answer yes, argue why there cannot be any other Nash equilibrium. If you answer no, find another Nash equilibrium and argue that it is indeed a Nash equilibrium.