2. (50 points) (E-mail game) This question shows how a higher-order beliefs may affect equilibrium behaviors. Suppose there are two players, 1 and 2, and that there are two states a and b: 1 A B State a: A 1,1 0, -1 B-1,0-1,0 A B State b: A 0,0 0, -1 B-1,0 1,1 As is obvious from the payoff matrix, action profile (A, A) ((B, B)) is unique equilibrium of state-a (state-b) game. Suppose that the probability that state a obtains is 0.5 and, on one hand, player 1 completely knows which state has ob- tained. On the other hand, player 2's information depends on the automatic communication tool set up in their computers (e-mails). If state b obtains, then player 1's computer automatically sends a message to player 2's computer. The message reaches player 2's computer with probability 1- , where & (0,1) is a probability that a message is lost. After receiving a message, player 2's computer automatically sends back a confirmation message to player 1's computer which, again, reaches with probability 1 . This process continues until a message is lost. Let the "type" t; be the number of messages sent from player i's computer. For example, (t, 12) (1,1) obtains when the state is b, player 1's first message successfully reach player 2, and player 2's first message fails to reach player 1. Observe that (t, t2) = (0,0) when the state a obtains. If t = k 1, then only (k, k 1) or (k, k) occurs, and if t = k 0, then only (k, k) or (k+ 1,k) occurs. If t 1 and t2 1, then both players know that the state is b while they are not sure how many order of beliefs their opponents have. Answer the following questions. 2 (a) (25 points) Let p(t, t2) be the probability that types (t, t2) obtains. Com- pute p(0,0), p(1,0), p(1, 1), p (2, 1), and p(2, 2). (b) (5 points) Derive p(t, 12). (c) (5 points) Let 0 (t) be player 1's strategy. Show that 1 (0) = A must hold in any Bayesian Nash equilibrium for any . = = (d) (5 points) Let 02 (t2) be player 2's strategy. By computing the conditional belief p(ti 0|t2 0) of player 2, show that 0 (0) must hold in any Bayesian Nash equilibrium for any . (Hint: Use the Bayes' rule to show that p(t = 0|t2 = 0) is larger than 1/2) A and 0 (1) = A must hold in any Bayesian = (e) (5 points) Show that 01 (1) Nash equilibrium for any . (f) (5 points) Show that the unique Bayesian equilibrium is a strategy profile 0 = (01,02) such that Page 2 for/angi and k 0. (Hint: Use induction.) 2. (50 points) (E-mail game) This question shows how a higher-order beliefs may affect equilibrium behaviors. Suppose there are two players, 1 and 2, and that there are two states a and b: 1 A B State a: A 1,1 0, -1 B-1,0-1,0 A B State b: A 0,0 0, -1 B-1,0 1,1 As is obvious from the payoff matrix, action profile (A, A) ((B, B)) is unique equilibrium of state-a (state-b) game. Suppose that the probability that state a obtains is 0.5 and, on one hand, player 1 completely knows which state has ob- tained. On the other hand, player 2's information depends on the automatic communication tool set up in their computers (e-mails). If state b obtains, then player 1's computer automatically sends a message to player 2's computer. The message reaches player 2's computer with probability 1- , where & (0,1) is a probability that a message is lost. After receiving a message, player 2's computer automatically sends back a confirmation message to player 1's computer which, again, reaches with probability 1 . This process continues until a message is lost. Let the "type" t; be the number of messages sent from player i's computer. For example, (t, 12) (1,1) obtains when the state is b, player 1's first message successfully reach player 2, and player 2's first message fails to reach player 1. Observe that (t, t2) = (0,0) when the state a obtains. If t = k 1, then only (k, k 1) or (k, k) occurs, and if t = k 0, then only (k, k) or (k+ 1,k) occurs. If t 1 and t2 1, then both players know that the state is b while they are not sure how many order of beliefs their opponents have. Answer the following questions. 2 (a) (25 points) Let p(t, t2) be the probability that types (t, t2) obtains. Com- pute p(0,0), p(1,0), p(1, 1), p (2, 1), and p(2, 2). (b) (5 points) Derive p(t, 12). (c) (5 points) Let 0 (t) be player 1's strategy. Show that 1 (0) = A must hold in any Bayesian Nash equilibrium for any . = = (d) (5 points) Let 02 (t2) be player 2's strategy. By computing the conditional belief p(ti 0|t2 0) of player 2, show that 0 (0) must hold in any Bayesian Nash equilibrium for any . (Hint: Use the Bayes' rule to show that p(t = 0|t2 = 0) is larger than 1/2) A and 0 (1) = A must hold in any Bayesian = (e) (5 points) Show that 01 (1) Nash equilibrium for any . (f) (5 points) Show that the unique Bayesian equilibrium is a strategy profile 0 = (01,02) such that Page 2 for/angi and k 0. (Hint: Use induction.)