July 13, 2006

World Cup Game Theory, Redux

Penalty kicks decided the World Cup. Gelf talks to the man who has turned them from an art into a science.

David Goldenberg

Penalty kicks were more important in this World Cup than in any other in history. Four of the knockout games were decided by PKs (tied with the 1990 Cup for the most ever) and the championship game went to kicks for only the second time in history (the first time, in 1994, Italy fell to Brazil).

Ignacio Palacio-Huerta
Ignacio Palacio-Huerta
Before the knockout stages of the World Cup began, Gelf interviewed Ignacio Palacios-Huerta about how economic theory influences decision-making on the soccer pitch. Palacios-Huerta is an economist at Brown University and the author of an exhaustive study [PDF] examining keepers' and shot-takers' strategic approach to penalty kicks. In the interview, he Palacios-Huerta said many of the world's best players appeard to be excellent game theorists, if unconsciously, by deciding where they aim the ball—or, in the case of keepers, their body—during PKs. In other words, they both spread their shots and bodies between left and right in order to maximize their success, and randomize their direction to prevent their opponent from predicting the strategy.

Now that the Cup is over, Gelf caught up with Palacios-Huerta to get his take on the drama of the last two weeks. We asked him to analyze the game-ending penalty kicks in the Germany-Argentina and Portugal-England quarterfinal games, as well as Zidane's bloop-kick that bounced off the cross-bar and in during the first half of the final. He also talked to us about the subpar performances of the French and Italian keepers during the PKs of the final—just one of nine kicks missed, and that one stayed out of goal because of the crossbar, not because of the keeper—and why England seems to have so much trouble when a match comes down to penalties. (You can watch the penalty kicks Palacios-Huerta discusses by clicking on the play button of the YouTube screens under the relevant answers.) The interview has been edited for clarity.

Gelf Magazine: What did you think about the fact that German keeper Jens Lehmann referred to detailed notes (based on his former manager's database ( between kicks during for the Argentina game?

Ignacio Palacios-Huerta: I thought it was absolutely great!

Had he known the players—had he been in front of German or English players (since he is German and since he plays for Arsenal in the English Premier league I am assuming he knows them well)—my guess is that he would not have needed those notes: The relevant information would have already been in his head. But in this situation he needed the information.

What he did shows how some preparation for the penalties, especially for penalty shoot-outs, can have a huge influence in a game, in a country (actually, in two countries) and—why not—in the world.

I would like to note that stand-alone PKs are typically shot by "specialists," who are for the most part unpredictable. But PK shoot-outs often involve several players who cannot be considered specialists. And in my dataset I can tell that these non-specialists are, more often than not, predictable.

GM: England's Jamie Carragher had to retake his kick (because he kicked it before the whistle blew). Did that give an advantage to Portuguese keeper Ricardo?

IPH: It probably did. It is difficult to say, but if it did, it will be because of the extra-nervousness that it implied. In my dataset, Carragher is one who shoots pretty much perfectly from the viewpoint of what Nash's game theory predicts.

GM: Did you know beforehand that Lehmann and Ricardo would be the better keepers?

IPH: First, I suspected it about Ricardo. He has always been good in PKs, although I do not have a very long dataset for him. (I do not have the Portuguese league in my database. I only have data on international games played by Portuguese teams, but with this dataset one can tell he was going to be very good.)

Second, with regard to Lehmann, he has always been pretty good guessing the correct side. On top of it, he is a very tall, long goalkeeper, forcing in some sense kickers to think that they have to shoot quite close to the post, which puts additional pressure on the kickers. For instance, in the very important quarterfinal game of the Champions League against Villarreal back in March, this is what Lehman did [on behalf of his Italian club, Inter Milan] against Argentinean Riquelme (YouTube).

GM: How would you have advised the teams involved?

IPH: From my data on Lehmann, it seems that when he faces players he has never or rarely faced before, as it was the case, he follows a pretty basic rule: "dive to the place where the kicker has shot more often lately."

This rule is optimal if in fact the kicker tends to follow it, of course. And the Argentinean players did. So, I would have given a note to Argentinean coach José Pekerman saying that that was what Lehmann tends to do, and hence advising his players to go the other way.

With regard to Ricardo, as I said, I do not have much data on him, but I do on the English players. If you take the three kickers together other than Jamie Carragher (i.e., Steven Gerrard, Frank Lampard, and Owen Hargreaves), the dataset clearly shows that their overall scoring rate is lower when shooting to the non-natural side. Hence, they should kick more to their natural sides (as they are right-footed, this is the right-hand side of the goalkeeper). Interestingly enough, this is what Gerrard and Lampard did NOT do and got the ball stopped, and this is what Hargreaves did, and he was the only one who scored.

My guess is that they were not very prepared for something so important, especially when they've failed so often in the past.

GM: Would you have considered Zidane's PK a Panenka [a famous goal (YouTube) described in the first Gelf interview]?

IPH: Yes, I think I would consider it that way. He tried to do a Panenka, although it ended up a bit too much to the left-hand side of the goalkeeper, and certainly too high, relative to a standard Panenka. Looking in my dataset, the data I have for him show that he has rarely shot that way. (I am sure he wanted to surprise Buffon, who was clearly familiar with Zidane's PKs; after all, Zidane played many years in the Italian league.) The unfamiliarity he had with a Panenka type of PK is probably why Zidane kicked it too high, hitting the crossbar, and not going quite to the center of the goal. This also shows what "pressure" may do, even to one of the top players ever in the world.

GM: During the PKs in the final, Barthez almost always guessed right, but seemed to be leaving late. Do you think he was waiting to dive until he saw the kick coming? Isn't that a stupid way to do things?

IPH: Yes, I agree with your conjecture. I think he was leaving late, and of course by leaving late it is not difficult to guess the correct side. But then again with a PK taking about 0.3 seconds on average, it is almost impossible to stop the ball unless it is poorly shot close to the center of the goal. Hence, I agree with you that waiting to see what the kicker does is a stupid way to do things. The optimal move should be pretty much at the point of football contact.

GM: Buffon always guessed wrong. From your dataset, it looks like he was bad at PKs in general. Why do you think he's bad at this?

IPH: Buffon is slightly below average in general, being somewhat worse when jumping to the non-natural side of the kicker (e.g., to his own left-hand side when the kicker is right-footed). There are some goalkeepers who are excellent during the game but have a poor record in PKs. Buffon reminds me a bit of Zubizarreta from Spain, who rarely stopped a PK in hundreds of PKs he faced in his career. Perhaps it has to do with the fact that he is indeed taller than the average, as Zubizarreta was, and takes slightly more time to react and then move.

However, from the perspective of Nash's game theory, the prediction for goalkeepers is that the stopping rate should be the same on the left and on the right. And from this perspective Buffon is good, albeit with a lower than average stopping rate on both sides. Also, although he is not great he is a somewhat decent "randomizer"; that is, it is difficult to predict which side he will jump at. Yet, he tends to switch sides from PK to PK a bit too often, hence being a bit predictable that way. The data shows that if he wants to improve his stopping rate he should not switch sides from PK to PK that often.

GM: In the other games that went to PKs, there were a lot of misses, which I thought was due to pressure. Were you impressed that, in the final, only David Trézéguet missed?

IPH: Not really. Missing one out of four is a 75 percent scoring rate, which is fairly close to the average scoring rate in stand-alone PKs, and also close to the average scoring rate in penalty shoot-outs, which in my dataset is about 68-70 percent. Missing one or two out of five is fairly standard in most penalty shoot-outs (that is, either a 60 or an 80 percent scoring rate, which on average are about 70 percent).

If you look at the other penalty shoot-outs, I think there were a bit too many misses, in all likelihood due to pressure plus the fact that PKs were shot by a number of non-specialists (like in the Germany-Argentina game). I was surprised the most by the excessive number of misses in England for example (3 out of 4), but England is a special case. They seem to be systematically ill-prepared for penalty shoot-outs, losing in almost every European and World Cup in the last few decades, and this probably caused an extra psychological pressure even in top players like Gerrard and Lampard.

Related in Gelf

Make sure to check out Gelf's original interview with Palacios-Huerta. Also, blogging referee Aaron Corman tells Gelf why the World Cup refs haven't been so bad after all, and Zooming In rounds up press coverage of the Cup from local papers around the world.

The Gelflog discusses things Materazzi could have said to set Zidane off, why SI columnists are wrong about Zidane, the future of the Serbia & Montenegro's side, Togo coach Otto Pfister's alter ego, and US commentators' factoid-finding mission.

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Article by David Goldenberg

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