3. The repeated prisoner's dilemma (PD) can be used as a simple framework to inves- tigate tacit collusion. Consider repeating the following stage game: A B A R, R 100, 24 B 24, 100 50, 50 where R is either 64 or 96. In each round, the players simultaneous choose an action (A or B). After both have made their choice the outcome is revealed, along with the payoffs for the round. Then a random number is drawn to determine whether the repeated game will continue for at least one more round. The probability the game will continue is c5, which is either ¹ or ³. 2 4 (a) Suppose the stage game were played just once (which is equivalent to c5 = 0). Find the Pareto efficient outcomes in the stage game. Does either player have a dominated strategy; if so, which? What is the one-shot Nash equilibrium pre- diction of the stage game? How does this prediction vary with the parameter R? (b) For the indefinitely repeated game, explain: i. Why both players choosing the strategy All B is a subgame perfect Nash equilibrium for all combinations of c5 and R. All B is the strategy that chooses B initially, then chooses B in any subsequent round, irrespective of the out- come from any previous round. ii. Why both players choosing the strategy Grim AB is a subgame perfect Nash equilibrium for for all combinations of c5 and R, except when c5 and R = 64. Grim AB is the strategy that chooses A initially, then chooses A again in a subsequent round if the outcome from every previous round was (A, A); otherwise (i.e. if someone, including themselves, choose B at some previous round), it chooses B in all subsequent rounds. = 1 2 (c) Prior experimental evidence suggests that the cooperation rate in such re- peated PDs is typically very low when cooperation is not part of a subgame perfect Nash equilibrium. Furthermore cooperation is greatly facilitated by larger rewards to cooperation (relative to the temptation to cheat) and a larger probability of continuation. Explain how you would design a set of treatments to test this hypothesis. Make sure you explain how you could use the data generated by the experiment to test the hypothesis.

3. The following is an interpretation of the rivalry between the United States (USA) and the Soviet Ünion (USSR) during the cold war. Each side has the choice of two strategies: Aggressive and Restrained. The payoff table is given as follows: USSR Restrained Aggressiveness Restrained 4,3 1,4 USA Aggressiveness 3,1 2,2 a) Consider this game when the two countries move simultaneously. Find all pure strategy Nash equilibria. b) Next consider three alternative ways in which the game could be played with sequential moves: (i) The USA moves first and the USSR moves second. (i) the USSR moves first and the USA moves second. (i) The USSR moves first, and the USA moves second, but the USSR has a further move after the USA moves. For each case, draw the game tree and find the subgame-perfect Nash equilibrium. c) What are the key strategic issues (commitment, credibility and so on) for the two countries. (Note: Be concise. Your answer should not exceed 300 words].

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I hope you can draw a picture and analyze it, thank you.