Which is true for the following game?1 \2 L C RT 2, 3 3, 4 4, 8M 5, 6 4, 3 7, 1B 6, 2 5, 5 2, 4(A) There is a mixed strategy Nash equilibrium in which Player 1 plays both M and B withpositive probability and Player 2 plays both L and C with positive probability(B) The best response of Player 2 to M is C(C) (B,C) is the only Nash equilibrium(D) (M,L) is a Nash equilibrium

To answer this question, we need to analyze the payoff matrix of the game. The payoff matrix is given as follows:

    L   C   R
T  2,3 3,4 4,8
M  5,6 4,3 7,1
B  6,2 5,5 2,4

Here, the first number in each cell represents the payoff for Player 1 and the second number represents the payoff for Player 2.

(A) There is a mixed strategy Nash equilibrium in which Player 1 plays both M and B with positive probability and Player 2 plays both L and C with positive probability.

To verify this, we need to check if there is a mixed strategy Nash equilibrium where Player 1 mixes between M and B and Player 2 mixes between L and C. A mixed strategy Nash equilibrium exists when a player is indifferent between two or more pure strategies because the expected payoff is the same. Here, Player 1's payoff for M and B is not the same in response to L and C, and Player 2's payoff for L and C is not the same in response to M and B. Therefore, this statement is not true.

(B) The best response of Player 2 to M is C

The best response of Player 2 to M would be the strategy that gives Player 2 the highest payoff when Player 1 plays M. Looking at the payoff matrix, when Player 1 plays M, Player 2 gets a payoff of 6 for L, 3 for C, and 1 for R. Therefore, the best response of Player 2 to M is L, not C. So, this statement is not true.

(C) (B,C) is the only Nash equilibrium

A Nash equilibrium is a set of strategies where no player can improve their payoff by unilaterally changing their strategy. Looking at the payoff matrix, when Player 1 plays B and Player 2 plays C, the payoffs are 5 for Player 1 and 5 for Player 2. Neither player can improve their payoff by changing their strategy unilaterally. Therefore, (B,C) is a Nash equilibrium. However, we need to check if there are other Nash equilibriums. By checking all the other strategy combinations, we can see that (T,L) is also a Nash equilibrium because the payoffs are 2 for Player 1 and 3 for Player 2, and neither player can improve their payoff by changing their strategy unilaterally. Therefore, this statement is not true because (B,C) is not the only Nash equilibrium.

(D) (M,L) is a Nash equilibrium

Looking at the payoff matrix, when Player 1 plays M and Player 2 plays L, the payoffs are 5 for Player 1 and 6 for Player 2. Neither player can improve their payoff by changing their strategy unilaterally. Therefore, (M,L) is a Nash equilibrium. So, this statement is true.

To answer this question, we need to analyze the payoff matrix of the game. The payoff matrix is given as follows: