Mixed Strategies in Soccer
In class, we considered simple versions of mixed strategy games applied to sports. In many competitive sports, players try to predict the actions of another. Whether baseball pitchers are choosing the type of pitch, tennis players deciding where to serve, or football offenses deciding between running and passing plays, the key to success is unpredictability. The mixed strategy Nash equilibrium offers a concrete way that unpredictability is achieved among rational players.
In this note, we consider soccer penalty kicks. A penalty kick is a game that looks a lot like the "odds-evens" game we played in class. From 12 yards away, a kicker strikes the ball at speeds that can exceed 100 miles per hour, reaching the goal in a quarter of a second. Because this is faster than a human can react, the goalie cannot wait for the kick to decide how to react. Effectively, the goalie must choose a direction and dive simultaneously with the kick. If the goalie 'matches' the kicker's direction, the chance of a save is higher. If they guess wrong, a goal is almost certain.
Using real data, we will calculate the mixed strategy equilibrium of the penalty kick game and compare this to observed behavior of professional goalies and kickers. Economist Ignacio Palacios-Huerta analyzed 1,417 penalty kicks from five years of professional soccer matches among European clubs.
The success rates of penalty kickers given the decision by both the goalie and the kicker to kick or dive to the left or the right are as follows:
| Goalie | |||
| Left | Right | ||
| Kicker | Left | 58% | 95% |
| Right | 93% | 70% | |
In all cases, "left" and "right" are from the kicker's perspective (i.e., Goalie Left means diving toward the kicker's left side). If the goalie guesses the kicker's direction correctly, he will block about 3 or 4 kicks out of 10. If the goalie guesses wrong, the kicker's chance of success is very high.
Next, we calculate the mixed strategy equilibrium. Let p be the probability that the goalie jumps to the left and 1-p be the probability he jumps right. To make the kicker indifferent between kicking left or right, we must solve:
| payoff from kicking left | = | payoff from kicking right |
| 58p + 95(1-p) | = | 93p + 70(1-p) |
The result is p = 42%. The goalie must jump left 42 out of 100 times to make the kicker indifferent between kicking left and right.
Next, we turn to the kicker's strategy that makes the goalie indifferent. First, note that the table above has only the kicker's payoffs represented by the probability of success. The goalie's payoffs are the opposite: the probability of a miss. We can rewrite the above table to represent the chance that the kicker misses by subtracting the numbers in the table from 100.
| Goalie | |||
| Left | Right | ||
| Kicker | Left | 42% | 5% |
| Right | 7% | 30% | |
Let q be the probability that the kicker kicks to the left and 1-q be the probability he kicks right. To make the goalie indifferent, we must solve:
| payoff from jumping left | = | payoff from jumping right |
| 42q + 7(1-q) | = | 5q + 30(1-q) |
The result is q = 39%. The kicker must kick left 39 out of 100 times to make the goalie indifferent between jumping left and right.
Surprisingly, the game is not very symmetric between kicking left and right which makes sense since humans aren't symmetric. This implies that the relative frequencies of left and right for the goalie and the kicker should not be 50-50. How well does game theory predict actual behavior? Here is the actual behavior of kickers and goalies in the 1,417 observed penalty kicks:
- Kickers:
- Predicted proportion of kicks to the left: 39%
- Observed proportion of kicks to the left: 40%
- Goalies:
- Predicted proportion of jumps to the left: 42%
- Observed proportion of jumps to the left: 42%
Quite remarkable! What's even more impressive is what was found when the players' ability to randomize without patterns was investigated. Despite throwing a lot of heavy statistical machinery at the data, no patterns were discovered. Neither the players' past kicks nor their opponents' past behavior is useful in predicting the next kick. Whether this is due to superhuman statistical prowess on the part of soccer players or the influence of the army of well-paid statisticians on each team's payroll is still uncertain.
The last item worth considering is the overall chance of scoring on a penalty kick, in equilibrium. We know from the first table above that if the kicker kicks to the right, his chance of scoring is either 93% or 70%, depending on whether the goalie jumps to the left or the right. But, we also know that the goalie jumps left 42% of the time. Thus, if the kicker kicks right, his chance of scoring is:
93 (0.42) + 70 (0.58) = 80%
If you considered the chance of scoring from kicking to the left, you would, of course, obtain the same expected payoffs (slight differences due to rounding):
58 (0.42) + 95 (0.58) = 80%
The success of game theory in predicting these outcomes suggests that experience and the high stakes involved push athletes toward optimal equilibrium behavior.
Citation: Ignacio Palacios-Huerta, "Professionals Play Minimax," Review of Economic Studies, vol. 70(2), April 2003, pp. 395-415.