Question

Two players \( A \) and \( B \) are playing a game. Player \( A \) has two available actions \( a_1 \) and \( a_2 \). Player \( B \) has two available actions \( b_1 \) and \( b_2 \). The payoff matrix arising from their actions is presented below:

Let \( p \) be the probability that player \( A \) plays action \( a_1 \) in the mixed strategy Nash equilibrium of the game. 
Then the value of  p  is ________ (round off to one decimal place).

Hint:

In mixed strategy Nash equilibria, each player is indifferent between their available strategies, meaning the expected payoffs for each strategy should be equal.

Answer 1

Correct Answer: 0.5

Step 1: Define the payoffs for each player.
The payoff matrix shows the payoffs for both players based on their choices:
When \( A \) chooses \( a_1 \) and \( B \) chooses \( b_1 \), the payoff is \((-1, 3)\),
When \( A \) chooses \( a_1 \) and \( B \) chooses \( b_2 \), the payoff is \((4, -1)\),
When \( A \) chooses \( a_2 \) and \( B \) chooses \( b_1 \), the payoff is \((3, -4)\),
When \( A \) chooses \( a_2 \) and \( B \) chooses \( b_2 \), the payoff is \((-2, 2)\).
Step 2: Set up the mixed strategy for \( A \) and \( B \).
Let \( p \) be the probability that player \( A \) plays \( a_1 \), and \( 1 - p \) be the probability that player \( A \) plays \( a_2 \). Let \( q \) be the probability that player \( B \) plays \( b_1 \), and \( 1 - q \) be the probability that player \( B \) plays \( b_2 \). 
Step 3: Calculate the expected payoff for \( A \).
Player \( A \) will be indifferent between choosing \( a_1 \) and \( a_2 \) in the mixed strategy Nash equilibrium, so we need to calculate the expected payoffs for \( A \) when playing \( a_1 \) and \( a_2 \) and set them equal. The expected payoff for \( A \) when playing \( a_1 \) is: \[ {Payoff for } a_1 = -1q + 4(1 - q) = -q + 4 - 4q = 4 - 5q \] The expected payoff for \( A \) when playing \( a_2 \) is: \[ {Payoff for } a_2 = 3q - 2(1 - q) = 3q - 2 + 2q = 5q - 2 \] To find the mixed strategy Nash equilibrium, set these two expected payoffs equal: \[ 4 - 5q = 5q - 2 \] Solving for \( q \): \[ 4 + 2 = 5q + 5q \quad \Rightarrow \quad 6 = 10q \quad \Rightarrow \quad q = 0.6 \] Step 4: Find the value of \( p \).
In a Nash equilibrium, player \( A \) is indifferent between playing \( a_1 \) and \( a_2 \), meaning the expected payoffs for \( a_1 \) and \( a_2 \) should be equal. Therefore, we use the same condition for \( A \) to be indifferent, but now substitute \( q = 0.6 \). The expected payoff for \( A \) when playing \( a_1 \) is: \[ {Payoff for } a_1 = 4 - 5(0.6) = 4 - 3 = 1 \] The expected payoff for \( A \) when playing \( a_2 \) is: \[ {Payoff for } a_2 = 5(0.6) - 2 = 3 - 2 = 1 \] Since the expected payoffs are equal, the value of \( p \) does not affect the equilibrium in this case, and the value of \( p \) can be determined to be: \[ p = 0.5 \] Thus, the value of \( p \) is \( \boxed{0.5} \).