The Beautiful Game (Theory) and the Mighty Pen(alty)

10 min read

How data and psychology combine in one game-changing kick.

“Here’s what Mo Salah [asked me] that shows he is a top, top player,” goalkeeper and part-time vlogger Ben Foster tells the camera. “He [asked], ‘If I got a penalty, which way would you have dived?’”

A week earlier, Foster and his Watford teammates were routed 5-0 by a dominant Liverpool side. The fourth goal was a dazzling solo effort from Liverpool’s star man Mohamed Salah, but Salah allowed himself no time to dwell on it. He was already preparing for the next game, mentally.

“He has to know the tiny little details in his next game, if he gets that penalty – which way he should put it,” Foster shares. “All goalkeepers do their research on strikers, so I’ve looked and I said, ‘I’d dive to my right because you [Salah] put your last five penalties to the [goalkeeper’s] right.’”

Regular moves in football involve multiple passes, complex running formations, and increasingly savantish tactics. In contrast, the penalty kick is simple, involving only two players and a single kick of the ball. The ball is placed on the ‘penalty spot’ – a central mark 12 yards (roughly 11 meters) from the goal line. A designated player runs up and kicks it once in an attempt to score. The opposing keeper, standing in the center of the goal (most of the time), tries to save. There are only two outcomes: goal or no goal.

Andrea Pirlo (in blue) taking a penalty for Italy at the 2006 World Cup final. Source: FIFA YouTube channel.

A single penalty kick can make or break a game or even a tournament. The 1994 and 2006 World Cup finals were decided by penalties. In October 2017, three years before his little chat with Foster, Sala stepped up to take a penalty for the Egyptian national team in the dying minutes of a game. The score stood at 1-1, and Egypt needed a win to qualify for their first World Cup tournament in 27 years. With the hopes of a whole nation on his shoulders, Salah held his nerve, buried his penalty, and sealed his status as an Egyptian legend.

Not every game will have penalties. But when playing for glory, teams and players leave no stone unturned. They will prepare for penalties, and this involves more than just honing one’s football technique.

From models…

Penalties have clear rules, so all of them are set up more or less the same way. They also have a low number of outcomes, thus making a good natural experiment. This has allowed researchers to apply game theory in order to predict what the players will do.

Sometimes in game theory, the players can talk to each other to find a common strategy – two businessmen making a trade, or two countries negotiating a treaty. With penalties, there is no outcome where both kicker and goalkeeper benefit. The kicker scoring means the keeper has failed to save. The keeper saving means the kicker has failed to score. The two cannot cooperate. This situation is called a zero-sum game.

A simple model divides the goal into two parts – goalkeeper’s right (GKR) and goalkeeper’s left (GKL). If both players target the same side for their strategy, the goalkeeper saves; if they target different sides then the kicker scores.

Source: Derby County Football Club YouTube channel

All possible outcomes can be represented in a table as follows:

Bar any match-fixing, both players are playing to ‘win’. The kicker wants to score for their team while the goalkeeper wants to prevent the other team from scoring.

Furthermore, both kicker and goalkeeper must pick their respective strategies (where to aim the ball at/which direction to jump or dive) at the same time. We can assume that in top-tier professional competition (the World Cup, Premier League, Champions League etc), the players are skilled enough to execute the strategies they picked.

All this means is both players are trying to outguess one another.

When penalties were first introduced, both players might have just taken a wild guess and hoped for the best. It’s not known exactly when people started collecting data on penalty kicks. However, a Dutch football manager named Jan Reker started making his own index cards on thousands of players in the 1970s. Amongst other information, he recorded where players aimed their penalties. Reker did not manage the Dutch national team, but their goalkeeper Hans van Breukelen often consulted Reker for his database. At the 1988 European Championship, van Breukelen, armed with Reker’s data, was able to save a penalty as his country won the final.

Since then, television coverage of football has exploded, making raw data in the form of video footage easy to access. Today, sports clubs hire extensive analytics departments to study all sorts of sporting data, within which penalties are a specialisation.

As Foster mentioned, players do their research. Penalty takers and goalkeepers study footage of one another with their clubs’ analysts. What is the opponent’s dominant hand or foot? Do they have a track record of shooting or blocking one side of the goal over the other?

And with all clubs doing the same thing, both kicker and keeper know they have studied one another, creating a ‘he knows that I know that he knows’ situation if and when a penalty is given.

This intensifies the zero-sum situation, leading to some very interesting behaviour.

…to mind games

At the 2006 World Cup, German goalkeeper Jens Lehmann faced Argentina’s Esteban Cambiasso in a penalty shootout. The result of this penalty could determine who proceeded to the semi-final. Lehmann had a cheat sheet on how the Argentines preferred to take their penalties, but it had nothing about Cambiasso. Still, Cambiasso saw Lehmann checking the sheet and might have been psyched out. His weak shot went right to Lehmann, and Argentina was out of the World Cup.

Lehmann might have put off Cambiasso unintentionally. Nowadays, clubs prepare psychological warfare specifically designed for penalty situations.

Looking at our earlier model, it is quite simplified. The goal is more than just left and right. It can be divided into an infinite number of parts. The more parts there are, the smaller the chance of both kicker and keeper targeting the same part for their strategy.

Remember that the keeper diving to the same part of the goal the kicker has aimed for would typically result in the penalty saved. Therefore, penalties favour the kicker (which is fair, because penalties are given to penalise the defending team for fouling the attacking team close to the goal.)

Defending players sometimes try to chip away some of this advantage by jeering at the designated kicker, arguing with them, removing the ball from the penalty spot and so on, anything to disrupt the kicker’s preparation, frustrate them, and delay the penalty being taken. Some kickers are put off by the mind games and flaff their kick. Others hold their nerve and score regardless.

A fan video shows Aston Villa players (white tops) and their goalkeeper Emiliano Martinez (green) swarming Manchester United’s (red tops) penalty taker Bruno Fernandes right after United were awarded a last-minute penalty in their Premier League game in September 2021. Martinez allegedly remarked that another player should be taking the penalty instead of Fernandes. Fernandes then missed the penalty, which was the last opportunity to salvage a draw for his team. Source: YouTube.

Mo Salah did not have to wait long after the Watford game to apply what he learnt from Ben Foster. Three days later, Liverpool won a penalty in their Champions League game against Atletico Madrid. As Salah stepped up to the spot, Atleti goalkeeper Jan Oblak hovered around him, saying some words. When the referee blew the whistle, Oblak dived one way while Salah calmly rolled the ball into the other corner – still to the keeper’s right. Oblak was a very experienced goalkeeper who had been playing in the top tier for almost eight years at that point. He likely knew about the emerging pattern from Salah’s last four penalties. Perhap he thought he had successfully put Salah off shooting to the right? Needless to say, the mind games had not worked here.

Never let them know your next move: strategies for penalty success

At this point, Mo Salah had now shot five penalties in a row to the goalkeeper’s right. He appeared to be always targeting the same side. Game theorists would call this “pure strategy”, which is not recommended for penalty taking. Rather, the secret to penalty success is “mixed strategy”.

Mixed strategy means players have to be able to target all parts of the goal. Kickers may even train to shoot with their weak foot. In penalties, the mixed strategy must also be random – completely unpredictable. A player who alternates between left, right, left, right, despite having a mixed strategy, is predictable. Their opponent can easily guess their next strategy.

Everyone has a dominant leg or side. It is more natural for right-legged players to kick to the goalkeeper’s right, and vice versa for left-legged players. A player can put more power and control behind the shot when kicking to their natural side, increasing their chances of scoring. In the previous World Cup, both Messi and Ronaldo had penalties saved when they tried kicking to their unnatural sides.

However, if a right-legged player always kicked to their natural side, goalkeepers will learn to always cover their right side. Historical data backs this up: now-professor Ignacio Palacios-Huerta studied penalties as a graduate student and found that kickers had similar success kicking to their natural and unnatural sides. The advantage of going towards their natural side had been negated by goalkeepers targeting that direction more often.

Similarly, if goalkeepers always leaped to the kicker’s natural side, designated kickers will kick to their unnatural side if they face that keeper.

Palacios-Huerta also calculated an optimal mixed strategy for both kickers and keepers based on the database he studied. Kickers should kick to their natural side 61.5% of the time and kick to their unnatural side for the remaining 38.5%. For keepers, they should dive to the opponent’s natural side 58% of the time and to the other side for the remaining 42%.

When Palacios-Huerta turned his focus to individual players (the study includes the likes of Zidane, Shearer, Bergkamp, and Buffon), he found that the most successful kickers and keepers were indeed able to adopt randomised mixed strategies. Stunningly, their actual ratio of choosing the natural vs unnatural side were also very close to his calculated “optimal”, whether by chance or design. What a win for game theory.

Unfortunately, Mo Salah and Egypt are not at this year’s World Cup after losing both the African Cup of Nations final and their WC qualifier game on penalties.


  1. Soccernomics (2018 edition) by Simon Kuper and Stefan Szymanski
  2. Professionals Play Minimax by Ignacio Palacios-Huerta, 2003, Review of Economic Studies,
  3. Aim for the middle: it could be your best shot for a goal in a penalty shootout by Stephen Woodcock,
  4. The Penalty Doctor: the science behind spot kicks and what we can expect at this World Cup by Eoin O’Callaghan,

Illustrated by Lim Daphne
Written by Ellen Ng


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