Game Theory & Soccer ⚽

Game Theory & Soccer ⚽

Intro

As we have seen throughout the semester, game theory can be applied to a variety of applications.  Let’s see how we can apply it to the game of soccer, focusing on penalty kicks.  What strategies should a striker use when taking a penalty kick and what can a goalkeeper do to prevent the ball from going in the net?  Let’s find out!  So grab your cleats and a ball, and let’s play some soccer!!

When is there a penalty kick?  If player 1 gets fouled inside the opposing player 2’s penalty box then player 1 will get a penalty kick.  Before we “dive” in, let’s first explain what a penalty kick is.

- The ball is placed on the penalty mark in front of the goal (12 yards away).

- The striker has a single kick and tries to put the ball in the net while the goalie tries to stop it.

It only takes 0.3 seconds for the ball to hit the back of the net so the goalie doesn’t have enough time to intercept the ball.  Instead, he must choose a side to dive toward or guess where the striker will shoot and hope it’s the same side. 

Figure 1: Penalty kick

http://gifrific.com/wp-content/uploads/2012/06/Andrea-Pirlo-Chip-Shot-Penalty-Shot-Euro-2012-Vs-England.gif

For our purposes, we’ll assume the following:

1. To make things interesting, we’ll say the goalie is a guy and the striker is a girl.  This obviously isn’t true in a real game (unless you play co-ed pick-up ;p) but it makes it easier to distinguish who we are talking about.

2. The goalie dives either left or right (we’ll use the striker’s perspective throughout).

3. The striker shoots left or right.  The shooter won’t ever kick it in the center to the goalie…that’s a big no no.  We won’t worry about the case where she kicks the ball in the air or on the ground either.

4. If the striker kicks the ball to the same side the goalie dives, we’ll assume the shot is blocked.  If the striker kicks the ball to the opposite side the goalie dives, we’ll assume it went in the net and is a goal.

Pure or Mixed Strategy?

cThis setup is similar to the game of matching pennies.  In matching pennies, you and an opponent simultaneously reveal a penny.  If both pennies show heads, your opponent pays you $1.  If one penny is heads and the other is tails then you pay your opponent $1.  In this case, you can be viewed as the goalie while your opponent is the striker in our soccer example.

Figure 2: Payoff matrices for matching pennies and penalty kicks

https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/

Figure 2 shows simplified payoff matrices for matching pennies and penalty kicks.  As we can see, this is a zero-sum game because each of the four instances in the matrix adds up to 0.  It turns out that there aren’t going to be any pure strategy Nash equilibrium.  Why?  I’ll go over this in my presentation and we’ll look at each instance to answer this question.  Feel free to comment on why that might be!

We learned from Nevin’s presentation on Nash’s Theorem that there must be at least one Nash Equilibrium for all finite games.  Since there is no pure strategy Nash equilibrium then one must exist for mixed strategies which involve a probability distribution over two or more pure strategies.   This means that the players will randomly choose among their two choices (left or right).

So, what should the players do?  What might be an optimal strategy?  It turns out that both players should pick a side with equal probability.  This means the striker kicks left ½ the time and right ½ the time and the goalie dives left ½ the time and right ½ the time.  Why is this an optimal strategy?  It is because the players are less likely to figure out what their opponent will choose.  It allows the players to be less predictable.  

Figure 3: Payoff matrix with probability 50%

https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/

This is, in fact, an example of a Nash equilibrium because both players can’t gain by changing their strategy.  If the striker has a 50/50 chance of shooting the ball either to the left or right then it doesn’t matter which way the goalie dives because he will have an equal chance of stopping the ball.  Similarly, if the goalie is diving left or right with equal 50% probability then it doesn’t matter which side the striker chooses to shoot the ball because she will have a 50% chance of scoring.  By randomizing your options, you become less predictable to your opponent.

This is easier to see if we look at other strategies that are not optimal where probabilities are different throughout.  If the striker kicked left 100% of the time then it would be easy for the goalie to stop the shot because he would then dive left 100% of the time.  This is also true if the striker shoots left 99% or 98% and so forth.  The goalie will always want to go to the left.

Conclusion

In actuality, both players have a choice of more than two tactics (going left or right). The shooter might aim low or high.  She might shoot to the left, right or center.  She might also go for either power or accuracy (if you didn’t already know, it is very easy to boot the ball with too much power or aim too high which forces the ball to go waaay over the net.  Not good my friends!).  The optimal strategy for both players will be mixed but it is best that players keep their probabilities of choosing left or right as close to 50% as possible.

We will see that game theory can be applied to many other sports including baseball and tennis.  Can you think of any other strategic interactions that are found in other sports?

Sources

Harford, Tim. "World Cup Game Theory." What economics tells us about penalty kicks. June 24, 2006. Accessed April 12, 2017. http://www.slate.com/articles/arts/the_undercover_economist/2006/06/world_cup_game_theory.html.

Spaniel, William. Game Theory 101: Soccer Penalty Kicks. June 16, 2010. Accessed April 12, 2017. https://www.youtube.com/watch?v=OTs5JX6Tut4.

Spaniel, William. "The Game Theory of Soccer Penalty Kicks." The Game Theory of Soccer Penalty Kicks. June 12, 2014. Accessed April 12, 2017. https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/.