World Cup Game Theory

Now the World Cup starts in earnest. The mini-leagues of the group stage are over, and the first knockout match takes place today between Germany and Sweden. Soccer being a famously low-scoring game, such matches often finish in a draw and must be decided by penalty shootouts—a competitive form at which my own team, England, has a particularly harrowing record. Perhaps England's players should study a little more economics.

In soccer, penalty kicks 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. A keeper who does not guess correctly has no chance.

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—because 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 he 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 favoring his stronger side again.