Final answer:
The expected payoff for player A when using strategy I with the given probabilities is 0.625. To find this, the payoff values from the matrix are multiplied by corresponding probabilities for the strategies, and then the products are summed up.
Step-by-step explanation:
To compute the expected payoff for each strategy in a game with a given payoff matrix, we use the probabilities associated with each strategy for both players. In this problem, we are given probabilities for strategy I for player A as P=[0 1/2 2/3] and for player B as Q= [ 0 ] [1/4] [3/4]. The payoff matrix A is given as:
A= [-3 1 1]
[0 -2 0]
[-1 0 2]
We calculate the expected payoff for player A when using strategy I by multiplying each payoff with the corresponding probabilities of P and Q, and then summing these products. Using the given probabilities, we can proceed as follows:
Expected payoff for Player A = (0 x -3 x 0) + (1/2 x 1 x 1/4) + (2/3 x 1 x 3/4)
Since the first value is multiplied by zero, it does not contribute to the sum:
Expected payoff for Player A = (1/2 x 1/4) + (2/3 x 3/4)
Expected payoff for Player A = (1/8) + (1/2)
Expected payoff for Player A = 0.125 + 0.5
Expected payoff for Player A = 0.625
Therefore, the expected payoff for player A when they use strategy I is 0.625. Similarly, we would compute the expected payoff for player B and then determine which combination of strategies is most advantageous for each player.