I am having some difficulty in solving the following problem. I was wondering whether someone would be kind enough to sketch a solution or even better to solve the whole game. Thanks
Suppose that 100 people live in a village of whom 51 support the conservative candidate and 49 support the liberal candidate. Villagers get a payoff of 10 if their candidate gets elected, -10 if the other one wins. Voting costs villagers 1 unit. Those who do not vote evade these costs but get the payoff related to the winning candidate ( +/- 10).
- Find a Nash Equilibrium (in mixed strategies) in which all conservatives use the same strategy and all liberals use the same strategy. In case of equal number of votes toss of a coin decides. (So for instance in the case of equal number of votes, if villager i is voting, then its payoff would be of 1/2*(10-1); while if villager i is NOT voting, then its payoff would be of 1/2*10)
- What is the expected number of villagers who will vote in this equilibrium?