Using Game Theory to improve goalkeepers chances on penalties

When the referee calls a penalty against your favorite team, the natural reaction is to complain and assume your team just conceded a goal. Rarely does a fan rationalize and consider that not all penalties are converted into goals. Although it is true that the goalkeeper does not hold the big end of the stick in that situation, he still has a chance to make a save, even at the Premier League level.

This whole situation wreaks of Game Theory. Game theory is the process of modeling the strategic interaction between at least two parties (Investopedia). For example, let’s say we have 2 players. They are both presented with a choice: press, or don’t, a red button. If both press it, they both lose 50$. If they both don’t press the button, nothing happens. If only one does press the button, he wins 50$. Game Theory studies how the players will play the game, how they will try to maximize their own gain.

In football, both parties are presented with three options: left, center and right (more options exist for shooters, like to shoot up or down, put for simplicity purposes, we will keep it at that). The goalkeeper wants to go the same way as the shooter, while the shooter wants the exact opposite.

To see how Game Theory could help goalkeepers in this situation, all 175 penalties from the past two Premier League seasons have been analyzed ( The direction of both the shot and the dive, as well as the end result, were collected. Below are the results, for the kickers.

Kickers LeftCenterRight
Success rate %79%77%82%

We see that there seems to be a slight tendency to shoot left more frequently, especially compared to shooting down the middle. Overall, the success rate is fairly similar, although shooting down the middle has been the slightly least effective approach for strikers.

In penalties, keepers are reactive to the action of the shooter, meaning that they are the ones trying to guess where the ball goes. Therefore, in theory, looking at the table above, they should be diving left all the time. This would mean diving on the correct side more frequently (66 out of 175) than if they dived on any other side. However, in real life, if they were always diving on the same side, strikers would adjust and shoot somewhere else. Therefore, what could be expected is a slight tendency from keepers to dive left. Below are the results.

Save rate %17%38%16%

Here is the difference in the goalkeepers dives as opposed to where penalty kicks go.

The data tells a lot. First, we do see that keepers dive to their left much more than shots go there (16 times more). We also see that staying in the center is not nearly as frequent (20 times less), even if it has given by far the best save rate (38%). This probably is because stopping a shot that goes in the center of the goal is much easier than saving a shot that goes in the top left corner!

Considering this, why are keepers in the Premier league not staying in the center more often, to at least stay there as much as penalty takers shoot there (18% compared to 30%). A reason could be that it might not be the best look for a keeper to stay put, as it makes it look like he is not even trying.

Nevertheless, it appears clear to me that goalkeepers should be trying to replicate more accurately the penalty kick direction distribution in the way they try to save penalties, especially by staying in the center much more frequently. Game Theory shows that this is their best chance of saving that crucial penalty that can change a Premier League season!

If you found it interesting and want to read more about the subject, check out this great article, that obviously takes the whole thing a step further, by looking at the impact of heterogeneity of all actors during penalties.

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