Soccer Penalty Kick as a Three-strategy Game
Both the lecture and the book Networks, Crowds, and Markets use penalty kick in soccer as an example of mixed strategy Nash equilibrium in a two-player, two-strategy game. The book computes the probabilities of pure strategies in mixed strategy equilibrium from empirical data, and finds out that the predicted outcome from mixed strategies is almost the same with the real penalty kicks. The book ignores the possibility of kicking up the center or remaining in the center, as such cases are “very rare”. However, as a study by Steve Levitt and others shows, the possibilities of kicking or defending the middle are not negligible in real life. Soccer penalty kick is actually a two-player, three-strategy game, and can also be modeled using mixed-strategy Nash equilibrium.
Some earlier posts, like “Left, Middle or Right? Game Theory of Soccer Penalty Kicks”, have also discussed penalty kick as a three-strategy game and mentioned Levitt’s study. However, a mixed strategy equilibrium hasn’t been modeled using Levitt’s data or compared to real life outcomes in posts before. From Levitt’s data on percentage of shots in which a goal is scored, we can construct the following matrix using the same method from book:
| Kicker | ||||
| L | M | R | ||
| L | -0.63, 0.63 | -0.81, 0.81 | -0.90, 0.90 | |
| Goalie | M | -1, 1 | 0, 0 | -1, 1 |
| R | -0.94, 0.94 | -0.89, 0.89 | -0.44, 0.44 |
As the matrix shows, a kicker even has different chances of scoring if the goalie dives in the correct direction. If both the goalie and kicker go left, the percentage of scoring is 0.63, higher than that if they both go right, which is 0.44; when the goalie dives in the opposite direction, the kicker has higher chances of scoring if the kicker chooses left, 0.94 over right, 0.90. This result comes from the fact that kickers are mostly right-footed, and their chances of scoring left are higher. When the goalie defends the middle, the situation is quite different, as the kicker can always score if he kicks left or right, but fail to score if he kicks in the middle.
Using the principle of indifference, we can calculate that in a mixed strategy equilibrium, for a goalie, the possibility of choosing L should be 0.60, choosing M should be 0.09, and choosing R should be 0.31; for a kicker, the possibility of choosing L should be 0.43, choosing M should be 0.24, and choosing R should be 0.33. From data in Levitt’s study for real shots taken, the real chances for a goalie to defend left, middle and right are 0.57, 0.02 and 0.41 respectively, while the possibilities for a kicker are 0.45, 0.17 and 0.38 respectively. The result is quite interesting. The real outcomes to kick right or defend left are similar to the predicted outcome in mixed strategy equilibrium (0.60 vs 0.57, and 0.43 vs 0.45, former is the predicted outcome while the latter is real). However, the possibilities of choosing middle and right are a bit off. Both the goalie and the kicker choose middle less than they should in Nash equilibrium (0.09 vs 0.02, and 0.24 vs 0.17), and both of them choose right more than they should in the equilibrium (0.31 vs 0.41, 0.33 vs 0.38).
There might be many causes behind the difference between predicted outcome from the model and the real outcome, like psychological and tactical reasons. For example, a goalie defending middle would be seen as not moving at all, similar to a give-up gesture, so goalies would be less likely to choose the middle. The real life situation is more complicated than an economic model.
Link:http://pricetheory.uchicago.edu/levitt/Papers/ChiapporiGrosecloseLevitt2002.pdf