We all may already know the game of penalty kicks, there are two players, the striker and the goalie. The striker must shoot either left or right to score a goal and the goalie needs to dive left or right to stop the goal. Suppose that the goalie always manages to stop the ball and the kicker is not always accurate in his kicks, he is accurate in kicking on left and can score a goal, however sometimes he misses and hits the right side with a probability of X. This occurs even if the goalie guesses incorrectly. The payoff matrix for this looks like:
Let Y axis be the goalie and X axis be the striker.
How does penalty kicks play out in reality?
That is the basic model of penalty kicks, but let’s delve deep into the reality of penalty kicks.
Real life penalty kicks are different in two ways, one that kicking to the right and kicking to the left is not always the same. For example, a player that is right footed can be more accurate in shooting towards the left rather than shooting towards the right. This tells us that if he hits towards the right he is more likely to miss the shot, as the goalie becomes more likely to stop the ball on the right.
The second way it differs is that the players can choose to shoot towards the middle, and likewise the goalie can also choose to defend the middle.
We know that a striker’s probability of scoring is equal no matter the direction he shoots in, because suppose it the probability was not equal. This means that if the striker is continuously kicking towards the right, then the goalie is also likely to dive towards the right, then there is a chance of exploitation, where the striker can suddenly kick left, which can result in a higher scoring percentage. To avoid this exploitation, the striker’s probability of scoring is equal. Just like the striker, the goalie also has an equal probability of defending where if one of the sides is more likely to be scored in then someone has a higher advantage than others.
This is data that shows the percentage of shots scored from the strikers. From looking at the Total column and row for both striker and goalie we notice that they all have an equal probability where the difference isn’t too significant.
Then the main question that remains is: How can we improve our chances of scoring?
Suppose you are very accurate in shooting to the right side but not as accurate when shooting to the left side, so you improve your accuracy towards the left side. From intuition, you think that you will now shoot more often to the left side. However this is incorrect because you must consider the weakness of the goalie, shooting towards the left side can decrease your chances of getting a goal, because it may be the side that is better defended by the goalie.
Hence, with this mixed strategy approach we can improve our chances of scoring.
References:
https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/