Player 1 has two options: L and R. If Player 1 chooses L, then Player 2 has two options; she can choose A or B. If Player 2 chooses A payoffs are (0; 1) and if she chooses B payoffs are (0; 0). If Player 1 chooses R, then Player 2 has two options; she can choose Y or N. If Player 2 shows Y payoffs are (0; x) and if she chooses N payoffs are (1; 2).Which of the following statements is true? A. 1 is the lowest level of x under which (L; A,Y ) is a subgame perfect equilibrium of this game B. 2 is the lowest level of x under which (L; B,Y ) is a subgame perfect equilibrium of this game C. No matter what the level of x is, (L; A,Y ) is not a subgame perfect equilibrium of this game D. 2 is the lowest level of x under which (L; A,Y ) is a subgame perfect equilibrium of this game
Question
Player 1 has two options: L and R. If Player 1 chooses L, then Player 2 has two options; she can choose A or B. If Player 2 chooses A payoffs are (0; 1) and if she chooses B payoffs are (0; 0). If Player 1 chooses R, then Player 2 has two options; she can choose Y or N. If Player 2 shows Y payoffs are (0; x) and if she chooses N payoffs are (1; 2).Which of the following statements is true? A. 1 is the lowest level of x under which (L; A,Y ) is a subgame perfect equilibrium of this game B. 2 is the lowest level of x under which (L; B,Y ) is a subgame perfect equilibrium of this game C. No matter what the level of x is, (L; A,Y ) is not a subgame perfect equilibrium of this game D. 2 is the lowest level of x under which (L; A,Y ) is a subgame perfect equilibrium of this game
Solution 1
The correct answer is C. No matter what the level of x is, (L; A,Y ) is not a subgame perfect equilibrium of this game.
Here's why:
A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in every subgame of the original game.
In this game, if Player 1 chooses L, Player 2 will choose A or B. If Player 2 chooses A, the payoff is (0;1), and if Player 2 chooses B, the payoff is (0;0). Therefore, Player 2 will always choose A when Player 1 chooses L because the payoff is higher.
If Player 1 chooses R, Player 2 will choose Y or N. If Player 2 chooses Y, the payoff is (0;x), and if Player 2 chooses N, the payoff is (1;2). The choice of Player 2 will depend on the value of x.
However, no matter what the value of x is, (L; A,Y ) cannot be a subgame perfect equilibrium. This is because if Player 1 chooses L, Player 2 will always choose A, and the choice of Y is not relevant in this case. Therefore, the strategy profile (L; A,Y ) cannot induce a Nash equilibrium in every subgame of the original game, and it is not a subgame perfect equilibrium.
Solution 2
The correct answer is C. No matter what the level of x is, (L; A,Y ) is not a subgame perfect equilibrium of this game.
Here's why:
A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in every subgame of the original game.
In this game, if Player 1 chooses L, Player 2 can choose either A or B. If Player 2 chooses A, the payoffs are (0; 1), and if she chooses B, the payoffs are (0; 0).
If Player 1 chooses R, Player 2 can choose either Y or N. If Player 2 chooses Y, the payoffs are (0; x), and if she chooses N, the payoffs are (1; 2).
For (L; A,Y ) to be a subgame perfect equilibrium, Player 2 would have to prefer A to B when Player 1 chooses L, and Y to N when Player 1 chooses R.
However, when Player 1 chooses L, Player 2 would always prefer B to A, because the payoff from B (0) is higher than the payoff from A (1).
Therefore, no matter what the level of x is, (L; A,Y ) cannot be a subgame perfect equilibrium of this game.
Similar Questions
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