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 or (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? [8 marks] (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 = 2 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. [12 marks] (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. [10 marks]