A penalty is a guess against a guess. Five short steps to the game theory of it, using every penalty in StatsBomb's open data, then a shootout against the textbook, the pros, or a bot that learns your habits.
The 2022 World Cup final finished 3–3 after extra time. Four penalties had already been scored, by Lionel Messi and Kylian Mbappé. Then came the shootout: Kingsley Coman's kick was saved, Aurélien Tchouaméni's went wide, and Argentina won it 4–2. Two kicks decided the biggest prize in the sport.
Shootouts are where penalties hurt most. Of the 1,481 penalties in StatsBomb's open data, 363 came in shootouts, and they went in only 67% of the time, against 76% during play. So it is worth knowing how to take one, and how to face one.
The taker picks a side: left, middle or right (height comes later). The keeper picks too: dive left, stay, or dive right. The ball arrives too fast to react to, so the keeper has to go as the ball is struck. Neither sees the other choose, which makes this a simultaneous game, even though one kicks and the other dives.
Last kick of a World Cup final. You're taking it. Where do you shoot?
How often a penalty goes in, for each side the taker picks and each thing the keeper does, from the real numbers (where they come from is below the shootout). The keeper's score is just the rest: a 60% chance of a goal is a 40% chance of a save or a miss.
Before you look: is there one side that's always best, whatever the keeper does?
With nothing always best, your best reply depends on what you think the keeper will do. Real keepers almost always dive, so set how likely you think it is they dive to your right (the rest dive left).
Textbook versions of this game often end with a rule: never shoot down the middle, because it is never the best reply to anything you could believe. The real numbers disagree. Keepers dive so reliably that when the keeper looks close to a coin toss between left and right, the middle is your best shot. Yet only 11% of real penalties go down the middle.
And whichever side you favour, a keeper who knows it will go that way. So the stable way to play is mixing: both sides randomise, at rates that leave the other nothing to gain by leaning. That is the equilibrium of this game, and it is what the textbook opponent plays below.
Every model leaves things out. These are the ones that matter for penalties, each with what the data says.
Real keepers stayed in the middle on only 6% of kicks; the equilibrium says 8%. Not a mistake but a missing payoff: conceding after standing still feels worse than conceding after a dive, so keepers dive.
Bar-Eli et al. 2007Shootout kicks went in 67% of the time, against 76% during play. Same takers, same goal, more to lose.
StatsBombTakers sent 59% of side kicks across their body, to their natural side, yet those scored barely more: 74% against 73%. Equal scoring across sides is exactly what good mixing predicts.
StatsBombKicks into the top third that stayed on target were almost never saved, but 23% of them missed. Real takers went high on only 16% of kicks. You get to aim high in the shootout.
StatsBombIn the 2006 World Cup quarter-final, Germany's keeper Jens Lehmann took a note from his sock listing the Argentina takers' habits, and saved two kicks. A keeper who knows your habits is the reader below.
Public recordOne large study found the team kicking first won 60% of shootouts; later studies found a smaller edge. Argued over, so treat it with care.
ContestedSoon you will be able to switch these on and watch the table change.
Where you aim and which way the keeper goes are both up to you and your opponent. Whether a kick then goes in is decided by real numbers. From 1481 penalties in StatsBomb's open data: how often a kick to that spot missed the target, and how often one that stayed on target was saved. From Bar-Eli and colleagues' study of 286 kicks: how much the keeper's choice matters, diving the right way, the wrong way or staying put. Played the way real takers and keepers play, the model scores 74%, as the real kicks did. Guessing the right way is no guarantee: keepers who went the right way still only stopped about one kick in three.
Because each side's best move depends on guessing the other's, the stable way to play is mixing: each side randomises at rates that leave the other with nothing to gain. Those rates are what the textbook plays.
When your two options differ in strength, equilibrium mixing is not fifty-fifty but is tuned so that the opponent gains nothing by leaning either way.
Why is still argued over.
In the 2006 World Cup quarter-final, with the match level at 1-1, the German goalkeeper Jens Lehmann pulled a folded note from his sock between penalties. It listed the Argentine takers' habits, side by side with their names, and he saved from Roberto Ayala and Esteban Cambiasso as Germany won the shootout 4-2. The note has since been auctioned for charity and is one of the most reproduced scraps of paper in football.
It is the clearest demonstration of what the game is about: the kicker's two sides are not equally good for him, so he leans to his stronger one, and the moment that lean becomes legible the keeper can profit from it, which is exactly why the equilibrium mix has to be deliberate rather than habitual.
A penalty is struck and the keeper dives at effectively the same moment, so neither can react to the other. Kickers are better on their natural side, and the studies of large samples of professional penalties find that both kickers and keepers favour that side, at rates that leave neither with anything to gain by leaning further either way. They also find little of the exploitable serial pattern you would expect from people trying to be random by hand.
The two options are not equally good for the kicker, so the equilibrium mix is deliberately lopsided rather than even, which is what separates this from a plain coin matching problem.