Consider the below payoff matrix. Player 1 chooses rows and Player 2 chooses columns. A BX 15 , 0 0 , 8Y 0 , 39 15 , 0Denote the probability of “X" for Player 1 as p, and the probability of “A” for Player 2 as q. What is the value of q in the mixed-strategy Nash equilibrium of this game? Round your answer to two decimal places (e.g. 0.15).
Question
Consider the below payoff matrix. Player 1 chooses rows and Player 2 chooses columns. A BX 15 , 0 0 , 8Y 0 , 39 15 , 0Denote the probability of “X" for Player 1 as p, and the probability of “A” for Player 2 as q. What is the value of q in the mixed-strategy Nash equilibrium of this game? Round your answer to two decimal places (e.g. 0.15).
Solution
To find the value of q in the mixed-strategy Nash equilibrium of this game, we need to determine the probabilities that Player 2 assigns to the two available strategies, A and B.
In a mixed-strategy Nash equilibrium, both players are indifferent between their available strategies, meaning they have no incentive to deviate from their chosen probabilities. This implies that the expected payoffs for each player are equal regardless of the strategy chosen by the other player.
Let's start by considering Player 1's strategy. Player 1 has two options: X and Y. The expected payoff for Player 1 when choosing X is 15p + 0(1-p), and the expected payoff for Player 1 when choosing Y is 0p + 39(1-p). Since Player 1 is indifferent between X and Y, these expected payoffs must be equal:
15p + 0(1-p) = 0p + 39(1-p)
Simplifying this equation, we get:
15p = 39(1-p)
Now, let's solve for p:
15p = 39 - 39p 54p = 39 p = 39/54 p ≈ 0.72
So, Player 1 assigns a probability of approximately 0.72 to strategy X.
Next, let's consider Player 2's strategy. Player 2 has two options: A and B. The expected payoff for Player 2 when choosing A is 15q + 0(1-q), and the expected payoff for Player 2 when choosing B is 0q + 8(1-q). Since Player 2 is indifferent between A and B, these expected payoffs must be equal:
15q + 0(1-q) = 0q + 8(1-q)
Simplifying this equation, we get:
15q = 8(1-q)
Now, let's solve for q:
15q = 8 - 8q 23q = 8 q = 8/23 q ≈ 0.35
Therefore, the value of q in the mixed-strategy Nash equilibrium of this game is approximately 0.35.
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