3) Consider a p-Beauty Contest with the following rules: Players pick a (real) number from...

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3) Consider a p-Beauty Contest with the following rules: • Players pick a (real) number from 0 to 200. • The average of the numbers is called X, and let Y = PX. • The player whose number is closest to y wins. Suppose that the population of players is large, and we are restricting attention to pure strategies. Assume that 0 < p < 1. Suppose that a fraction of the players are perfectly rational game theorists with much p-Beauty Contest experience, but a fraction 1- μ of the players are new and always choose 100. Denote rational player i's chosen number by si, and denote the average of the other rational players' numbers by s_j. a) (10pts) Solve for the unique symmetric Nash equilibrium chosen number by the rational players s; when p = 2/3 and μ = 0.9. (A symmetric equilibrium is one in which s; = S-;-) Give your response to the nearest 2 decimal places. b) (10pts) What happens to s; as μ approaches 1? Explain. c) (10pts) Show that s; approaches 100 as p approaches 1. Thus, when p is close to one, the irrational players completely dominate the equilibrium played by the rational players. Explain the intuition. 3) Consider a p-Beauty Contest with the following rules: • Players pick a (real) number from 0 to 200. • The average of the numbers is called X, and let Y = PX. • The player whose number is closest to y wins. Suppose that the population of players is large, and we are restricting attention to pure strategies. Assume that 0 < p < 1. Suppose that a fraction of the players are perfectly rational game theorists with much p-Beauty Contest experience, but a fraction 1- μ of the players are new and always choose 100. Denote rational player i's chosen number by si, and denote the average of the other rational players' numbers by s_j. a) (10pts) Solve for the unique symmetric Nash equilibrium chosen number by the rational players s; when p = 2/3 and μ = 0.9. (A symmetric equilibrium is one in which s; = S-;-) Give your response to the nearest 2 decimal places. b) (10pts) What happens to s; as μ approaches 1? Explain. c) (10pts) Show that s; approaches 100 as p approaches 1. Thus, when p is close to one, the irrational players completely dominate the equilibrium played by the rational players. Explain the intuition.

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a To solve for the unique symmetric Nash equilibrium chosen number by the rational players s we need to find the value of s that maximizes their expec... View the full answer