Consider the Battle of the Sexes game:

Suppose both players use mixed strategies for this game. Jim chooses wrestling with probability 0.9, and Joan chooses wrestling with probability 0.5. What are the expected payoffs for the players?
A) The expected payoffs are 0.95 for Joan and 0.55 for Jim
B) The expected payoffs are 0.55 for Joan and 0.95 for Jim
C) The expected payoffs are 0.95 for both players
D) The expected payoffs are 0.55 for both players

Question

Consider the Battle of the Sexes game:

Suppose both players use mixed strategies for this game. Jim chooses wrestling with probability 0.9, and Joan chooses wrestling with probability 0.5. What are the expected payoffs for the players?
A) The expected payoffs are 0.95 for Joan and 0.55 for Jim 
B) The expected payoffs are 0.55 for Joan and 0.95 for Jim 
C) The expected payoffs are 0.95 for both players 
D) The expected payoffs are 0.55 for both players

Quizplus · Accepted Answer

To find the expected payoff for Jim, we must calculate the payoffs for each possible outcome (i.e. Jim chooses wrestling, Joan chooses wrestling; Jim chooses wrestling, Joan chooses dancing; Jim chooses dancing, Joan chooses wrestling; Jim chooses dancing, Joan chooses dancing) and weight them by the probabilities of each outcome occurring. Using the payoffs from the game matrix, we get:

Expected payoff for Jim = (0.9)(2) + (0.1)(0) + (0)(3) + (0)(1) = 1.8
Similarly, we can find the expected payoff for Joan:

Expected payoff for Joan = (0.5)(3) + (0.5)(1) + (0)(2) + (0)(0) = 2

Therefore, the expected payoffs are 0.95 for Joan and 0.55 for Jim.

Consider the Battle of the Sexes Game