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) What guess(es) survive iterated elimination of dominated strate-gies?
From now on, suppose that a fraction 1 - μ of the players is rational, and a fraction μ of the players is irrational. It is known that irrational players like high numbers, so they always guess 200.
(b) Solve for the rationals' equilibrium guess as a function of p and μ.
(c) What happens to the rationals' guess as p approaches 1 from below? Explain the intuition.
(d) What happens to the rationals' guess as μ approaches 0? Explain the intuition.