Which is true for the following two-player game:1 \2 L C RT 4, 4 1, 6 4, 2M 6, 0 6, 6 4, 2B 3, 4 5, 9 2, 2(A) T is a best response to L(B) (M,C) is the only Nash equilibrium(C) (M,R) is a Nash equilibrium(D) There is a mixed strategy Nash equilibrium in which player 2 chooses both C and R withpositive probability
Question
Which is true for the following two-player game:1 \2 L C RT 4, 4 1, 6 4, 2M 6, 0 6, 6 4, 2B 3, 4 5, 9 2, 2(A) T is a best response to L(B) (M,C) is the only Nash equilibrium(C) (M,R) is a Nash equilibrium(D) There is a mixed strategy Nash equilibrium in which player 2 chooses both C and R withpositive probability
Solution
To answer this question, we need to understand the concept of Nash Equilibrium and best response in game theory.
A Nash Equilibrium is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy.
A best response is a strategy that maximizes a player's payoff given the strategies chosen by the other players.
Let's analyze each option:
(A) T is a best response to L Looking at the payoff matrix, if player 1 chooses L, player 2 gets the highest payoff (6) by choosing M, not T. So, T is not a best response to L.
(B) (M,C) is the only Nash equilibrium If player 1 chooses M and player 2 chooses C, both players get a payoff of 6. Neither player can increase their payoff by unilaterally changing their strategy, so (M,C) is a Nash equilibrium. However, we need to check the other strategies to see if it's the only Nash equilibrium.
(C) (M,R) is a Nash equilibrium If player 1 chooses M and player 2 chooses R, player 1 gets a payoff of 4 and player 2 gets a payoff of 2. However, player 1 could increase their payoff to 6 by switching to L or B, and player 2 could increase their payoff to 6 by switching to C. So, (M,R) is not a Nash equilibrium.
(D) There is a mixed strategy Nash equilibrium in which player 2 chooses both C and R with positive probability A mixed strategy Nash equilibrium is a strategy in which players randomize their actions. To determine if such an equilibrium exists, we would need to calculate the expected payoffs for each player given different probabilities of choosing C and R. This requires more information than is provided in the question.
So, the correct answer is (B) (M,C) is the only Nash equilibrium.
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