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Football Rules

My nephew, Paul, loves watching football. Until quite recently, I found this hard to understand. Where is the fun in sitting through a 90-minute match in which close to zero entertainment is offered, hardly any goals are scored (2.5 goals per match, on average), and random mistakes by referees can have a disproportional influence on the outcome of the match? Even watching Bridget Jones’s Baby, where random mistakes at least give rise to new life, would seem more attractive to me.

But nowadays, when nephew Paul invites me to watch a football match, I happily accept the invitation. The reason is that I have learned to appreciate football as a rich and openly accessible test bed for social sciences. Take penalty kicks. Penalty kicks offer a great opportunity to test predictions from game theory. Why? First of all, penalty kicks boil down to a well-defined game between two players: the football player taking the penalty and the goalkeeper. Their actions (the corner they choose) and the outcome (goal or no goal) are clearly observable. Second, the interaction between the player and the goalkeeper is close enough to a “simultaneous-move” game: the player and the goalkeeper must move at the same time because the ball only takes about 0.3 seconds to travel the distance between the penalty mark and the goal line. Third, both the player and the goalkeeper have great incentives to perform well during penalty kicks. A goal scored in a penalty kick has a high chance of being decisive for the outcome of the match—keeping in mind that only about 2.5 goals are scored in an average match. Also, the financial stakes in football are several orders of magnitude higher than in a typical lab experiment (where participants can earn about 10 euros an hour).

Ignacio Palacios-Huerta, a professor at the London School of Economics, is as thrilled as I am by penalty kicks. He studied 1,417 penalty kicks from professional football matches that took place between September 1995 and June 2000 in Spain, Italy, England, and other countries. Palacios-Huerta modeled a penalty kick as a game in which the players have two possible actions: The player can only choose to shoot to the left or to the right and the goalkeeper must choose whether to dive to the left or to the right (from the player’s viewpoint). The table below gives an example of the resulting game.

In the table, the payoffs for the goalkeeper are 1 if he stops the ball and 0 if the player scores. For the player, it is exactly the other way around. Just for illustrative purposes, I have chosen the numbers in the table assuming that if both players choose the same side, the goalkeeper will always stop the ball, while if they choose opposite sides, the shot will always result in a goal.

Game theory predicts the following. First, both players randomize in that they do not choose one corner with certainty. In the table above, both players optimally choose either corner with a 50 percent probability. True randomization implies that a player’s choice does not depend on his choices in previous penalty kicks. Second, for players to be willing to randomize between corners, the probability of a player scoring a goal (or a goalkeeper preventing it) is independent of the corner chosen. Palacios-Huerta found support for both hypotheses in his dataset. The predictions by game theory seem to hold water.

Palacios-Huerta’s analysis is not without criticism, though. For example, researchers from Israel observed that game theory does not perform so well if “stay in the goal’s center” is added to the goalkeeper’s strategy set. Perhaps surprisingly, they observe that goalkeepers are better off staying in the goal’s center instead of jumping right or left. This is inconsistent with game theory which is built on the assumption that a player optimally responds to the other player’s strategy. The authors then argue that the goalkeeper may suffer from “action bias”: he feels worse when a goal is scored when staying in the center (inaction) than when jumping (action).

People may also suffer from action bias outside the football field. For instance, it would be hard for policymakers to not “do something” in a recession. Even if the intervention does not work, at least they can say that they tried to improve the situation. In the next elections, perhaps you should look for a candidate who voted against such interventions. For the same reason, company owners may want to protect managers who decide to retain the status quo even when things turn bad for the company. On a more personal level, the next time you find yourself lost in a Malaysian rainforest with your GPS tracker out of battery, zero bars on your cell phone, and no trail of bread crumbs marking your route, you are well advised to remain where you are (inaction) instead of moving deeper into the rainforest (action) to maximize the probability that the rescue team will find you.

All in all, I now happily join my nephew Paul to watch football matches. Not only do they offer me an excuse not to watch Bridget Jones’s Baby, they also teach me a lot about game theory and psychology.


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