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Midterm Examination EC 703 Spring 2016 Instructions. Answer any two of the following three questions. Each question is worth 50 points. 1. Suppose two players have $10 to divide. The procedure: They simultaneously choose demands which have to be in pennies. If the demands are consistent, they split the money. Otherwise, they get nothing. More formally, the strategy set for player i is Si = {. . . , −.02, −.01, 0, .01, .02, . . .}. The negative “demands” can be thought of as offers to pay. Note that each player can demand any amount of money, so long as it is a multiple of .01. The utility function for i is si , if s1 + s2 ≤ 10 ui (s1 , s2 ) = 0, otherwise. (a) Find all pure strategy Nash equilibria. (b) Find all (weakly) dominated strategies. Find all pure strategy Nash equilibria in the game remaining after we remove all dominated strategies. (c) Do another round of removal of (weakly) dominated strategies. Now find all pure strategy Nash equilibria in the remaining game. 1 2. Find all pure strategy Nash, subgame perfect, weak perfect Bayesian, and sequential equilibria of the following game. For sequential, prove consistency. (For weak perfect Bayesian and sequential, you don’t need to give all possible beliefs — one belief for each strategy profile is sufficient.) 1 a 4 2 2 3 4 -3 c b 2 d -2 0 3 d e e 3 f -3 0 1 g f 0 3 0 1 1 0 2 g 2 1 1 3. Consider the following signaling game with infinitely many messages. Player 1 is one of two types, ta or tb , where the common prior is that these types are equally likely. Player 1’s set of messages M = [0, ∞). Player 2 has two possible actions A = {α, β}. The payoff function for player 1 takes the form u1 (t, a) − m while the payoff function for player 2 takes the form u2 (t, a) + m. The table below gives the values of u1 (t, a) and u2 (t, a) where the first number in each cell is u1 and the second is u2 : ta tb α β 2, 1 0, 0 1, 0 0, 3 Find all pure strategy weak perfect Bayesian equilibria. (As usual, you don’t need to give all possible beliefs — one belief for each strategy profile is sufficient.) 3