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Penalty Kicks and Game Theory in the 2008 Champions League

In class, we recently discussed the concept of applying game theory to soccer penalty kicks, as an example of mixed Nash equilibria: examples where the choices made by each player must be randomized as not to be exploited by others.  In penalty kicks, the direction a goalie jumps is crucial towards winning, for if the player kicks in the opposite direction of their jump they have no chance of saving the shot.  Thus, both the goalie and player must try to predict the direction the other will go, leading to more random choices that make predictions harder.

An interesting example of this game occurred in the 2008 Champions League Game between Manchester United and Chelsea.  The game was tied 1-1, leading to a series of penalty kicks that would decide the game.  What is so interesting about this case, however, is that United had done research prior, utilizing game theory.  Prior to the game, management in Chelsea had collaborated with Basque economist Ignacio Palacios-Huerta, who had published numerous papers on just this topic: game theory and penalty kicks.  Palacios-Huerta informed them that Edwin Van de Sar, the United goalkeeper they would be shooting on, tended to jump in the natural direction of the kicker, for example, if the shooter kicked with his right foot, Van de Sar would jump right.  Armed with this knowledge, Chelsea hoped to subvert the mixed game, and gain an advantage on United they could use to win.

United and Chelsea traded shots, and Chelsea followed this exact advice, shooting on the opposite sides, and into the corners.  While Chelsea was somewhat successful, a punted shot from John Terry put United at the advantage. As Chelsea striker Nicholas Anelka took the pitch for the deciding shot of the game, a very interesting game developed.  As Anelka prepared his shot, Van de Sar called him out. “You’re going to put it there, aren’t you? I’m onto you guys and your little plan. Not this time son, not this time,” he yelled, pointing the left corner, on Anelka’s opposite shooting side.  Anelka now faced a very interesting decision: should he shoot left to continue with the plan to exploit Van de Sar’s weakness, now that he knew that they knew?  Or should he account for this, and go on the right?

In the end, Anelka shot right, and Van de Sar blocked it, winning the game and making history.  While this situation was truly amazing for soccer fans everywhere, it also offers a very interesting look into game theory: how should you act when your opponent knows what you are planning on doing?  And from the perspective of Van de Sar, should you inform your opponent you know their plan, and can you use that to our advantage?  

https://www.sportskeeda.com/football/a-closer-look-at-the-2008-champions-league-final-penalty-shoot-out/2

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