Measured Games · Paper 08 of 07 · Sports economics
Professionals play minimax
Abstract
In a strictly competitive game with no pure equilibrium, unpredictability is itself the strategy, and the minimax theorem prescribes an exact mixture. Testing that prescription requires a setting where payoffs are observable, strategies are few and outcomes are immediate. Penalty kicks in professional football supply one. Across 1,417 kicks, scoring probabilities are statistically identical across strategies and choices are serially independent, making this the first natural setting in which both implications of the theorem hold. The serial independence is the harder result: in laboratory and psychological studies people switch too often, and professionals here do not.
Keywordsminimax theorem · mixed strategy equilibrium · serial independence · randomisation
Setup
Some games have no stable choice at all. If your opponent could predict you, you would lose; if you could predict your opponent, they would lose. Nothing you can commit to survives being known. In that situation the strategy is not a move but a probability, and von Neumann’s minimax theorem says exactly which one.
Testing that prescription on people has been difficult. It needs a setting where payoffs are observable, strategies are few, outcomes follow immediately, and the players are expert rather than confused. A penalty kick supplies all four. The ball takes about three tenths of a second to cross the twelve yards, which is less than the goalkeeper’s reaction plus movement time, so both players commit simultaneously whatever it looks like.
Sides are labelled by the kicker’s foot rather than by the pitch. R is the kicker’s natural side, L the non-natural one, so a left-footed and a right-footed kicker are playing the same game with the labels swapped.
The model
The equilibrium yields two predictions sharp enough to fail. First, each player’s success rate should be the same whichever side they choose, because a difference would be worth exploiting. Second, choices must be serially independent: no alternating, no streaks, no reacting to the last outcome. The second is much the harder test, and it is the one nearly every study of human randomisation had failed.
Take the kick. Your side against the keeper's, most recent at the right
Your scoring rate against the value of the game
Randomisation test · runs in your sequence against what chance would give. A run is an unbroken stretch of the same choice, so more runs means more switching
Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.
Results
Palacios-Huerta (2003) tested both implications on 1,417 penalty kicks from professional leagues in Spain, Italy and England, taken between September 1995 and June 2000, using the 22 kickers and 20 goalkeepers who appeared in at least 30 penalties each.
Aggregate behaviour sits almost exactly on the prediction. The theory says keepers should go non-natural 41.99 per cent of the time and kickers 38.54 per cent; the observed figures are 42.31 and 39.98. On equal success rates, the joint Pearson test across all 42 players gives 43.944 on 42 degrees of freedom, p = 0.389, which does not reject equilibrium play. Individually the hypothesis fails for three players at the 5 per cent level where 2.1 rejections are expected by chance.
The serial independence result is the striking one. A runs test rejects independence for two players at the 5 per cent level against an expected 2.1, and logit models find no player whose current choice depends on their own previous choices, the opponent’s previous choices, or past outcomes. These players neither repeat nor alternate more than chance. Walker and Wooders (2001) had found the same equal-win-rate result in tennis serves but also found that servers switch too often; the psychological literature on randomisation has found the same bias against repetition for decades. Professionals in this setting do not show it.
The tests have real power to detect failure. Monte Carlo simulation shows that if opponents had simply mixed fifty-fifty, the null would be rejected with probability 0.70 for kickers and 0.92 for goalkeepers.
Evidence
The empirical work is the paper here, rather than something the model was later applied to. Chiappori, Levitt and Groseclose (2002) approached the same play from the other direction: with too few kicks per player to test individuals, they derived predictions that survive aggregation across heterogeneous players and found those supported too. Palacios-Huerta reports that their aggregate predictions also hold in his data.
Two features make the setting unusually clean. Payoffs are directly observable as scoring rates rather than inferred from preferences, which is the obstacle that has blocked most natural tests of game theory. And the players are experts with career-long stakes rather than subjects meeting a game for the first time, which is the standard objection to laboratory rejections of minimax play.
Later work with larger samples has revisited the result. Analyses using more than 12,000 kicks from the 2011 to 2017 seasons reproduce the key findings on a data set nearly ten times the size.
Limitations
The most serious challenge is about generalisation rather than about the football. Palacios-Huerta and Volij (2008) reported that professional players carried this ability into the laboratory, playing minimax in abstract experiments where students did not. Levitt, List and Reiley (2010) then ran the comparison again and could not reproduce it, concluding that what happens in the field stays in the field. Their exchange is unresolved. Read the field result as solid and the transfer claim as contested.
Within the study, the two-action simplification does some work. Kickers do shoot centre, about 7.5 per cent of the time, and the paper folds centre into the natural side on the grounds that players describe the two as equally natural to strike. That is a defensible modelling choice supported by a robustness check rather than a fact about the game.
The payoff matrix is also an average across players. Individual kickers and keepers differ, and the equilibrium frequencies quoted here are the ones implied by the pooled matrix rather than by any one player’s. The individual-level tests in the paper address this directly, but the simulator above necessarily plays the average game.
References
- Chiappori, P.-A., Levitt, S. D., & Groseclose, T. (2002). Testing mixed-strategy equilibria when players are heterogeneous: The case of penalty kicks in soccer. American Economic Review, 92(4), 1138–1151. https://doi.org/10.1257/00028280260344678
- Levitt, S. D., List, J. A., & Reiley, D. H. (2010). What happens in the field stays in the field: Exploring whether professionals play minimax in laboratory experiments. Econometrica, 78(4), 1413–1434. https://doi.org/10.3982/ECTA7405
- Palacios-Huerta, I. (2003). Professionals play minimax. The Review of Economic Studies, 70(2), 395–415. https://doi.org/10.1111/1467-937X.00249
- Palacios-Huerta, I., & Volij, O. (2008). Experientia docet: Professionals play minimax in laboratory experiments. Econometrica, 76(1), 71–115. https://doi.org/10.1111/j.0012-9682.2008.00818.x
- von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele. Mathematische Annalen, 100(1), 295–320. https://doi.org/10.1007/BF01448847
- Walker, M., & Wooders, J. (2001). Minimax play at Wimbledon. American Economic Review, 91(5), 1521–1538. https://doi.org/10.1257/aer.91.5.1521
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