Problem 4
If its stage game has exactly one Nash equilibrium, how many subgame perfect equilibria does a two-period, repeated game have? Explain. Would your answer change if there were \(T\) periods, where \(T\) is any finite integer?
Problem 8
Consider an infinite-period repeated game in which a "long-run player" faces a sequence of "short-run" opponents. Formally, player 1 plays the stage game with a different player 2 in each successive period. Denote by \(2^{t}\) the player who plays the role of player 2 in the stage game in period \(t\). Assume that all players observe the history of play. Let \(\delta\) denote the discount factor of player 1 . Note that such a game has an infinite number of players. (a) In any subgame perfect equilibrium, what must be true about the behavior of player \(2^{t}\) with respect to the action selected by player 1 in period \(t\) ? (b) Give an example of a stage-game and subgame perfect equilibrium where the players select an action profile in the stage game that is not a stage Nash equilibrium. (c) Show by example that a greater range of behavior can be supported when both players are long-run players than when only player 1 is a long-run player.
Problem 10
Consider a repeated game between a supplier (player 1) and a buyer (player 2). These two parties interact over an infinite number of periods. In each period, player 1 chooses a quality level \(q \in[0,5]\) at cost \(q\). Simultaneously, player 2 decides whether to purchase the good at a fixed price of 6 . If player 2 purchases, then the stage-game payoffs are \(6-q\) for player 1 and \(2 q-6\) for player \(2 .\) Here, player 2 is getting a benefit of \(2 q\). If player 2 does not purchase, then the stage-game payoffs are \(-q\) for player 1 and 0 for player 2 . Suppose that both players have discount factor \(\delta\). (a) Calculate the efficient quality level under the assumption that transfers are possible (so you should look at the sum of payoffs). (b) For sufficiently large \(\delta\), does this game have a subgame perfect Nash equilibrium that yields the efficient outcome in each period? If so, describe the equilibrium strategies and determine how large \(\delta\) must be for this equilibrium to exist.
Problem 12
Consider the infinitely repeated prisoners' dilemma and recall the definition of the grim-trigger strategy. Here is the definition of another simple strategy called Tit-for-tat: Select \(\mathrm{C}\) in the first period; in each period thereafter, choose the action that the opponent selected in the previous period. \({ }^{8}\) Is the tit-for-tat strategy profile a Nash equilibrium of the repeated game for discount factors close to one? Is this strategy profile a subgame perfect Nash equilibrium of the repeated game for discount factors close to one? Explain.
