Consider a Bayesian game with incomplete information in which player 1 may be either type a or type 13, where type a has probability .9 and type 13 has probability .1 (as assessed by player 2).

Player 2 has no private information. Depending on player l's types, the payoffs to the two players depend on their actions in C, = {x,,y,} and C2 = {x2 ,y2} as shown in Table 3.16.

Show that there exists a Bayesian equilibrium in which player 2 chooses x2 .

b. (A version of the electronic mail game of Rubinstein, 1989.) Now suppose that the information structure is altered by the following preplay communication process. If player 1 is type 13, then he sends no letter to player 2. Otherwise, player 1 sends to player 2 a letter saying, "I am not type p." Thereafter, each time either player receives a message from the other, he sends back a letter saying, "This is to confirm the receipt of your most recent letter."

Suppose that every letter has a probability VI o of being lost in the mail, and the players continue exchanging letters until one is lost. Thus, when player 1 chooses his action in fx₁,y₁, he knows whether he is 13 or a; and if he is a, then he also knows how many letters he1 got from player 2. When player 2 chooses her action in {x₂ ,y₂}, she knows how many letters she got from player 1.

After the players have stopped sending letters, what probability would player 2 assign to the event that a letter sent by player 1 was lost in the mail (so the total number of letters sent by 1 and 2 together is odd)? If player 1 is not type ß, then what probability would player 1 assign to the event that a letter sent by player 1 was lost in the mail? Show that there is no Bayesian equilibrium of this game in which player 2 ever chooses x2.