Tit-for-Tat Reputation: Consider the following Prisoner's Dilemma: ...

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Tit-for-Tat Reputation: Consider the following Prisoner's Dilemma:
                         Player 2
                         c         d
                  C    I. I    - I. 2
Player I      D   2.- I    0, 0

Player 2 has payoffs given by the matrix while player I has one of two types: he is either a strategic type or a tit-for-tat (TFT) type. The TFT type uses the following strategy: in the first period he always cooperates; in every period / I thereafter he plays what his opponent played in period t - I. Let P = ΒΌ be the probability that Nature chooses player I to be the TFT type. Assume that the players do not discount future payoffs.
a. What is the perfect Bayesian equilibrium if the game is played twice?
b. If the game is played three times, is there an equilibrium in which the strategic type of player I defects in the first period?
c. What is the perfect Bayesian equilibrium if the game is played three times?

17.2
Building Trust: A graduating senior (player 1) has an idea for two start-up ventures, a small one and a large one. He can trust (T) his ideas with his roommate (player 2), who is the only one who can implement these ideas, or he can not trust (N) player 2. forget his dreams, and get a boring job, resulting in the status quo payoffs of (0.0) for players I and 2. respectively. There are two types of player 2: an honest type that always honors trust (H) and a normal type that can choose to honor trust (H) or abuse trust (A). It is common knowledge that a quarter of the people in the world are honest, but only player 2 knows whether he is honest or not.
The sequence of play is as follows: First, player I can choose T or N with respect to sharing his small idea. A choice of N (no trust) ends the game and the payoffs are (0, 0). Following T player 2 can honor the trust, resulting in payoffs of (1, 1), or abuse the trust and expropriate the idea, resulting in payoffs of (-1, 2). (Recall that an honest player 2 cannot abuse trust.) If trust was offered in the first stage then first-stage payoffs are determined (i.e., the outcome is known to both players), and player I can then choose whether or not to trust player 2 with his big idea. (If trust was not offered in the first stage then the game ended.) No trust in the second stage will result in no payoffs in addition to those obtained in the first stage. Following trust in the second stage, if player 2 honors trust then the additional payoffs will be (7.7), while if he abuses trust and expropriates the idea the additional payoffs will be (-5, 11). There is no discounting between these stages.
a. Draw the whole game tree and write down the pure strategies of each player for the whole game. How many pure strategies does each player have?
b. Using sequential rationality argue that following an abuse of trust in the first stage, there will be no trust in the second stage, effectively ending the game. Write down the reduced game tree that is implied by this argument, and the resulting pure strategies for each of the two players. From now on focus on the reduced game you derived, in which the game ends after an abuse in the first stage.
c. Can there be a pooling perfect Bayesian equilibrium in which the normal type in the first period honors trust for sure?
d. Can there be a separating perfect Bayesian equilibrium in which the normal type in the first period abuses trust for sure?
e. Find the perfect Bayesian equilibrium of this game. Is there a unique one?

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