NB: this is an award-winning project I did on my course on Game Theory, hence the heavy mathematical slant, and the patronising explanation of football things.

Introduction

In several game theory models, it is assumed that both sides make their choice simultaneously, or on a turn-by-turn basis. In such areas as politics, economics, biology, this means that the models used in game theory aren’t very realistic.

For those of you unsure what a penalty shootout is: it's a method used at the end of a knockout fixture which has ended in a draw, where each team takes turns to score a penalty, until at the end, Germany win.

**The Model**

**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.

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 the Basque economist 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:

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.

The Strategies

Therefore, both A and B need to find the best mixed strategy to give them the best chance in the long run. This means that both players will need to work out what proportion of the time, on average, they should choose each strategy.

Say Player A wants to find x, which is the fraction of times he should choose A1. Of course that means he’ll go with A2 (1-x) times. And similarly, Player B wants to know what fraction of the time he should choose B1, let’s call this y. He will choose B2 (1-y) times. Both players want to find the right mixed strategy where it doesn’t matter what their opponent chooses, they will always have the same chance of success. So we can set up the following equations.

0.70x + 0.92(1-x) = 0.95x + 0.58(1-x) 70x + 92 – 92x = 95x + 58 – 58x 92 – 22x = 58 + 37x 34 = 59x 34 ÷ 59 = x 0.576 = xSo Player A should shoot to his natural side 57.6% of the time. | 70y + 95(1-y) = 92y + 58(1-y) 70y + 95 – 95y = 92y + 58 – 58y 95 – 25y = 34y + 58 37 = 59y 37 ÷ 59 = y 0.627 = ySo Player B should dive to the taker’s natural side 62.7% of the time. |

In Reality

In Huerto’s research, 60% of penalties taken were hit towards the taker’s natural side. Goalkeepers dived that way around 58% of the time3. These are both fairly close to the optimal strategy.

With the figures provided, we can work out what the chances are that a random given penalty will go in. Using one of the formulas from the previous page, we can substitute x for 0.576 to find the percentage of penalties we can expect to result in goals.

0.70(0.576) + 0.92(0.424)

0.793

The success rate of 78.59% is very close to what we expected. However, one problem with this data is that the 341 penalties taken from the World Cup and European Championships are from penalty shootouts. Due to the rules of a penalty shootout, at least one penalty must be missed for it to end. This means that the data will be slightly biased by the fact that a certain proportion of the penalties

*must*be missed.

As we may have expected, the success rate has increased marginally. At 80.02%, it is still very close to our estimate.

Clearly, footballers are better game theorists than expected.

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.

- 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.

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.

And finally, the percentage of penalties in this shootout to result in a goal was… 78.57%.

Sources

Online papers

Action bias among elite soccer goalkeepers: The case of penalty kicks - Bar-Eli, M, et al - this looks at why goalkeepers avoid staying in the middle for a penalty kick.

Professionals Play Minimax - Palacios-Huerta, I - this paper by gives the information we converted into our outcome matrix.

Books

Why England Lose & Other Curious Football Phenomena Explained - Kuper, S & Szymanski, S - this book contains information on Huerta’s research, as well as details on his involvement in the 2008 Champions League final.

The Hidden Mathematics of Sport - Eastaway, R & Haigh, J - this book contains similar research to that which we have done, except they look at going to a side versus staying in the middle. It also contains an explanation for the formula we used in the Strategies section.

Websites

My Football Facts - penalty success records for the Premier League seasons 2001/02 – 2010/11.

Wikipedia - articles on the World Cups and European Championships were used to get the penalty shootout records.