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Penalty Shoot-Outs: How to Take the Perfect Penalty

Published after the 2009 Champions League Final, during which Chelsea Football Club lost to Manchester United in a penalty shoot-out, an article from The Sunday Telegraph analyzes the penalty kick scenario in the context of game theory. Indeed, sports economist Stefan Szymanski and journalist Simon Kuper indicate that the player striking the soccer ball and the goalkeeper attempting to make a save constitute the two actors in a two-player game. With the actors, they characterize the directions in which the shooter aims and the goalkeeper dives as the two choices each actor faces, thus creating the conditions of a matching pennies game.

Szymanski and Kuper point out that there is no pure strategy in a penalty kick situation. That is, if a kicker always chooses to direct the ball in one direction, the goalkeeper would always know where he should dive in order to make a save (a goalkeeper diving only in one direction would also create a similar situation). To point the similarities between a matching pennies game and a penalty kick scenario, the researchers point to empirical evidence in their study, pointing out that “there are no kickers in our [their] sample with at least four kicks who always kick in one direction.”

Consistent with the material we have covered in class, the actors in the penalty shootout do not have a pure dominant strategies and thus must pursue mixed strategies. Syzmanski and Kuper further elaborate on the matching pennies model by indicating that a kicker has a higher accuracy rate of scoring if he directs his shot to his natural side. For example, right-footed players favor shooting to the right of the keeper and thus would score more often by aiming in that direction.

The researchers draw from the database of 1,417 penalties collected by former University of Chicago graduate student Ignacio Palacios Huerta, and incorporate the phenomenon of natural sides into the calculation of the game’s mixed strategies. They report that the optimal strategy of the kicker would be to aim 61.5 percent of his kicks to his natural side and 38.5 percent of his kicks to his unnatural side; the keeper’s ideal strategy would be to dive to the kicker’s natural side 58 percent of the time and dive to the opposite direction 42 percent of the time. Drawing from actual figures accrued by Palacios Huerta and Steven Levitt, author of Freakonomics, the researchers find that in over 95 percent of cases, the decisions of real players and the optimal mixed strategies were virtually indistinguishable.

Such a finding, in addition to the conclusion that soccer players are able to construct truly random sequences in their penalty kick decisions, indicates that the behavior of professional soccer players matches up with the matching pennies model. More broadly, this finding supports the notion that the game theory models we learned in class can be used to accurately model real-world situations. Utilizing game theory to point out the absence of pure strategy Nash Equilbria enables theorists to apply the notion of mixed strategies, a concept covered in class, to determine mixed strategies. This article, moreover, points to the possibility of extending the models in the course by introducing outside information. By incorporating the phenomenon of natural and unnatural sides, we can understand how the payoffs magnitudes can be calculated to better model real-world situations.



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