Game Theory of Penalty Kicks
By Ritwik Khanna
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Penalty kicks in football are high stake moments which are more about the psychological aspects of the game rather than the technical ones. Often, a penalty shootout is used to decide the outcome of a game in a knockout tournament, and thus, a lot of importance is given to both scoring penalties and saving them. The game of penalty kicks is a simultaneous game because there is a reaction time of microseconds which can safely be ignored without contaminating the model. At a professional level, the reaction time is effectively zero, and the goalkeeper guesses one way and the taker decides which way he will shoot before the kick being taken. Thus, the two actions are simultaneous. Penalty kicks are a zero-sum game, i.e. the favorable outcome for one agent coincides with an unfavorable outcome for the other agent.
In this article, we will study the game theory of penalty kicks under four circumstances with varying assumptions on the capability of the striker and the goalkeeper. The first common assumption for all models is that the striker can only go right or left. He cannot shoot in the center. Right or left here is from the view of the goalkeeper. The second common assumption is that a goal being scored is assigned a payoff of +1 for the striker and -1 for the goalkeeper and vice-versa for a goal not being scored.
The first model is what I like to call the superhuman players model. Under this model, both the goalkeeper and striker have perfect skills. This means that if a goalkeeper guesses the correct way he saves all the shots and the striker hits the target 100% of the time. Thus, if the goalkeeper goes the wrong way then the striker is bound to score.
On creating the payoff matrix (above) and solving for the mix strategy Nash equilibrium we get the result that the best strategy for both agents in this model is to completely randomize their actions in such a way that they go to each side exactly half the time. An important note here is that their action should not be dependent on their previous action, i.e. there should be no pattern to kicking and diving.
The second model with tweaked assumptions is what I like to term superhuman striker, human goalkeeper. Borrowing terminology from Ignacio Palacios-Huerta (Beautiful Game Theory, 2014) players have a natural and an unnatural side. The natural side for a right-footed striker tends to be the goalkeeper’s right and for a left-footer, it tends to be the goalkeeper’s left. The natural side for a goalkeeper is the side with the stronger hand - a right-handed goalkeeper’s natural side is his right.
For simplicity, it is assumed that right is the natural side for all goalkeepers and strikers. The goalkeeper is weaker on his left side and saves only the ‘q’ portion of the shots even when he goes the correct way. Thus, (1-q) shots are goals even when he goes the correct way on his left side. The striker as earlier has 100% accuracy.
In the table above we can see that the striker gets a score of plus one each time he kicks the opposite way the goalkeeper is diving but the goalkeeper only gets plus one when he goes the same way as the striker on the right side. On the left side, he saves q times so the payoff is q from the times he can save and -1 times 1-q from the times he cannot save so the total payoff for guessing correctly on the left side becomes
q + (-1) (1-q) = 2q-1.
Similarly, the striker gets a payoff of 1-q when both go left but the keeper cannot save and -1 times q when the keeper does save, so the payoff becomes
1-q + (-1) (q) = 1-2q
On solving for the Nash equilibrium we get the result that the portion of times the goalkeeper goes to his left side decreases as q increases which means that as the goalkeeper’s ability to save on that side increases, he goes that way fewer times. This happens because the goalkeeper compensates for a lower save percentage by going more towards that side to keep the striker guessing. If he doesn’t do that and goes only to the stronger side, then the striker will know that the goalkeeper prefers the side and is stronger there as well.
For the striker, the portion of kicks towards the keeper’s left side decreases as q increases because the goalkeeper starts saving more penalties on the weaker side and the striker must increase the randomization.
The third model is the opposite of the second one - superhuman goalkeeper, human striker. · The striker has perfect accuracy only on one side, i.e. his right side. The striker scores the ‘p’ portion of the times on his left side. If he chooses to shoot towards his left side, he misses (1-p) times even if the goalkeeper dives right.
In the above table, we can see that the goalkeeper saves all the shots if he guesses the correct way and earns a +1 score. The striker earns a +1 score for all the shots he takes on the right side when the keeper goes left. However, when the keeper dives right and he shoots left he gets a score of p when he hits the target and -1 times (1-p) when he misses the target, so the payoff becomes
p + (-1) (1-p) = 2p-1
In the same situation, the keeper will get -1 times p for the times the striker scores the goal and (1-p) for the times he misses. the payoff thus becomes
(-1) p + (1-p) = 1-2p
On solving for the Nash equilibrium we get the result that as p increases, the goalkeeper goes to the striker’s left side more. This is because when p is low then he knows that there is a chance of him getting a higher payoff from that side even when he goes to the striker’s right side. So, he will prefer going to the striker’s right side when there is low p to increase the total number of saves.
However, the striker will go more to the left side as p decreases to compensate for the lack of accuracy. If p is low and he keeps going to his right side, then the goalkeeper will model the game accordingly as mentioned in the previous paragraph. Therefore, the striker compensates for a lower accuracy by increasing the randomization.
The fourth model is an amalgamation of the previous two - human goalkeeper, human striker. The goalkeeper saves all the penalties, if he guesses correctly, only on his right side. On the left side, he saves only a ‘q’ portion of penalties even when he goes the correct way. Some portion of shots (1-q) on the left side is scored even when he guesses correctly. Similarly, the striker has perfect accuracy only on one side, i.e. the right side. The striker scores ‘p’ portion of the times on the left side. If he chooses to shoot left, he misses ‘1-p’ times even if the goalkeeper dives right.
In the above table, the first cell earns the goalkeeper a +1 score as he guesses the correct way and he can save all the shots on the right side. The second cell earns the striker a +1 score because the goalkeeper goes the wrong way and the striker always hits the target on the right side. In the third cell, the striker hits the target p times and misses 1-p times, so he gets a payoff of
p + (-1) (1-p) = 2p-1.
Here, the goalkeeper gets (-1) times p when the striker hits the target and (1-p) when the striker misses. His payoff becomes
-p + 1-p = 1-2p.
In the fourth cell, the striker hits the target p times out of which q shots are saved and 1-q are scored. 1-p times the striker misses altogether. So, the payoff for the striker becomes
p(1-q) +(-1) (1-p) + (-1) pq = 2p-2pq-1
Similarly, the goalkeeper’s payoff becomes
pq + (-1) p(1-q) + (1-p) = 2pq + 1 – 2p
On solving for the mixed strategy Nash equilibrium and plugging in various values for p and q we get the following result table
When the accuracy of both goalkeeper and striker increases on the left then the goalkeeper will go to the left more while the striker will go to the left less. This is because of the goalkeeper’s payoff on going to the left increases as for a low p and q. He will save more shots on his right and the striker will miss more on the left meaning the goalkeeper’s payoff will be high even if he doesn’t go to the left. The striker will go to his left less in this situation as he reduces his compensation for a low p. As we saw in the third model the striker goes to his left more when p is low to randomize and the reverse takes place here as combined with an increasing q.
When p increases and q decreases, the striker will go to his left more to randomize his behavior and we will see the same scenario as we saw in the second model. The striker will go this left less because the goalkeeper knows his accuracy is more on his left and if he doesn’t compensate for that then the goalkeeper will read the striker’s shots.
When p falls and q increases, the goalkeeper will go to his left less as his payoff for right increases relative to the payoff to the left due to the falling accuracy of the striker on the left. He is rewarded more without going more on his left and thus he will prefer going to the left. The striker will go more to the left to randomize and compensate for his lower accuracy as was the case in the third model.
When both p and q fall the keeper will go to the left less because of falling p and his reward will increase more while he can save more shots on the right. The striker will go to the left more not only to compensate for a lower accuracy but also because the goalkeeper is saving fewer shots on the left.
Conclusion
The best strategy is not clear on the surface and we need to delve deep into the patterns and behavior of the striker and the goalkeeper to maximize each one’s payoff. Complete randomization is practically the best strategy for penalty kicks. Empirically, it has been found that professional players fulfill this strategy as verified using the Runs test by Huerta (Beautiful Game Theory, 2014)
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Ritwik is a final year economics undergraduate Hansraj College. His academic interests lie at the intersection of sports and economics, development economics, and data visualization