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 2t 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 d 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 2t 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.