EXAMPLE 3 A Penalty Kick Shootout

After a tie overtime period in a FIFA World Cup soccer match, penalty shootouts are used to determine which team advances to the next round. A penalty shootout consists of five kicks for each team, in alternating order. If at the end of the five kicks, the teams are still tied, then the two teams continue to alternate penalty kicks until the tie is broken—the team that scores when the other doesn’t advances to the next round. Penalty shootouts have even decided the World Cup winner, as when Italy defeated France 5-3 in a shootout in 2006.

In soccer, the goal is 8 feet tall by 24 feet wide. A kick in a penalty shootout is taken at 36 feet from the goal, and the ball travels upwards of 100 miles per hour. Because of the wide goal and the distance and speed that the ball travels, the goalie typically guesses either left or right. Likewise, the kicker decides on either kicking the ball to the goalie’s left or to the goalie’s right. Players vary their kicking techniques; if they don’t, then a well-prepared goalie can learn players’ tendencies to kick to one side or the other. Indeed, Petr Cech saved all five penalty kicks in a penalty shootout by guessing the correct direction, thereby helping Chelsea Football Club of England defeat Bayern Munich of Germany to win the 2012 UEFA European Champions League. Cech had prepared by watching every penalty kick the Bayern Munich players attempted since 2007.

Assume that a particular left-footed kicker can kick the ball either to the goalie’s left or to the goalie’s right and so has two strategies: goalie’s left (denoted by ) or goalie’s right (). The goalie decides between diving to the left or to the right, and likewise has two strategies: left (denoted by ) and right (). Being left-footed, the kicker’s strong side is to kick to the goalie’s left, resulting in a more powerful, but less accurate, shot. The kicker is more accurate when kicking the ball to the right, which results in a less forceful shot. Assume that the kicker’s success rates for the different strategies by the players are given below and that both players know these rates:

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The kicker’s success rate is the number of goals scored divided by the number of attempts. Hence, 0.2 means that the kicker succeeds 20% of the time, or 2 out of every 10 shots, on average.

This game is summarized in Table 15.4. The payoffs in the table, or matrix (called a payoff matrix), represent the likelihood that the kicker is successful. Because the goalie wishes to minimize this likelihood, the game is zero sum. We see from the right-hand column in the table that the kicker’s maximin is 0.2, which is realized when the kicker chooses . Always going with the power side, the kicker is assured of successfully scoring at least 20% of the time—hardly a successful outcome.

We see from the bottom row of the table that the goalie’s minimax is 0.9, which is obtained by guessing right (). The maximin and minimax are not equal, which means that the penalty shootout game does not have a saddlepoint. By varying his or her strategies, the kicker’s success rate should be somewhere between 0.2 and 0.9. In a sense, the strategies are to determine how much of the difference is split between the two players. The goalie wishes to push down the success rate as far as possible, whereas the kicker wants to raise it from 0.2 as much as possible.