With the 2024 European Football Championship underway, I’m doing a couple of posts on the quirks of football analysis, and what they can tell us about wider statistics and science, based on my research for The Perfect Bet. The previous one was about match prediction, and now we move on to game theory and penalties, via poker…
John von Neumann was a brilliant researcher, from nuclear physics to computing, but he wasn’t so good at poker. At first glance, poker might seem like the perfect game for a mathematically minded scientist. Surely it’s just a game of probabilities: the chance you get a decent hand, and the chance your opponent gets a better one.
But von Neumann realized there’s more to it than that. In poker, there’s an element of hidden information: you want to guess what your opponent has while they're trying to guess what you have, and you want to guess what they're guessing.
Von Neumann wanted to understand this back-and-forth process, so he looked at some really simplified forms of the game. One of these was a game where there are just two players, each gets a single card, and there’s one round of betting. In this situation, von Neumann realized that there was a tug-of-war going on. In poker, everything you win comes out of your opponent's pocket, so both players are simultaneously trying to maximize their winnings while minimizing what they hand to their opponent.
Studying this game, von Neumann showed that in this tug-of-war, there’s an equilibrium point where the forces are balanced and each player has an optimal strategy. This means they wouldn’t expect a better outcome if they did something different. In game theory, this is known as a minimax strategy.
When he worked out this optimal strategy for his very simple poker game, he found that the player who goes first should take the following approach:
If they get dealt a high card, they should bet. Intuitively, that makes sense: if you have a decent card, you put money on it.
If they get dealt a middling value card, they shouldn’t bet; they should ‘check’ and see what happens. Again, this makes sense.
If they get dealt a low-value card, they should bet – in other words, they should bluff.
For centuries, players have been bluffing in games like this, but it was often thought of as a quirk of psychology. Von Neumann showed that bluffing was a mathematical necessity in these games if you wanted the optimal strategy.
In this very simple game, von Neumann could write down fixed rules for each player. For example, if the first player gets dealt a high or low card, they bet; if they get dealt a middling value card, they check. By following these rules in the long run, they get the best expected outcome. This is what is known as a pure strategy in game theory.
From cards to kicks
But real life is rarely this simple. Which brings us to penalty kicks in football. In 2003, economist Ignacio Palacios-Huerta analysed data from over 1400 penalties. Overall, kickers scored 80% of them. But he found that all things being equal, a right-footed player will have a better chance of scoring if they put the ball on the left side of the goal (i.e. the right-hand side from the keeper’s viewpoint), what he calls their ‘natural’ side1. So perhaps players should always shoot on this side?
The problem is that if the keeper works out what the player is doing, they’ll always go that way and reduce the chance of scoring. So perhaps players should always shoot on their non-natural side? But again, if the keeper guesses what they’re doing, they’ll go that way and reduce the chance of scoring.
We have this back-and-forth second-guessing, just like von Neumann was interested in. And in this situation, von Neumann showed the optimal strategy is to play with an element of randomness. In other words, players shouldn’t always put the ball on the same side; they need to mix it up (in game theory, this is called a mixed strategy).
Analyzing penalties, Palacios-Huerta found that the mathematically optimal thing to do as a kicker is to shoot on the natural side about 58% of the time, and on the non-natural side the remainder of the time. And the optimal choice for a keeper is to dive to the kicker’s natural side 61% of the time.
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Remarkably, when Palacios-Huerta looked at real-life penalty data, this was almost the exact ratio players used: kickers picked the natural side 58% of the time, and keepers 60% of the time. This doesn’t mean players sit in the changing rooms working through the mathematics, of course. But it does mean that, by trial and error, they’ve converged to the optimal strategy2.
At least, on average. When it comes to individual penalties, the quality of decision making can still vary depending on the situation. For example, Palacios-Huerta found that players are less likely to score penalties in the final 10 minutes of the match, when pressure is presumably higher (73% vs 83% for penalties in the first half). Similarly, a 2020 study, which looked at all penalties taken in European and World Cup matches since 1976, found that England score 90% of penalties in open play. However, they only scored 61% in penalty shoot-outs. In contrast, Germany scored 75% of penalties in open play, but 85% in shoot-outs.
For simplicity, he included shooting down the middle in the ‘natural’ side of the goal, because only 4% of penalty shots target the centre of the goal.
This isn’t the case in all sports. One 2009 study suggested that baseball pitchers throw too many fastballs compared to the optimal game theory strategy, and NFL football teams pass less than they should.
Cover image credit: Jannes Glas via Unsplash
Interesting. What about if the penalty taker sends a false sign to the keeper. I look as if I'm going to kick it one way but kick it the other or vice versa but this time diving.
My hope is that Southgate is sending out a false message about England. The team plays very mediocre passing football to fool the opposition. When it matters they turn it on.
As I am not a politician so I'm not going to put any money on it.
There a good book out "The Liars of Nature and the Nature of Liars." Lixing Sun. Princeton UP.
Just started it.