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Game Theory: Penalty Kicks and Shootouts

Soccer is a passion of mine; I was intrigued by the idea that the game I had been playing my whole could be applied/explained through game theory economics.

My team had made it to the state championship. After ninety minutes of regular play and twenty minutes of over time, the game remained tied up, and the fate of the team depended on a shootout – a total of ten penalty kicks (five from each team). Penalty kicks involve just two players (the kicker and the goalie) and thus allows for a test of the economic game theory. The kicker should attempt to maximize his chance of scoring, while the goalie should try to maximize his chance of saving. Each player has to decide which directing to kick/jump, either to the left, the center, or to the right. In this game one person wins and one person loses. Therefore, there is no pure strategy Nash Equilibrium – the optimal strategy is randomization. The goalies strategy will depend on the kicker’s past kicks, while the kickers strategy is independent of the goalie. It is important to note that kickers tend to be more successful when they are kicking to their natural side (their natural side is usually the left, because most players kick with their right foot).

Both players want to maximize their utility (their value/happiness/benefits) . In order for the goalie to maximize his utility, he is going to want to jump in the same direction that the kicker kicks: (left, left), (center, center), or (right, right). This is because if he jump in the same direction as the kicker kicks the probability that he will make the save is much higher than if he were to jump in the wrong direction. In order for the kicker to maximize his utility he is going to want to kick in the opposite direction that goalie jumps: (left, right), or (right, left). Again, this because if he kicks in the opposite direction than the goalie jumps, the probability that he will score is increased. The kicker will still have some utility if the goalie doesn’t go in the complete opposite direction. For example, if the goal jumps left and the kicker kicks to the center, he will still have some utility because there is still a chance he will score, but that probability is smaller than if the goal jumped in the complete wrong direction. This applies to (center, left), (left, center), (right, center), and (center, right). The same is true for the goalie; he will always have some positive utility when he remains in the center of the net, because he won’t jump in the complete wrong direction, leaving him some chance of saving the kick.

After all these years of going through penalty kicks I never realized I was playing a mental game of economics, rather than simply playing soccer.

The following is a link to an interview with economist/soccer fan about the strategies behind penalty kicks and how they apply to game theory economics:

http://www.gelfmagazine.com/archives/the_game_theory_of_penalty_kicks.php

This is another link (with diagrams and a video) that helps explain how game theory economics apply to penalty kick:

http://philosophicaldisquisitions.blogspot.com/2011/06/game-theory-part-6-penalty-kick-game.html

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