Let’s compute players’ expected payoffs if the wife chooses the mixed strategy (1/9, 8/9) and the…

Let’s compute players’ expected payoffs if the wife chooses the mixed strategy (1/9, 8/9) and the husband (4/5, 1/5) in the Battle of the Sexes. The wife’s expected payoff is

To understand Equation 8.4, it is helpful to review the concept of expected value from Chapter 2. Equation (2.176) indicates that an expected value of a random variable equalsthe sum over all outcomes of the probability of the outcome multiplied by the value of therandom variable in that outcome. In the Battle of the Sexes, there are four outcomes,corresponding to the four boxes in Table 8.3. Since players randomize independently, theprobability of reaching a particular box equals the product of the probabilities that eachplayer plays the strategy leading to that box. So, for example, the probability of (boxing,ballet)—that is, the wife plays boxing and the husband plays ballet—equals (8/9)x(4/5). The probabilities of the four outcomes are multiplied by the value of the relevant randomvariable (in this case, player 1’s payoff) in each outcome.Next we compute the wife’s expected payoff if she plays the pure strategy of going toballet [the same as the mixed strategy (1, 0)] and the husband continues to play the mixedstrategy (4/5, 1/5). Now there are only two relevant outcomes, given by the two boxes in the row in which the wife plays ballet. The probabilities of the two outcomes are given by theprobabilities in the husband’s mixed strategy. Therefore,

Finally, we will compute the general expression for the wife’s expected payoff when she plays mixed strategy (w, 1 – w) and the husband plays (h, 1 – h): if the wife plays ballet with probability w and the husband with probability h, then