### Brief summary

Statistically the best spot to aim a penalty is the middle of the goal because the goalkeeper nearly always dives left or right. Yet, despite footballer players being aware of this statistical fact, they seldom shoot down the middle. This article uses game theory to explain why.

In a 2016 episode of British comedy quiz show *QI*, presenter Stephen Fry asks panellists where they should shoot a penalty for the best chance of success. Statistically the answer is that penalty takers should shoot down the middle because the goalkeeper nearly always dives left or right. Yet, despite footballer players being aware of this statistical fact, they seldom shoot down the middle.

To understand why we must first understand that football is a spectator sport. Footballers care about more than just putting the ball in the back of the net. We can use an area of maths called *game theory* to capture this thinking so that we can more accurately model a player's approach to penalty kicks, helping us understand why penalty takers transgress from the advice of statisticians.

### Estimating the payoffs

There are five strategies the penalty taker can follow: they can shoot top left (TL), bottom left (BL), middle (M), top right (TR), and bottom right (BR). Similarly, the keeper has three strategies: they can dive left (l), right (r), or stay in the middle (m). The outcome of any pair of strategies has an associated reward/cost for each player called a *payoff*. We can estimate the payoffs for these strategies by taking into consideration things like personal accomplishment, embarrassment, guilt, the reaction of fans, and so on. There's not some strict system of measuring these, so instead we give them arbitrary values that enable us to consider the payoff of one strategy relative to another.

Generally, if something pleases/displeases the spectators, football players care about it and this ultimately affects their payoffs for each strategy. We can lay out the respective payoffs for each of the players, given all their possible strategies, in a *payoff matrix*. Each cell in the payoff matrix contains two sets of payoffs, the first set for the taker and the second set for the keeper. And each set of payoffs contains three entries for when a shot is scored, saved, and missed respectively.

For example the central cell (M,m) shows the payoffs if the penalty taker shoots in the middle of the goal (M) and the keeper remains in the middle of the goal (m). In that case, the payoffs for the taker are (0, -400, 0): 0 if they score, -400 if the shot is saved, and 0 if they miss. While the payoff for the keeper is (0, 400, 0): 0 for a successful penalty, 400 if they save the penalty, and 0 if the shot misses. For simplicity, I assume that this situation (M, m) will always lead to a shot being saved, hence why the payoffs for a shot being scored or missed are 0 for both players.

l | m | r | |

BL | (200, -200, -300), (-200, 300, 200) | (100, 0, -400), (-200, 0, 100) | (100, 0, -400), (-100, 0, 100) |

TL | (400, -300, -500), (-100, 500, 200) | (200, 0, -600), (-200, 0, 100) | (200, 0, -600), (-100, 0, 100) |

M | (100, 0, 0), (-100, 0, 0) | (0, -400, 0), (0, 400, 0) | (100, -0, -0), (-100, 0, 0) |

TR | (200, 0, -600), (-100, 0, 100) | (200, 0, -600), (-200, 0, 100) | (400, -300, -500), (-100, 500, 200) |

BR | (100, 0, -400), (-100, 0, 100) | (100, 0, -400), (-200, 0, 100) | (200, -200, -300), (-200, 300, 200) |

Payoff matrix showing the payoffs for all the possible outcomes of a penalty (scored, saved, missed) for the five strategies of the taker and three strategies of the keeper.

I've generated the numbers used in the matrix in line with common intuitions about sport. For example for the taker, it's better to score in the top corner than the bottom corner: so their payoffs for scoring in the top left corner (TL) are always greater than their payoffs for scoring in the bottom left corner (BL), regardless of the keeper's strategies. The possible payoffs for scoring TL are represented by the first entries in the first sets of payoffs across that row (400, 200 and 200). These are always greater than the payoffs for scoring BL (200, 100 and 100), the first entries in the first sets of payoffs across the BL row. However, it's worse for the taker if they miss in the top corner rather than the bottom corner, especially if the keeper dives the wrong way. This can be seen by comparing the value of -600 in the final entry for the taker in (TL, m) and (TL, r), with the value of -400 in the final entry for the taker in (BL, m) and (BL, r)

Another useful comparison to make is between the second entry for both players (when a shot is saved) in (TR, r) and (BR, r). If the taker aims for the top right corner and the keeper saves the shot by diving right (TR, r), there is a payoff of -300 for the taker and 500 for the keeper. However, the keeper saving a shot to the bottom right corner only gives a payoff of -200 for the taker and 300 for the keeper. Therefore, it is worse for the taker and better for the keeper when a shot is saved in the top corner than the bottom corner because the shot was more difficult to save, meaning the keeper outplayed the taker.

### How likely are the payoffs?

We now must consider the statistical likelihood that these payoffs are achieved. As previously mentioned, a shot saved in the top corner is worse for the taker than one saved in the bottom corner, but it is also less likely to be saved, and this fact must be accounted for. We can estimate these by using data from past games, generating a probability of how likely it is for any payoff to happen.

The following probability matrix has been generated mostly using empirical data from Chiappori (2002) and Bruin Sports Analytics. Each cell contains the probability that a shot is scored, saved, and missed respectively. (These three entries add together to 1 as these are the only three possible outcomes for each penalty.)

l | m | r | |

BL | 0.29, 0.7, 0.01 | 0.99, 0, 0.01 | 0.99, 0, 0.01 |

TL | 0.34, 0.635, 0.025 | 0.975, 0, 0.025 | 0.975, 0, 0.025 |

M | 1, 0, 0 | 0, 1, 0 | 1, 0, 0 |

TR | 0.975, 0, 0.025 | 0.975, 0, 0.025 | 0.23, 0.745, 0.025 |

BR | 0.99, 0, 0.01 | 0.99, 0, 0.01 | 0.21, 0.78, 0.01 |

The probability matrix showing the likelihood of the penalties being scored, saved and missed for each combination of strategies.

I assume that a shot taken in the middle will not be missed, so the final entries for row M are all zero. And I've assumed that shots taken in the top corners are 2.5 times more likely to miss, as you can see in the final entries for the top corners compared with the bottom corners. (This multiplier may be higher, but this has a negligible effect on the final game matrix.) Chiappori’s study found the probability of scoring a penalty kick in a given location depends on the direction in which the keeper dives. However, their study only had 3 strategies for the taker: left, middle, and right. To adjust for this, I used Bruin Sports analytics and Bryson et al (2009) to approximate the probability of scoring in the top and bottom corners. This approximation generated an average expected goal within 2% of the empirical observations of Chiappori (2002).

Which way?

We can multiply the payoffs by the respective probabilities to give the *expected* *payoff* to each player for each strategy. For example, consider the first set of entries in the cell (BL, l) in both the payoff and the probability matrix, showing the possible payoffs for the taker and the likelihood of the three outcomes. The taker has a 29% chance of scoring, which would give them a payoff of 200, a 70% the penalty is saved giving them a payoff of -200, and a 1% chance they'll miss their shot, with a payoff of -300. So the expected payoff for the taker shooting at the bottom left corner, when the keeper dives left, is:

0.29 x 200 + 0.7 x (-200) + 0.01 x (-300) = -85.

Multiplying the payoffs by the respective probabilities gives us the following matrix representing the expected payoffs (for the taker and the keeper respectively) of a penalty kick:

l | m | r | |

BL | -85, 154 | 95, -197 | 95, -98 |

TL | -674, 289 | 180, -192 | 180, -95 |

M | 100, -100 | -400, 400 | 100, -100 |

TR | 180, -95 | 180, -192 | -144, 355 |

BR | 95, -98 | 95, -197 | -117, 194 |

Before solving and analysing the game, I have 5 sanity checks to test if the expected payoffs are in line with common sense intuitions about the game:

- If the taker knows where the keeper dives, they should always prefer to shoot elsewhere
- If the keeper knows where the taker shoots, they should always prefer to attempt to save it
- When the keeper dives the wrong way, there is a preference for the taker to shoot in the top corner
- If the taker does not shoot down the middle, the keeper always prefers to dive, otherwise they look as though they have not attempted to save the shot.
- The taker has no preference between TR and TL when the keeper dives the wrong way

Given the payoffs are in line with these common-sense assumptions, we can begin to solve the game.

### Solving the game

Game theory allows us to ‘solve’ strategic interactions by predicting how they will be played. If strategy A yields a payoff of 1 and strategy B yields a payoff of -1, we would predict a player to choose strategy A. Nothing is stopping a player from choosing B, but game theorists assume that players are rational and that they always try to maximise their payoff. Arguably the most famous solution is a *Nash equilibrium*, which is a set of outcomes where no player can improve their payoff by changing their strategy. In other words, both players are satisfied with their choice given their opponent’s strategy. With this in mind, we can now attempt to solve the penalty kick.

Intuitively, there is no pure strategy Nash equilibrium in this game because at least one player will always wish to change. If the taker shoots left and the keeper dives right, the keeper would have improved their payoff by changing their strategy to dive left. If the keeper did change their strategy to dive left, then the taker would improve their payoff by changing their strategy to shoot right, and so on.

There is, however, a *mixed strategy Nash equilibrium* where each player plays certain strategies with a certain probability, rather than a single strategy all the time. For example, they might shoot TL some percentage of the time and TR another. To find the mixed strategy Nash equilibrium, I'll first see if there are any strategies I can ignore.

If one strategy always yields a lower payoff than another strategy, it is *strategically dominated*. Assuming players maximise their payoffs, this means no rational player would choose a dominated strategy as they would always receive a greater payoff by choosing the more dominant strategy. In this case, it is easy to observe that BL is dominated by TL and BR is dominated by TR. While it is less obvious, M is also dominated by playing a combination of TL and TR, since M is never the best response to any strategy of the keeper. Because of this, we can eliminate these strategies, reducing the matrix to the following:

l | m | r | |

TL | -67, 289 | 180, -192 | 180, -95 |

TR | 180, -95 | 180, -192 | -144, 355 |

We can reduce the matrix even further. Regardless of whether the taker plays TL or TR, the keeper receives a greater payoff by playing r or l, rather than playing m. This means it's always a better choice for the keeper to dive left or right, so we can eliminate m from the matrix, leaving us with the final matrix:

l | r | ||

TL | -67, 289 | 180, -95 | q |

TR | 180, -95 | -144, 355 | 1-q |

p | 1-p |

Here, *p* is the probability of the keeper playing l and *q* is the probability of the taker playing TL. If the opponent plays their mixed Nash equilibrium strategy, then no single strategy for the player can lead to a higher expected payoff (EP). We can write this in the following way for the taker, where the keeper is playing their mixed strategy Nash equilibrium of diving left with probability *p*, and right with probability (1-*p*):

Expected Payoff of shooting TL = Expected Payoff of shooting TR

EP(TL, l) x *p* + EP(TL, r) x (1-*p*) = EP(TR, l) x *p* + EP(TR, r)x(1-*p*)

-67*p*+180(1-*p*) = 180*p*+(-144)(1-*p*)

571*p* = 324

*p* = 0.57

So the Nash equilibrium strategy for the keeper is to dive left 57% of the time (a probability *p*= 0.57) and right 43% of the time.

And for the keeper, when the taker is shooting top left with probability *q*, and top right with probability (1-*q*):

Expected Payoff of diving left = Expected Payoff of diving right

EP(l, TL) x *q* + EP(l, TR) x (1-*q*) = EP(r, TL) x *q* + EP(r, TR) x (1-*q*)

289*q*+(-95)(1-*q*) = -95*q* + 355(1-*q*)

834*q* = 450

*q* = 0.54

Giving the taker a Nash equilibrium strategy of shooting top left 54% of the time (a probability *q* = 0.54) and top right 46% of the time.

### Conclusion

When taking into consideration the spectacular nature of football, we understand that there is more to a penalty than putting the ball in the back of the net. While it’s clear why shooting in the top corner may be preferred to the bottom corner, it was not initially clear why penalty takers refrain from shooting down the middle – especially given this provides the highest chance of success.

In my model, although shooting down the middle was not strictly dominated by one strategy alone, it was strictly dominated by a mixed strategy of shooting in the top right and top left. In other words, shooting down the middle is never the best response for the taker. After eliminating BL, BR, and M as strategies, the optimal choice for a penalty taker is to play a mixed strategy: shooting top left 54% of the time and top right 46% of the time.

However, in real life, penalty takers don’t confine their shooting strategy to the top corners. Perhaps this is because these shots are more difficult – you’re more likely to miss – and because of this, emotional factors like fear or risk aversion dissuade penalty takers from playing their optimal strategy. Though penalty takers may not strictly conform to the predictions of my model, it nevertheless helps us to understand why they often ignore the statistics and avoid shooting down the middle.

### About the author

Billy Watson graduated from King's College London in 2024 with a degree in Philosophy, Politics, and Economics. His research interests include behavioural economics and normative ethics.>