Mixing it up
Can maths help you to do better in a penalty shoot-out? Lucy Bronze certainly thinks so.
As we reach the latter stages of any major football tournament, penalty shoot-outs between evenly matched teams seem almost to be an inevitability. An absorbing spectacle for the neutral, agonising for the fans of the teams involved, and potentially career defining for the players, the penalty shoot-out offers a form of sporting drama almost unrivalled in its acute tension.
England take on Italy tonight in the first Euro 2025 semi-final. Italy have outperformed expectations based on their FIFA world ranking of 13th, while England have arguably underperformed for much of the tournament. What odds then on the game going the full distance and ending up in a penalty shoot-out?
England have already had to endure the stress of one shoot-out - in their battling victory against Sweden last Thursday night after the teams couldn’t be separated over normal and then extra time in their Euro 2025 quarter final. First up stepped Allessia Russo for England to take the initial kick, putting it just beyond the reach of the diving Swedish keeper Jennifer Falk. But England would not score again for the next three kicks. Falk dived the right way and saved all three. When the camera cut to Falk before England’s fifth penalty she could be seen consulting her water bottle on which the names of the England penalty takers and their preferred penalty direction were listed.
Fortunately for England, Sweden were equally wayward with their penalties – goalkeeper Hannah Hampton saving two while two others missed the goal completely - leaving the sides tied on two scored and four missed after 6 penalties each. Up stepped stalwart defender Lucy Bronze to take what would transpire to be England’s last penalty; blasting it right down the middle over the legs of a diving Falk. Teenager Smilla Holmberg then skied Sweden’s final spot kick to spark scenes of jubilation amongst the England players.
In her interview after the game, Lucy Bronze (whose mum is a maths teacher and herself achieved a very apt bronze award in the UK Mathematics Trust challenge) discussed why she chose to blast her penalty straight down the middle. “I watched the goal keeper, and every single penalty she dived quite early. Statistically it’s risky for the keeper to stand still… So yeah, I love maths.”
And Bronze’s comments raise an important question: can mathematics help players determine how they should take their penalties? What can they do to make it more likely that they hit the back of the net? Should they go high or low, left or right or perhaps down the middle? Should they put their boot through it as hard as they can or should they aim for placement, or should they have the audacity to take a ‘Panenka’ – a chipped penalty kick down the centre of the goal relying on the keeper diving one way or the other. Audaciously brilliant if it works, but embarrassing if the keeper doesn’t buy the dummy and the ball floats gently into their open hands.
When all the rivalries, the personalities and the occasion are stripped away, penalty taking is a very simple sort of game with just two players; the taker and the keeper. Fortunately there is a branch of mathematical sciences, known as Game Theory, dedicated to understanding and interrogating the dynamics of such simple competitions. To use the language of that subject, penalty taking is a zero-sum game: for the taker to score the keeper has to concede and for the keeper to triumph they must ether save the shot or rely on the taker missing. Either way, there is only one winner.
As outlined above, the kicker has a variety of strategies from which to choose when taking the penalty and the keeper has a corresponding range of options to choose from in attempting to block the shot. For the sake of simplicity though, let’s just think about three options for the placement of the penalty kick – to the left, to the right or down the middle – and the same three options that the keeper can use to attempt to save. Let’s also imagine, as is usually the case, that both the taker and the keeper make up their minds about what their strategy will be in advance.
As four of the seven English penalty takers in the quarter-final shoot out discovered, no single strategy like “always aim right” is going to be optimal, precisely because it makes you predictable and predictability can be exploited, as so expertly demonstrated by Sweden’s goalkeeper Jennifer Falk. There is no one best place to put your penalties. In fact, the optimal strategy to use when taking penalties might just be what game theorists call a ‘mixed strategy’. A mixed strategy is an approach in which a player deliberately introduces unpredictability into their decisions to prevent opponents from exploiting patterns. In the context of penalty taking, this would involve the taker aiming at a different, randomly chosen position each time.
A 2002 study found that, rather than consistently favouring one side of the goal, penalty takers in two of Europe’s top leagues chose randomly between kicking to the left, the right or down the middle. Remember, though, that this is not the same as simply alternating sides, which is an entirely non-random and easily predictable strategy.
Football is by no means the only sport in which a mixed strategy might pay dividends. In tennis, the serve is one of the most potent weapons a player has in their arsenal. Pro players might expect to win about two-thirds of the points on their serve which translates to winning 86% of their service games. For the elite this percentage is even higher. In the recent men’s Wimbledon final, reigning Champion Carlos Alcaraz won 73% (121 of 165) of his service points, whilst world number one and eventual winner Jannik Sinner won 76% (116 of 152) of his.
Part of the advantage of the serve comes from its speed, leaving the receiving player relatively little time to react. Typically there are two options for serves in elite level tennis, either ‘down the middle’ or ‘out wide’. Being able to predict which of these two strategies an opponent will use on their service games would give the returner an advantage, enabling them to anticipate, prepare and execute a return of serve.
So again the optimal strategy for the server is to mix things up by randomly selecting between the two options to keep the receiver literally on their toes. Analysing almost half a million serves from over 3000 matches, Romain Gauriot and his colleagues found that professional tennis players adopt a near-optimal strategy when it comes to switching between the different serve types. Interestingly they also found that junior level players diverge quite substantially from the optimal strategy – serving in a more predictable manner, suggesting that perfecting a mixed strategy is something that pro tennis players have either learned to do or at least something that has facilitated their passages into the professional ranks.
Beyond the world of sport, mixed strategies have been adopted in a variety of different arenas ranging from politics to business to hunting. It is believed that for hundreds of years, the Naskapi people of eastern Canada have been using a randomised strategy to help them hunt. Their direction-choosing ceremony involves burning the bones of previously caught caribou and using the random scorch marks which appear in order to determine the direction for the next hunt. It is thought that divesting the decision to an essentially random process circumvents the inevitable repetitiveness of human-made decisions. Some academics have suggested that this reduces both the likelihood of depleting the prey in a particular region of the forest and the probability of the hunted animals learning where humans like to hunt and deliberately avoiding those areas, although others have cast doubt on how effective this mixed strategy would be in practice.
In recent experiments into the impacts of emotional unpredictability, management students were asked to negotiate a hypothetical venture with each other according to some pre-specified rules. In one scenario, negotiators were asked to be relentlessly negative and angry, while in another, they were asked to frequently change their emotional tone between positive and negative. The students whose counterparts displayed emotional unpredictability were made to feel as though they lacked control over the negotiations, leading them to make larger concessions and irresolute demands.
Political leaders employing mixed strategies aim to appear erratic and volatile to manipulate adversaries. One particular mixed strategy, a form of brinkmanship known in political science as the Madman Theory, was the basis of much of Richard Nixon’s foreign policy in the late 1960s and early 1970s. The aim, as the name would suggest, was to convince Nixon’s communist opponents that he was more than a little unhinged. He reasoned that if his opponents judged him to be an irrational actor, they would not be able to predict his plays and would thus have to make more concessions to avoid the risk of accidentally triggering him into retaliation. More recently, observers have argued that Donald Trump is also harnessing the power of unpredictability by resurrecting Nixon’s Madman ideology in his trade negotiations as well as his dealings in the Middle East and Ukraine.
Whether we are on the football pitch, the tennis court or the boardroom, game theory teaches us that predictable decisions can give our opponents the upper hand. Mixed strategies remind us that raw talent isn’t everything and that randomness in how our skills are deployed can have a big impact; staying unpredictable might be our most predictable path to victory.
This post is adapted from my BBC Future article “The mathematics of the perfect penalty shootout”
I suppose you've read Luck by Ed Smith? Worth reading if you haven't.