3. (14 points) Consider the infinitely repeated game in which the stage game is shown here. Each players payoff is the present value of her payoff stream, where the discount factor is . Also assume that the two players share the same discount factor of future payoffs. (a) (1 point) What is the pure-strategy Nash equilibrium of the stage game? (b) (1 point) If Player 1 chooses A, what is the best response of Player 2 ? (c) (1 point) If Player 1 chooses B, what is the best response of Player 2 ? (d) (5 points) Consider the following punishment mechanism: - Each player will choose B in stage 1 and in any later stages as long as B was chosen by both players in all past stages. - If a player chooses any strategy other than B in one stage, the opponent will chooses A in all later stages. For what range of does this punishment mechanism support (B, B) to be played in every stage? (Hint: You can start by using your answers in parts (b) and (c) to figure out these two things: (1) If Player 2 defects in a stage by choosing something other than B, what specific strategy would she choose? (2) For this defecting player, what would be her resulting payoff stream in later stages?) (e) (1 point) If Player 1 chooses C, what is the best response of Player 2 ? (f) (5 points) Consider an alternative punishment mechanism as follows: - Each player will choose C in stage 1 and in any later stages as long as C was chosen by both players in all past stages. - If a player chooses any strategy other than C in one stage, the opponent will chooses A in all later stages. For what range of does this punishment mechanism support (C,C) to be played in every stage? (Hint: You can start by using your answers in parts (b) and (e) to figure out these two things: (1) If Player 2 defects in a stage by choosing something other than C, what specific strategy would she choose? (2) For this defecting player, what would be her resulting payoff stream in later stages?) 3. (14 points) Consider the infinitely repeated game in which the stage game is shown here. Each players payoff is the present value of her payoff stream, where the discount factor is . Also assume that the two players share the same discount factor of future payoffs. (a) (1 point) What is the pure-strategy Nash equilibrium of the stage game? (b) (1 point) If Player 1 chooses A, what is the best response of Player 2 ? (c) (1 point) If Player 1 chooses B, what is the best response of Player 2 ? (d) (5 points) Consider the following punishment mechanism: - Each player will choose B in stage 1 and in any later stages as long as B was chosen by both players in all past stages. - If a player chooses any strategy other than B in one stage, the opponent will chooses A in all later stages. For what range of does this punishment mechanism support (B, B) to be played in every stage? (Hint: You can start by using your answers in parts (b) and (c) to figure out these two things: (1) If Player 2 defects in a stage by choosing something other than B, what specific strategy would she choose? (2) For this defecting player, what would be her resulting payoff stream in later stages?) (e) (1 point) If Player 1 chooses C, what is the best response of Player 2 ? (f) (5 points) Consider an alternative punishment mechanism as follows: - Each player will choose C in stage 1 and in any later stages as long as C was chosen by both players in all past stages. - If a player chooses any strategy other than C in one stage, the opponent will chooses A in all later stages. For what range of does this punishment mechanism support (C,C) to be played in every stage? (Hint: You can start by using your answers in parts (b) and (e) to figure out these two things: (1) If Player 2 defects in a stage by choosing something other than C, what specific strategy would she choose? (2) For this defecting player, what would be her resulting payoff stream in later stages?)