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 xm
(1) Let π (xA,xB) denote the probability that party A wins the election. Write down party B’s expected payoﬀ for any arbitrary xA, xB.
(2) Argue that both parties choosing xm 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.