EXAMPLE 6 The Penalty Shootout Revisited

In Table 15.7, we add probabilities, which we explain next, to Table 15.4. The goalie should use a mixed strategy , where represents the probability that the goalie guesses left (i.e., dives to the left) and represents the probability that the goalie guesses right. Notice that , as required, assuring that the goalie chooses a side! The probabilities and (where ) are indicated below the game matrix and under the corresponding strategies, and , for the goalie. If the goalie plays a mixed strategy against the two pure strategies, and , for the kicker, then the respective expected values for the kicker are

As in the second matching game, the solution to this game occurs at the intersection of the two lines given by and . Setting the equations of these lines equal to each other so that yields or , giving .

Thus, the goalie should use the optimal mixed strategy by choosing with probability and with probability . This choice limits the success rate of the kicker to 0.55, or 55%, which is the value of the game. The value of the game is statistical in nature in that it means that if the goalie follows the prescribed strategy, then the kicker will, on average, successfully score 55% of the time. Remember that this doesn’t provide information about what will happen on any one particular kick.

Assume that the kicker uses a mixed strategy , as indicated to the right of the game matrix in Table 15.7. This means that the kicker kicks the ball to the goalie’s left with probability and kicks the ball to the goalie’s right with probability . This mixed strategy, when played against the goalie’s pure strategies, and , results in the following expected values: