Where to shoot in penalty kicks
An introduction to bayesian thinking
Why is Bayesian thinking the right way of thinking about things?
To me, Bayesian statistics is summed up as thinking about probability of events occurring as follows, the probability that an event occurs is updated based on the information you perceive in the world. This kind of thinking makes a lot of sense, suppose you are in Southern California consider you just stroll out of bed and you see your mom grab an umbrella, without yet looking outside that day how confident are you that it is raining or going to rain that day? Well normally it doesn’t rain in Southern California but given the fact that your mom grabbed an umbrella (a low probability event) your probability for it raining goes up drastically.
The Penalty Kick Strategy Nash Equilibrium
An ok way to think about penalty kicks but not really Bayesian or correct.
A very rudimentary yet interesting study of how penalty kicks work are as follows, and lead to a significant result, the better a player is at shooting to their weaker side the less likely they are to shoot to their weaker side.
The counter intuitive result comes from the following example. Consider there is a perfect goalkeeper who when they guess correctly automatically saves a penalty kick. The shooter, however, is perfect shooting to his strong side but is not great shooting to his weak side. Namely the shooter scores with probability 100% if he shoots to his strong side and the goalie dives the wrong way and the shooter scores with probability X% < 100%. (note we use the notion of 1 to represent 100% because when talking about probability a probability of 1 is assumed to be 100%). We can then represent the payoffs for the shooting team and saving team as follows.
Dive Strong Side | Dive Weak Side | |
---|---|---|
Shoot Strong Side | (0, 0) | (1, -1) |
Shoot Weak Side | (X, -X) | (0, 0) |
Note the goal keepers team has a negative payout of 1 goal when the opposition scores by shooting the opposite way as the goalkeeper. We can solve for the equilibrium condition such that the goalkeeper and striker chose a strategy that maximizes their payouts. (to solve this use a nash equilibrium condition), but the interesting result is that the striker shoots to his weak side with probability 1/(1 + X). This is to say the best strategy for the striker is to shoot to the weak side based on how good he is at shooting to the weak side. So then the natural question arises, if the striker is perfect at shooting to his weak side X = 1 then he only shoots to the weak side with probability 1/2 since he is indifferent, but if the player misses half the time when they shoot to their weak side and the goalie goes the wrong way they shoot to that side with probability 2/3. The logic why they should shoot more to where they are weaker at comes from the fact that the goalie has information about how good of a shooter his opponent is. If the goalie thinks you are awful at shooting to one side and much stronger at shooting to the other side, then they are much more likely to dive to your strong side, because they think you are more likely to score that way and want to prevent you. So in this fake game you should shoot more to your weak side to compensate for that weakness. Pretty Cool huh?
Well their are multiple reasons why this analysis doesn’t work so let’s be clear about them here
- The goalie and the shooter are not perfect, so the probabilities given in that chart are all wrong
- The goalie is not certain in it’s beliefs about how much stronger the shooter is in shooting one way
- The striker and coaches are not confident they know exactly how good the striker is shooting to either side
Bayesian Penalty Kicks
What happened in the previous kick mattered.
Let’s be clear, I do not envy any of the modern penalty kick takers. Every single professional soccer team hires video analysts to look at the film of the opposition and study them. Immediately one of the highest reward activities to analyze is the opponents penalty kick taker. Now let’s say our opponents penalty kick taker does not know we will watch him, and we recognize that he has a favored penalty style that he is more confident with. We realize that the probability he shoots to his favored side is 90%. We would be idiots to dive any other way and our probability of saving the penalty is higher. This Bayesian way of thinking that previous data effects the next shot makes our analysis much more robust and gives us a better sense of what the actual probability distribution is. Now his team has educated him and he knows that we are watching him. This will immediately change the distribution to which side he shoots the ball. Specifically he will remember his last 5 penalty kicks and try to “vary it up.” Well even if they attempt to do this they are in trouble, because if they “vary it up” against a lot of teams we can study the pattern by which they decide to vary up their shooting methods unless it is completely random. The gist of the story here is if you don’t want to get read by a goalkeeper shoot completely randomly.
Ok let’s say the opposing player is now completely shooting randomly. Well our goalkeeper again can win this shootout by diving randomly but more frequently to the sides in which we are more likely to make a save. So what does this mean for the shooter, it means that the shooter must shoot randomly according to his own ability, using the results we got above, because the goalkeeper is more likely to dive to the side in which the striker is a better shooter.