One of the most common errors to plague all sorts of statistical analyses is the assumption that uncertain future events occur according to a Gaussian (or “normal”) distribution. While this assumption can make our analysis simpler by letting us take advantage of the familiar bell curve, there are broad classes of phenomena for which the normal distribution does not reflect the real probabilities we observe. A new example of this type of deviation is the scoring of goals in soccer matches. A naive assumption would predict the number of goals scored to be spread around a mean in a typical bell curve fashion. In reality, each goal scored seems to increase the odds of another goal being scored–a phenomenon referred to as “football fever”, where the scoring of goals takes on a “contagious” nature.

According to a news report in the June 15, 2006 Nature, it has been established mathematically that soccer goals are contagious, statistically speaking: scoring one goal increases the probability that your team will score more. Michael Hopkin, who write the piece, calls this “one of soccer’s classic clichés,” and attributes the result to Martin Weigel (Herriot-Watt University, Edinburgh) and his colleagues Elmar Bittner, Andreas Nussbaumer and Wolfhard Janke, all at Leipzig University. The four have posted a preprint on arXiv.org with the title “Football fever: goal distributions and non-Gaussian statistics.” As they put it: “modifying the Bernoulli random process underlying the Poissonian model to include a simple component of self-affirmation seems to describe the data surprisingly well and allows to understand the observed deviation from Gaussian statistics.” They analyzed “historical football score data from many leagues in Europe as well as from international tournaments, including data from all past tournaments of the ‘FIFA World Cup’ series” and concluded: “The best fits are found for models where each extra goal encourages a team even more than the previous one: a true sign of football fever.”

Read more:

• Math at the World Cup
• Football fever: goal distributions and non-Guassian statistics

In today’s Slate, Tim Harford writes about using game theory to analyze the optimal strategies for strikers and goalies in a penalty kick situation.

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.

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 mathematically analyzed situations of strategic interaction—that is, any situation where participants have to take into account the other side’s responses. A free throw in basketball is not a strategic interaction, but a soccer penalty is. A “game” is a mathematical description of how all the possible payoffs to the different players vary with their different strategies—so if the goalkeeper jumps to his left while the striker shoots to the keeper’s right, the striker will get a high payoff and the goalkeeper will get a low one.

In fact, at least one economist has recently studied the strategies players use in penalty kicks and found them to be nearly perfect in a game-theoretic sense. In this complex subgame of anticipation, feigned signals, and reaction time, top soccer players are extremely skilled at maximizing their chances of scoring a goal.

Ignacio Palacios-Huerta, an economist at Brown University, found that individual strikers and keepers were, in fact, master strategists. Out of 42 top players whom Palacios-Huerta studied, only three departed from game theory’s recommendations—in retrospect, they succeeded more often on one side than the other and would have been better altering the balance between their strategies. Professionals such as the French superstar Zinédine Zidane 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 among the different options. These geniuses do not just think with their feet.