Question

4. Political Competition Each of 2 political candidates chooses a policy position x e [0,1: Voters (there is a continuum of them) are uniformly distributed on [0,1]; each votes for whichever candidate chooses a position closest to him. (So for example, if candidate l chooses X1 and candidate 2 chooses x2,then all voters above1 vote for candidate 2, all below with positive probability. A candidate's payoff is 1 if he wins, 0 if he loses. Find all NE of the game. (Note: a NE is a pair (x1;x2) such that, given the opponent's policy choice, neither candidate wishes to change his position).

Answer 1

This is a famous game known as the median voter theorem game.

The Nash equilibrium is for both candidates to select the median location 1/2, doing this guarantees each candidate half the votes, but deviating to any other point generates strictly less.

This problem is similar to the hotelling problem which is easier to understand theoretically,therefore we will try to understand our original problem through this. the basic idea is that there are two ice cream vendors are on a beach that stretches the 0-1 interval. Customers are uniformly distributed along that interval. The vendors simultaneously select a position. Customers go to the closest vendor and split themselves evenly if the vendors choose an identical position. Each vendors want to maximize its number of customers.

Imagine a line from 0 to 1.

Suppose that vendor 1 (V1) selects the position 1/2. now whatever be the position of vendor 2 (V2), vendor 1 is definitely going to get half of the customers because if V2 is to the left of V1, the customers to the right of 1/2 will be captured by V1 and vice versa. Same strategy goes for vendor 2. Therefore in equilibrium both V1 and V2 must get greater than or equal to 1/2.

Now suppose there were other nash equilibria possible. Let V1 start at 1/3 and V2 at 1/2. now V2 has an incentive deviate that is if he moves a little closer to its left towards 1/3, he can capture more customers. Same goes for vendor 1. As it turns out,for every other point there is an incentive to the vendor to deviate and thus those points are not nash equilibria.

So the answer to our problem is that there exists only one nash equilibrium which is candidate(1) and candidate(2) chooses 1/2.