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).

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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).

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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} $.

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