To Vote or Not to Vote: Two candidates, D and R, are running for mayor in a town with n residents. A total of 0 < d < n residents support candidate D, while the remainder, r = n − d, support candidate R. The value for each resident for having his candidate win is 4, for having him tie is 2, and for having him lose is 0. Going to vote costs each resident 1. Each resident simultaneously decides where to vote for her cadnidate or not to vote.
a. Let n = 2 and d = 1. Write down this game as a matrix and find all the NE. Are equilibrium outcomes Pareto optimal?
b. Let n > 2 be an even number and let d = r = n/2. Find all PSNE. Are equilibrium outcomes Pareto optimal?
c. Consider the case of n = 3 with players 1 and 2 supporting D and player 3 supporting R: Does a PSNE exist? Explain.