Question 5: Consider the following situation. Two childhood friends, Ann and Bob, have been trading Christmas cards for years and years. However, they have now grown apart, each with their own lives to live, and each of them feels somewhat awkward and obligated to send the annual Christmas card. But since each of them received one last year, they feel uncomfortable being the first one to stop sending cards and the exchange of cards is continuing. Both of them would prefer to stop sending cards, but neither one wants to be the first one to stop sending cards and paying the cost of embarrassment. If they stopped at the same time, they could avoid this cost, but neither one wants to even broach the subject of stopping sending cards together because by doing that, the person would effectively still be the first to admit that they do not want to continue the card exchange. Every year, the payoffs of the players are given by the following payoff matrix:

		Bob
		Send	Don't
Ann	Send	-1,-1	-1,-3
	Don't	-3,-1	0,0

Note that this game constitutes a "trust game" in the terminology of class 3. There are two pure-strategy Nash equilibria, {Send, Send} and {Don't,Don't}, where the players would prefer the {Don't,Don't} equilibrium but may fail to get there due to the fear of the other person not reciprocating. In addition, the game has a mixed-strategy Nash equilibrium, where the players will send the Christmas card probabilistically, and that is the equilibrium of interest for this question.

Suppose, for simplicity, that there are two periods left in the game (after that, Ann or Bob will move somewhere where they cannot get mail, and so the game is known to be over). The payoffs of the whole game are constructed as follows. In the current year, if at least one player does not send a card, the players realize the payoffs of the above matrix, plus the game is over (so that next year's payoff is zero). But if both players send a card this year, then the game is repeated next year, with the payoffs as given by the table. The players discount the next period's payoff by b (so that 1 tomorrow is worth b today). Using backward-induction, solve for (i) the mixed-equilibrium probabilities p_A,2, p_B,2(for Ann and Bob, respectively) that they will not send a card during the second year, and (ii) the mixed-equilibrium probabilities p_A,l,p_B,l that they will not send a card during the first year. Briefly explain how the probabilities between this and next year differ, and how it depends on the discount rate b.

(If you are mathematically inclined and want to challenge yourself, instead of considering just two periods, you can consider this as an infinitely repeated game that will continue indefinitely until at least one of the players does not send a card).