n players are playing the guessing-the-average game. Each player simultaneously chooses a number from the interval [0; 100] (note that a player is allowed to choose a number that is not an integer). Let xi denote the guess of player i. Given a strategy profile (x1; x2; :::; xn), player i's payoff is

-(x_i - (x₁+x₂+....+x_n/2n))²

(a) Suppose n = 2. Find the best reply function of each player. Plot the players' best reply functions in one figure and find all the Nash equilibrium in pure strategies. Explain

your answer. (10%)

(b) Suppose n = 3. If player 2's guess is x2 and player 3's guess is x3, what is the best reply of player 1? Find a symmetric Nash equilibrium. Explain your answer. (10%)

(c) Now assume that n > 3. Find a symmetric Nash equilibrium. Explain your answer. (5%)