Question

Two candidates A and B are running for election. There are 5 voters who vote for one of the two candidates (no abstention). The candidate with most votes wins. A majority of voters prefer candidate A over candidate B.
a) Describe formally this environment as a game (players, strategies, payoffs)
b) Show that to vote for your preferred candidates is a weakly dominant strategy.
c) Show that there exist Nash equilibria in which candidate B gets elected and that most voters play a weakly dominated strategy.

Answer 1

a. The environment can be written as a game with {n=5, strategies-A or B, payoffs- (1 if the preferred candidate wins, 0 otherwise.}

b. Consider the case of a voter who prefers A. When , 2 candidates each are voting for A, the voter prefers to vote for A as then A wins. In any other case, the voter cannot change the outcome by voting for A. Either A will definitely win or lose. So the voter is indifferent between voting or not voting for A as in both cases he gets 0. So no matter what others do, this voter will vote for A. A symmetric argument applies for voters who prefer B. Therefore voting for the preferred candidate is a Nash Eq.

c. To show a Nash Eq. we have to show that there does not exist any unilateral profitable deviation for anyone. Consider a case where all 5 voters vote for B. In this case, no voter has an incentive to deviate as they cannot singlehandedly change the outcome. B will still win. Therefore no voter has an incentive to deviate. So this is a Nash Eq. with most voters playing their weakly dominated strategy. There is another Nash Eq. when 4 voters vote for B and 1 votes for A