Here’s a quick followup to my post on the game theory of penalty kicks.
During today’s World Cup match between Switzerland and France, Karim Benzema took a penalty kick versus Swiss goalkeeper Diego Benaglio. Benzema shot left; Benaglio guessed left and successfully stopped the shot. Immediately thereafter, the ESPN broadcasters explained why this outcome occurred: Benaglio “did his homework,” insinuating that Benaglio knew which way the kick was coming and stopped it appropriately.
This is idiotic analysis for two reasons. First is the game theoretical issue. It makes no sense for Benzema to be predictable in this manner. Imagine for a moment that Benzema had a strong tendency to shoot left. The Swiss analytics crew would pick up on this and tell Benaglio. But the French analytics crew can spot this just as easily. At that point, they would tell Benaglio his problem and instruct him to shoot right more frequently. After all, the way things are going, the Swiss goalie is going to guess left, which leaves the right wide open.
In turn, to avoid this kind of nonsense, the players need to be randomizing. The mixed strategy algorithm gives us a way to solve this problem, and it isn’t particularly laborious. Moreover, there is decent empirical evidence to suggest that something to this effect occurs in practice.
The second issue is statistical. Suppose for the moment that the players were not playing equilibrium strategies but still not stupid enough to always take the same action. (That is, the goalie sometimes dives left and sometimes dives right while the striker sometimes aims left and sometimes aims right. However, the probabilities do not match the equilibrium.) Then we only have one observation to study. If you have spent a day in a statistics class, you would then know that the evidence we have does not allow us to differentiate between the following:
- a player who successfully outsmarted his opponent
- a player who outsmarted his opponent but got unlucky
- a player who got outsmarted but got lucky
- a player who got outsmarted and lost
- players playing equilibrium strategies
I can’t think of a compelling reason to make anything other than (5) the null hypothesis in this case. Jumping to conclusions about (1), (2), (3), or (4) is just bad commentary, pure and simple.
The embarrassing thing about this kind of commentary is that it is pervasive and could be reasonably stopped with just a tiny bit of game theory classroom experience. Even someone who watched the first 58 minutes of my Game Theory 101 (up to and including the mixed strategy algorithm) playlist could provide better analysis.
I really enjoy your commentary and instruction on game theory. I have some questions on your goalie/kicker scenario above. Your left/right choices amount to a zero-sum game in which mixing strategies in equal proportions is the best for both and yields a Nash equilibrium in mixed strategies. However, the kicker and the goalie would have to be truly and utterly random in their moves. Isn’t this what you’re assuming in conclusion #5? However, I don’t think it’s possible for humans to be totally random. We are always compelled to make some semblance of a deliberate choice. As such, I think players can outsmart each other. Consider a batter in baseball. World-class hitters can actually see the ball even though it’s traveling between 90-100 mph. Wouldn’t it be possible for a world-class goalie to have the capability to detect a split second detail in kicker body language? …Just wondering what your take is.
By the way, I think you are missing the best statistical shot for a kicker: kick down the middle. 🙂