The Game Theory of Penalty-Taking
Why it sometimes pays to aim straight
02 July 2016
The penalty shootout between Italy and Germany at Euro 2016 had all the drama you could want. At 1-1 Zaza of Italy blasted the ball over the bar. Then Muller of Germany's poor strike was saved. Italy scored to put them 2-1 up - and Germany missed. The game was Italy's for the taking, but then Pelle hit the ball wide....and so it went on, with Germany eventually winning 6-5. In total, only 11 of the 18 penalties were scored.
One interesting feature of the shootout was that the German players always aimed for one corner or the other. The Italians, however, chose twice to aim straight - and on both occasions they scored. Aiming straight has the huge advantage that you are unlikely to miss the target, whereas aiming for the corner can mean hitting the post (as Ozil of Germany did) or going wide.
What's also interesting is that for all 18 penalties, the goalkeeper made a pre-meditated decision to dive to the left or the right. This made aiming straight an even better strategy, since it meant there was always an empty net to aim at.
My general observation of penalty shootouts is that goalkeepers do indeed almost always opt to dive for one of the corners rather than stand still. I think one reason why they do this is that diving is what goalies do. Standing still as the ball sails into the corner doesn't look very smart, so psychologically there's probably some pressure to be dynamic. If this is the case, why don't strikers opt to aim for the centre more often? The answer is that, again, a striker who aims straight looks a bit feeble if the keeper does stand still and save the ball.
But let's suppose strikers decide to start aiming straight more often. In the short term they will score more goals. But goalkeepers will quickly learn. They will change strategy, and decide occasionally (and at random) to stand still, double-bluffing the strikers. Eventually aiming/diving for the corner and going straight/standing still will reach an equilibrium that gives neither the striker nor the goalie a particular advantage.
What is that equilibrium?
In our book How to Take a Penalty*, John Haigh and I proposed a model for penalty taking. We suggested that if a striker aims straight and the goalie stands still, the chance of a goal is (say) 30%. (You can still score if you blast the ball hard enough). And if the goalie dives, the chance of a straight kick going in is (say) 90%. It's not 100% as you might still kick over the bar, or hit the diving goalie's feet. We suggested that the chance of a goal if you aim for the corner and the goalie dives randomly left/right is (say) 50%, and if the goalie stands still it's (say) 80% - you might kick wide.
You might want to argue with some of those probabilities, but for the sake of argument, let's suppose they are correct. For those probabilities of scoring, it turns out that there is a strategy that guarantees the striker a minimum 63%** chance of scoring. It is as follows: randomly aim for one of the corners 2/3 of the time, and aim straight 1/3 of the time. Meanwhile the goalie can ensure that the striker can't do better than 63% if he stands still 4/9 of the time and dives 5/9 of the time. This is the equilibrium strategy that strikers and goalies should adopt.
That 63% figure may sound familiar. Yes, it's exactly the proportion of penalties that Germany and Italy scored between them in the match: 12 out of 19 = 63% (remember that Italy scored a penalty in the 78th minute.)
Of course there are some who might say this is just a coincidence.
* In later editions this became Beating the Odds and most recently The Hidden Maths of Sport.
** The success rate of penalties in real tournaments is 70-75% - the chance of scoring when aiming for the corner is probably a bit higher than in our very simple model.