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The World’s Favorite Game and a Nash Equilibrium

An exciting, nail-biting 90 minutes of soccer comes to an end, and no team has scored; now, the shootout – perhaps even more exciting than the game itself.  These penalty kicks provide us with another example of game theory.  The kicker must decide to kick left, right, or in the middle of the goal.  The goalie must make a simultaneous decision in an attempt to block the shot – he can also play to the right, left, or middle.  In this game, each player has 3 strategies, and there are 2 possible outcomes.  Assuming the kicker and goalie are at similar ability levels, we can predict that if the kicker and goalie choose different sides, there is a goal (+1 for the kicker, 0 for the goalie).  If they choose to play the same direction, there is no goal (0 for the kicker, -1 for the goalie).

Daniel Altman explains the importance of “mixing up” one’s strategies in this game.  If a player always chooses the same side, he will not be successful in this game, as he is too predictable.  Thus, there cannot be a dominant strategy for each player in the shootout game.  Altman looks to statistics for insight into the game.  He explains that each player must mix up his strategies so that his chance of success is the same for each of his strategies.  Altman’s article provides us with the following data on penalty kicks in the French and Italian leagues over a few years:

Kickers scored 77 percent of the time when they went to their natural side (right foot kicking to left side, for example), 70 percent of the time on the opposite side and 81 percent of the time in the middle — pretty similar success rates, or at least within a statistical margin of error. Goalkeepers are even closer in their success rates. When they guessed center or the opposite side, kickers failed to score 27 percent of the time. When they guessed the kicker’s natural side, there was no goal 24 percent of the time.

This exemplifies the mixed Nash equilibrium.  Each player is indifferent between his three strategies, as his success rate is approximately equal for each of his strategies.  Altman also states that this “randomness” and indifference between strategies is observed for individual players, as well as players collectively.  This concept of a mixed Nash equilibrium contributes much to the excitement of a shootout, making it random and unpredictable.



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October 2012