Economics can also be used to analyze politics. Suppose two political parties are competing for votes. Each party i chooses a policy position xi 2 [0; 1], i = 1; 2. Voter preferences uj are uniformly distributed on [0; 1] with uj(xi) = ABS(vj - xi). This means that a voter j will vote for the party with a policy position closest to vj on the unit interval. Whatever party gets the most votes wins the election. If both parties choose the same policy position, the election is tied and each win with probability 0.5.
(a) What is the Nash equilibrium of the game? (I.e. what are the equilibrium policy positions for the two parties.)
(b) Suppose a third party enters the game. Does a pure strategy Nash equilibrium exist? If yes, find it. If no, show that it cannot exist.