Suppose the following normal form game is played twice. Players observe the actions chosen in the first period prior to the second period. Each player's total payoff is the sum of his/her payoff in the two periods.Consider the following strategy: Play A in period 1, play C in period 2 if the action profile in period 1 is (A;A), otherwise play B. What is the highest value of x>0 for which playing the stated strategy by both players is a subgame perfect equilibrium of the twice repeated game? A B CA 1, 7 0, 4 0, 0B 4, 0 x x 0, 0C 0, 0 0, 0 7, 1[Write your answer as a decimal number like 0.33]
Question
Suppose the following normal form game is played twice. Players observe the actions chosen in the first period prior to the second period. Each player's total payoff is the sum of his/her payoff in the two periods.Consider the following strategy: Play A in period 1, play C in period 2 if the action profile in period 1 is (A;A), otherwise play B. What is the highest value of x>0 for which playing the stated strategy by both players is a subgame perfect equilibrium of the twice repeated game? A B CA 1, 7 0, 4 0, 0B 4, 0 x x 0, 0C 0, 0 0, 0 7, 1[Write your answer as a decimal number like 0.33]
Solution 1
The game described is a two-period game with observed actions. The strategy stated is to play A in the first period, then play C in the second period if the action profile in the first period was (A;A), otherwise play B.
To find the highest value of x for which this strategy is a subgame perfect equilibrium, we need to ensure that no player has an incentive to deviate from this strategy in any subgame.
In the first period, the strategy is to play A. The payoff for playing A is 1 for player 1 and 7 for player 2. If player 1 deviates and plays B, his payoff would be 4, but this would lead to a payoff of 0 in the second period since the strategy dictates to play B in the second period if the first period's action profile is not (A;A). Therefore, playing A in the first period is a best response for player 1.
In the second period, if the action profile in the first period was (A;A), the strategy is to play C. The payoff for playing C is 7 for player 1 and 1 for player 2. If player 1 deviates and plays B, his payoff would be x. Therefore, for playing C to be a best response for player 1, we need x < 7.
If the action profile in the first period was not (A;A), the strategy is to play B. The payoff for playing B is x for player 1. If player 1 deviates and plays C, his payoff would be 0. Therefore, for playing B to be a best response for player 1, we need x > 0.
Therefore, the highest value of x for which the stated strategy is a subgame perfect equilibrium is x = 6.99 (or any value slightly less than 7).
Solution 2
The game described is a two-period game with observable actions. The strategy stated is to play A in the first period, then play C in the second period if the action profile in the first period was (A;A), otherwise play B.
To find the highest value of x for which this strategy is a subgame perfect equilibrium, we need to ensure that no player has an incentive to deviate from this strategy in any subgame.
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