Soccer Penalty Kicks Article Flawed?

In my last post on this subject, I made a quick reference to an article on a mathematical examination of soccer penalty kicks. In this article, Tim Harford gives a brief survey of the findings of a paper by Ignatio Palacios-Huerta of the Brown University. The paper (pdf) is quite an interesting read, and I really recommend anyone with some knowledge in statistics and game theory to read it. However, I do believe I've detected a flaw in it. Though, before proceeding any further, I should include all the standard disclaimers, including, but not limited to, the fact that I'm in no way an authority nor an expert in this area, and that there is a chance that I've misunderstood things. I have all due respect for Mr Palacios-Huerta as a scientist and for Mr Harford as a writer, and I'm merely a layman myself.

Anyways. After writing my first entry on the article by Tim Harford, I got to thinking. Quoting from Tim Harford's article:
Professionals such as the French superstar Zinédine Zidane and Italy's goalkeeper Gianluigi Buffon are apparently superb economists: Their strategies are absolutely unpredictable, and, as the theory demands, they are equally successful no matter what they do, indicating that they have found the perfect balance among the different options. These geniuses do not just think with their feet.
At first, this seemed to be a good indication that Zidane and Buffon are indeed playing optimal strategies. But what hit me after writing my first entry, is that their playing optimal strategies doesn't make themselves indifferent between their strategy choices. That is, their playing optimally doesn't make them succeed equally often no matter what they do. It does, however, make their opponents indifferent between their strategy choices.

The optimal strategy is about making your opponent indifferent between his strategy choices. Recall, from my previous post on this subject, how the indifference equations for each player included the strategy choices for the other player, but not his own strategy choices. This relationship works two ways: Your playing optimally doesn't make you indifferent, and your indifference is not an indication that you're playing optimally.

So the Harford article is wrong. The fact that Zinédine Zidane and Gianluigi Buffon seem to be indifferent does not indicate that they play optimal strategies. It does, however, indicate that their opponents, on an aggregate level, are playing optimally.

Now, is this Tim Harford's or Ignatio Palacios-Huerta's mistake? In order to find out, I read the original paper by Mr Palacios-Huerta.

In the paper, Palacios-Huerta starts by formulating a hypothesis saying that professional players are indeed playing a minimax strategy. In order to test this hypothesis, he examines a sample of 1417 penalty kicks. I have no objections to his examination on all the players on an aggregate level. However, when testing the hypothesis for individual players, he seems to be looking at each individual players' strategy choices and their corresponding outcomes. Using Pearson statistics and p-values, based on those figures, the hypothesis is rejected for five players.

On an aggregate level, we can look at the overall figures of both sides of the game. According to the hypothesis, both goalies and kickers should have equal sucess rates, no matter their choices. This can be tested and the hypothesis rejected with the tools used by Palacios-Huerta. But when testing the hypothesis for individual players, we should look at that individual player's aggregated opponents' sucess rates for their strategy choices, which, it seems to me, is not what he's done.

So what hypothesis should we reject when the individual figures used by Palacios-Huerta don't give a good enough match with the hypothesis? Well, not the one that that particular player is playing minimax, but rather the one that his opponents, on an aggregate level, play minimax. This is not, in itself, an uninteresting hypothesis to examine, but, as far as I can see, it's not the one intended by Palacios-Huerta.

Unfortunately, the tables provided in the paper don't allow for the data to be rearranged so that we can perform this test on our own. There is no information on the strategy choices of the opponents of each individual player and their corresponding outcomes, so we can't examine the hypothesis that a specific individual player plays optimally, without accessing the underlying data.

So, what do we know about Zinédine Zidane and Gianluigi Buffon? Not much, but it seems they've been playing against superb economists.
Again, let me remind you of the disclaimers. I'm really a laysman, and I may very well be wrong. Either all wrong or just in my interpretation of the paper.

External links in this post:
World Cup Game Theory - What economics tells us about penalty kicks by Tim Harford. The quoted article in Slate Magazine.
Ignatio Palacios-Huerta at the Brown University website
Professionals Play Minimax by Ignatio Palacios-Huerta of the Brown University (pdf format)

Other resources:
Tim Harford - The Undercover Economist


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