Suppose that two people are playing a guessing game with a prize going to the person closest to one-half of the average. Guesses are required to be between 0 and 100 (could be integers or decimals). Show that none of the following are Nash equilibria:

(a) Both choose 1. (Hint: consider a deviation to 0 by one person, so that the average is 1/2, and half of the average is 1/4.

(b) One person chooses 1 and the other chooses 0.

(c) One person chooses x and the other chooses y, where x > y > 0. (Hint: If they choose these decisions, what is the average, what is the target, and is the target less than midpoint of the range of guesses between x and y? Then use these calculations to figure out which person would win, and whether the other person would have an incentive to deviate.)