You may know that a penalty kick is when a player gets a free shot at goal from a point about 12 yards (or 11 meters) from it with only the goalkeeper allowed to block the shot. Since this is such a close distance, the goalie just has to take a guess as to whether the kicker will shoot to the left or the right. The kicker also has to choose to kick to his left or his right, since kicking to the center is kicking right to the goalkeeper.
So, we know what each player's strategies are. Let's create a grid to represent it, like we do in most game theory examples.
|Dive Left||Dive Right|
Next, we need to figure out what percent of the time the kicker scores in these four outcomes. So, a professor from the London School of Economics named Ignacio Palacios-Huerta figured these statistics out by taking data from over 1400 penalty kicks. This was his result:
|Dive Left||Dive Right|
|Kick Left||.58, .42||.95, .05|
|Kick Right||.93, .07||.7, .3|
Now, we will use some game theory techniques to figure out what the optimal strategy of each player is. In this case, it would be best to find the mixed-strategy equilibrium
Goalkeeper's Optimal Strategy (Diving Left)
.58x + .95(1 - x) = .93x + .7(1 - x)
.58x + .95 - .95x = .93x + .7 - .7x
.95 - .37x = .23x + .7
.25 = .6x
.42 ≈ x
Kicker's Optimal Strategy (Kicking Left)
.42x + .07(1 - x) = .05x + .3(1 - x)
.42x + .07 - .07x = .05x + .3 - .3x
.35x + .07 = .3 - .25x
.6x = .23
x ≈ .39
So, the goalie's optimal strategy is to dive to the left 42% of the time and the kicker's optimal strategy is to kick to the left 39% of the time.
This alone is pretty cool, that we can determine the best way for a soccer player to handle this situation. However, we don't know if this actually works. So, Ignacio Palacios-Huerta took data from the best kickers and goalies in the world to see how their strategies matched up with the math.
Surprisingly enough, they were using the exact same strategy. The fact that even though these soccer players don't know game theory, but happened to stumble upon this perfect strategy was really impressive.
Bonus: While on the topic of game theory in soccer, I thought I would mention a famous soccer game that involved some game theory. In the 1994 Caribbean Cup, Granada faced Barbados. Because of some unique rules, a very interesting thing happened. Click here to read the Wikipedia article about the match.