A large population of players are playing the following version of the beauty-contest game. Each player guesses a number between 100 and 200 (inclusive), and the player closest to p < 1 of the average wins a given prize. If multiple players are closest to p < 1 times the average, the prize is allocated randomly between them.

(a) [5pts] What guess(es) survive iterated elimination of dominated strategies? From now on, suppose that a fraction 1 of the players is rational, and a fraction of the players is irrational. It is known that irrational players like high numbers, so they always guess 200.

(b) [10pts] Solve for the rationals' equilibrium guess as a function of and .

(c) [5pts] What happens to the rationals' guess as approaches 1 from below? Explain the intuition.

(d) [5pts] What happens to the rationals' guess as approaches 1? Explain the intuition.