With the Scottish Cup finals just around the corner, its time you better start practicing your spot kicks in case you and your opposition can’t be separated in normal time. Many describe the penalties procedure as a lottery. But just how much chance is there? A mathematical approach can be applied to the supposed ‘randomness’ of the shootout.
There has been a lot of research (however, apparently not by the English FA) into penalty kicks. In competitions such as the World Cup or the Champions League these kicks will have millions of pounds resting on them, so it is no wonder that more and more teams are looking for the best method to come out on top at the crucial moments.
We can simplify the decisions made by the penalty taker and the goalkeeper to the following:
Taker (Player A):
A1: shoot to his natural side
A2: shoot to his unnatural side
Goalkeeper (Player B):
B1: dive to the taker’s natural side
B2: dive to the taker’s unnatural side
We can eliminate the option of shooting down the middle; not only will this make the calculations a little more complicated, but research has shown that goalkeepers tend to avoid staying still, because if the ball goes to the side and the keeper stays still, his fans think he’s not trying. Evidence has also shown that a surprisingly small amount of takers opt to shoot down the middle, too.
Since for any given kick one player will win, whilst the other loses, this is a two-player zero-sum game.
The diagram on the left shows how the four possible combinations of the strategies play out. Clearly, the penalty taker prefers scenarios 2 and 3, whilst the goalkeeper prefers 1 and 4.
A study by Ignacio Palacios Huerta looked at penalty kicks between 1995 and 2000, calculating the percentage of successful kicks for each of the combinations of strategies detailed above. We can use his data to make a pay-off matrix:
Player A wants the maximum percentage of penalties to be goals, so he wants the maximin (the highest of each row's minimums). The column on the right shows that this is to go with strategy A1.
Conversely, Player B wants the lowest amount of shots to go in, so he wants the minimax (the lowest of each column's maximums). The bottom row shows this is B1. So initially, the two players will settle on (A1, B1).
However, this isn’t a stable saddle point, as A will then want to use strategy A2, then B will change to B2, and so on.
In 2008, Chelsea were about to face Manchester United in the Champions League final in Moscow. Huerta sent a report to Chelsea manager Avram Grant, with information on Manchester United’s penalty takers. It included, amongst others, these pieces of advice:
• Manchester United’s Cristiano Ronaldo was 85% likely to shoot to his natural side if he paused during his run-up. However, if the goalkeeper moved too early, he was also capable of changing the direction of his shot.
• The Manchester United goalkeeper, Edwin Van der Sar dives to a kicker’s natural side more often than most goalkeepers.
These seem to have been acted upon:
• When Ronaldo made is run-up, he paused. Chelsea goalkeeper Petr Cech stayed very still, before diving to Ronaldo’s natural side to save the shot.
• Of Chelsea’s seven penalty takers, five shot to their unnatural side, Van der Sar saving none of them.
We will have a closer look at the Chelsea penalties.
Michael Ballack, Juliano Belletti, Frank Lampard and Salomon Kalou, all right-footers, all shot to their unnatural side. All four scored.
John Terry (also right-footed) shot to this side, however his infamous slip saw the shot go wide. Van der Sar had dived the other way.
Ashley Cole, a left-footer, shot to his natural side, which was nearly saved, however it was hit well enough to go in.
As the diagram above shows, Chelsea had hit all their penalties thus far to the right-hand side of the goal. By now, Van der Sar had become wise to Chelsea’s strategy, so before the seventh Chelsea penalty, he can be seen to be pointing towards that side of the goal (see right). This caused Chelsea’s taker, Nicholas Anelka, to shoot to his natural side, where the kick was saved by Van der Sar, and the match won by Manchester United.
So whilst Chelsea were intelligent enough to do their research before the match, they fell into the classic game theory trap of following a pure strategy.