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Suppose that each player wants to maximize their own payoff in questions (i)-(iii) below.
(i) Suppose this game is played twice, and player 2 commits to a “tit-for-tat” strategy: she will start with C in the first period, and in the second period she will take whatever action player 1 played in the first period. How should player 1 play the game to maximize his own payoff? Does player 2 benefit from her commitment to tit-for-tat, compared to the case with no commitment?
(ii) Suppose now the game is played three times, and player 2 still commits to tit-for-tat.
Suppose player 1 chooses D accidentally in the first period. Then what should player 1 do in the next two periods?
(iii) Now consider the standard repeated game without explicit commitment. Suppose the above game is played for an infinite number of periods, and each player has a discount factor < 1. Find the condition for the two players to cooperate every period by using the grim trigger strategy.
Social preferences may make players cooperate with each other even without a long-term relationship. Consider a different scenario. Let us call the numbers in the above matrix “standard” payoffs. Suppose now players also care about the joint payoff (i.e. the sum of the two players’ standard payoffs). In particular, when a player has a standard payoff x and a joint payoff y, her “real” payoff is x+k×y, where k is a parameter which captures the degree of social preferences. For example, if player 1 chooses D and player 2 chooses C, then player 1’s real payoff is 5+6k, and player 2’s real payoff is 1+6k.
(iv) Suppose the game is played only once. Draw the matrix of real payoffs, and find the condition of k for (C,C) to be the only NE.
(v)* Suppose now the game is played twice. k starts with 0.4 in the first period. It will change in the second period, depending on the outcome of the first period. If both choose to cooperate in
the first period, k will increase from 0.4 to 0.6. (In a cooperative environment, people become to care more about each other.) If only one player defects in the first period, k will drop from 0.4 to 0.2. If both defect in the first period, k will drop to 0. What’s your prediction of the outcome of the game? Explain your answer formally.
Consider the following Prisoners' Dilemma game: Player 2 (she) с F 4,4 Player 1 (he) C 5,1 D 1,5 2,2