Consider the electoral competition game presented in Lecture 6. In this game there are two candidates who simultaneously choose policies from the real line. There is a distribution of voters with median m and the candidate whose policy is closest to the median wins the election and the winning candidate's policy is implemented. If the two candidates are an equal distance from the median, then the average of the two policies is implemented. For this problem we suppose that both candidates care about both the implemented policy and winning the election. That is, the payo to each candidate has two parts. The first part is the utility from the implemented policy a*. That is, each candidate has utility u(a* ; x_i), where x_i is the ideal policy of candidate i and utility decreases to the left and right of xi. We suppose that x_i < m < x_j . The second part is the value of winning office, which we denote w_i > 0 for candidate i. Putting these two parts together, we de ne the payoff to candidate i by

Find all Nash equilibria to this game.