Electronic Mail game Let L>M> 0 be two positive real numbers. Two players play a game in which the payoff function is one of the following two, depending on the value of the state of nature s, which may be 1 or 2:

The probability that the state of nature is s = 2 is p 1/2. Player I knows the true state of nature, and Player II does not know it. The players would clearly prefer to coordinate their actions and play (A, A) if the state of nature is s = 1 and (B,B) if the state is s = 2, which requires that both of them know what the true state is.

Suppose the players are on opposite sides of the globe, and the sole method of communication available to them is e-mail. Due to possible technical communications disruptions, there is a probability of ε > 0 that any e-mail message will fail to arrive at its destination.

In order to transfer information regarding the state of nature from Player I to Player II, the two players have constructed an automated system that sends e-mail from Player I to Player II if the state of nature is s = 2, and does not send any e-mail if the state is s = 1.

To ensure that Player I knows that Player II received the message, the system also sends an automated confirmation of receipt of the message (by e-mail, of course) from Player II to Player I the instant Player I's message arrives at Player II's e-mail inbox. To ensure that Player II knows that Player I received the confirmation message, the system also sends an automated confirmation of receipt of the confirmation message from Player I to Player II the instant Player II's confirmation arrives at Player I's e-mail inbox.

The system then proceeds to send an automated confirmation of the receipt of the confirmation of the receipt of the confirmation, and so forth. If any of these e-mail messages fail to arrive at their destinations, the automated system stops sending new messages. After communication between the players is completed, each player is called upon to choose an action, A or B. Answer the following questions:

(a) Depict the situation as a game with incomplete information, in which each type of each player is indexed by the number of e-mail messages he has received.

(b) Prove that the unique Bayesian equilibrium where Player I plays A when s = 1 is for both players to play A under all conditions.

(c) How would you play if you received 100 e-mail confirmation messages? Explain your answer.