(Guessing two-thirds of the average) Each of three people announces an integer from 1 to K. If the three integers are different, the person whose integer is closest to 2 of the average of the three integers wins $1. If two or more integers are the same, $1 is split equally between the people whose integer is closest to 2 of the average integer. Is there any integer k such that the action profile (k, k, k), in which every person announces the same integer k, is a Nash equilibrium? (If k ≥ 2, what happens if a person announces a smaller number?) Is any other action profile a Nash equilibrium? (What is the payoff of a person whose number is the highest of the three? Can she increase this payoff by announcing a different number?)

Game theory is used widely in political science, especially in the study of elec- tions. The game in the following exercise explores citizens' costly decisions to vote.