Game theory. Kickers could score more goals if they kept track of the pattern of goalkeepers' dives.

Game theory. Kickers could score more goals if they kept track of the pattern of goalkeepers' dives.

RIA Novosti/Alamy

Gambler's fallacy trips up goalies

You see it happen in all the major soccer tournaments: As a penalty shot is being taken, the goalkeeper leaps in one direction and the striker deftly kicks the ball into the opposite corner. Such fruitless dives are inevitable because the goalie has no time to see which way the kick is going—he has to start the dive before then if he is to have any chance of saving it. The best strategy would be to choose randomly, but, according to researchers in the United Kingdom, goalkeepers aren't very good at that. Instead, they fall victim to a misjudgment called the gambler's fallacy. Strikers taking penalties should take note, the researchers say, because they could score more goals against goalies that make this mistake.

Penalty kicks, as the name implies, are normally a punishment for a serious infringement of the rules, such as a deliberate foul on an attacking player in the goal area. The ball is placed on a spot 11 meters from the goal. A player from the team that was fouled runs up and has a free shot at goal. In knockout tournaments, such as the World Cup, if the score is tied at the end of a match, the winner may be decided by a penalty shootout. The two teams take turns shooting at goal—using a different player for each kick—and the winner is the side with the highest score after five shots each. (If they’re still tied, they keep shooting.) 

Previous studies on penalty kicks have indicated that goalkeepers must make up their minds which way to move before they see the ball fly off the kicker's foot, either by watching the body movements of the kicker to anticipate his kick or simply by committing to one direction or the other. Cognitive neuroscientist Patrick Haggard of University College London says that because a kicker may try to disguise his true intentions, by and large the goalkeeper's decision is a simple guess. Game theory—a branch of mathematics that deals with competitive systems such as sport, economics, and ecology—says the best strategy is to decide randomly, as any regularity could be exploited by the kicker.

Haggard and his Ph.D. student Erman Misirlisoy looked at keepers' dive patterns in all 37 penalty shootouts in World Cup and European Cup matches between 1976 and 2012. They noticed that, after a ball had been aimed to one side, keepers were more likely to dive the other way for the next ball, and the odds increased if a second or third ball had gone to the same side. For example, after three balls had been directed to one side, goalkeepers directed more than 70% of their dives for the fourth ball the opposite way, the team reports online today in Current Biology. This is a classic statistical error called the gambler's fallacy. If two outcomes are equally likely, a long run of one outcome makes people instinctively expect the other in the next trial.

If kickers anticipated this behavior, they could score more penalties, Haggard says. But they don't seem to do so. When the researchers looked at patterns in the directions of the incoming shots, they appeared entirely random. One reason for this, they suggest, may be that in a penalty shootout successive kicks are taken by different players, but the same goalkeeper faces them all. So perhaps players, waiting for their turn at the penalty spot, should take note of which way their teammates shoot before them.

Haggard says that the gambler's fallacy may have a role in other contests on and off the sports field, such as in business decisions, but the penalty shootout is an unusually pure example of where “the gambler's fallacy is one of the few things that could allow one side to have an advantage over the other.”

Sports psychologist Michael Bar-Eli of Ben-Gurion University of the Negev, Beer-Sheva, in Israel, who was not involved in the research, describes the paper as “very interesting” but says that, if a kicker does his job properly, there is little a goalkeeper can do anyway. In two studies, one by his own group, not one penalty kicked into the upper third of the goal was saved. “Everybody talks about 'a game' in a game-theoretical sense, which is going on between the shooter and the goalkeeper,” he explains. “But the shooter can take the goalkeeper out of the game by shooting to the two upper corners. There, the chances of goalkeepers are literally 0%—they are simply unable to reach the ball.”