How to take the perfect penalty
Murad Ahmed reports on how game theory influences the way the world’s best footballers shoot from 12 yards.
Penalty kicks matter. Football is a low scoring game. So whether it's Beckham or Zidane, a penalty has the power to transform losers into winners.
A surprising amount of academic research has been dedicated to understanding the science of the penalty kick. This is because they provide a real world example of game theory, a field of mathematics famously used to predict US/Soviet military conflict. But it applies just as much to penalties. A penalty kick is a zero-sum game. What's good for the kicker is bad for the keeper and vice versa.
The kicker needs to work out a strategy before shooting. They could take every shot in the same direction, but the keeper would soon work out this pattern. So players need to use what economists call a mixed strategy. But that doesn't mean a 50-50 approach. When a player kicks a ball, they naturally have much more power and accuracy when they kick across their body. But to fool the goalkeeper, unpredictability is essential.
It's rational to favour the side you can strike better. So what's the best strategy? Game theory predicts that players should aim for their natural stronger side 61.5% of the time. But another thing kickers must do is make sure their choice is truly random. Imagine a coin being flipped over and over. Write down what you predict will happen. Heads or tails? Most people would never write down more than four heads in a row, but in reality, five heads in a row should happen at least once in a series of 50 coin flips.
Humans find this hard because we are slaves to "The law of small numbers." We can't see the difference between probabilities in the short and the long term. But can the best footballers overcome this? Over thousands of kicks, shooters do choose their stronger side 60% percent of the time just as game theory predicts. And the very best players, such as Lionel Messi and Franck Ribery, are also great at randomising their kicks. Nobody's saying that they're the world's greatest mathematicians, but consciously or not players are applying the science of game theory to the art of football.