Consider the Hotelling model of spatial competition. The set of voters is denoted by V = {1, . . . , n}, with n odd, and each voter has an ideal point, vi ∈ [−1, 1]. These ideal points are common knowledge to two candidates, c ∈ {1, 2}, each of whom simultaneously commits to a platform, pc ∈ [−1, 1]. After this, each voter i observes (p1, p2) and votes for the candidate who chose the platform closest to his or her ideal point. If the candidates’ platforms are equidistant from vi, then the voter flips a fair coin to determine who he or she votes for. Each candidate’s payoff is simply how many votes he or she receives.
(a) Find a pure strategy Nash equilibrium of this game.
(b) Is this the only pure strategy equilibrium? If so, why? If not, either answer why not or provide another one.
(c) Suppose that n is even and all voters have distinct ideal points (i.e., vi ≠ vj for all i, j ∈ N). Find a pure strategy Nash equilibrium of this game. (d) Again supposing that n is even, how many pure strategy equilibria are there?