It may surprise casual observers of the beautiful game to discover that one of the heroes of English football this year was a German. Jens Lehmann, the Arsenal goalkeeper, saved a last-minute penalty against the Spanish side, Villarreal, thus earning Arsenal a coveted spot in the Champions League final.
Penalties pit the goalkeeper against a lone striker in a mentally demanding contest. Once the penalty-taker strikes the ball, it takes 0.3 seconds to hit the back of the net - unless the goalkeeper can somehow get his body in the way. That is simply not enough time for the keeper to pick out the trajectory of the ball and intercept it. He must guess where the striker will shoot and move just as the ball is being struck. If Lehmann had not guessed correctly, he wouldn’t have been a hero.
Both striker and keeper must make subtle decisions. Let’s say a right-footed striker always shoots to the right. The keeper will always anticipate the shot and the striker would be better off occasionally shooting to the left - even with a weaker shot it is best to shoot where the goalie isn’t. In contrast, if the striker chooses a side by tossing a coin, the keeper will always dive to the striker’s left: since they can’t guess where the ball will go, best to go where the shot will be weak if it does come. But then the striker should start favouring his stronger side again.
So what to do? The answer comes from a wartime collaboration between economist Oskar Morgenstern and mathematician John von Neumann. They produced a “theory of games” which, applied to this problem, says the strategy of the striker and the keeper cannot be predicted. The striker might shoot to the right two times out of three, but we cannot then conclude that it will have to be to the left next time.
Von Neumann and Morgenstern also say that each choice of shot should be equally likely to succeed, weighing up the advantage of shooting to the stronger side against the disadvantage of being too predictable. If shots to the right score three-quarters of the time and shots to the left score half the time, you should be shooting to the right more often. As you do, the goalkeeper will respond: shots to the right will become less successful and those to the left more successful. It might sound strange that at this point any choice will do, but it is analogous to saying that if you are at the summit of the mountain, no direction is up.
Von Neumann and Morgenstern did not produce game theory to help footballers: they believed it could illuminate anything from pay negotiations to waging war. The trouble is that for these applications the wrinkles of reality always obscure whether ordinary people actually follow the strategies that game theory predicts they should. Yet penalty taking is different. The objective is simple, the variables easy to observe, and the results immediate.
Ignacio Palacios-Huerta, an economist at Brown University, found that individual strikers and keepers were, in fact, master strategists. Out of 42 top players that Palacios-Huerta studied, only three departed from game theory recommendations. Professionals such as the Brazilian Rivaldo and Italy’s goalkeeper Gianluigi Buffon are apparently superb economists: their strategies are absolutely unpredictable and, as the theory demands, they are equally successful no matter what they do, indicating that they have found the perfect balance between the different options. These geniuses do not just think with their feet.