June 29, 2006

The Game Theory of Penalty Kicks

Economist and soccer fan Ignacio Palacio-Huerta talks to Gelf about the Minimax strategy behind penalty kicks and why the most famous PK is called a "Panenka."

David Goldenberg and Rachel Bialik

As the World Cup reaches its final stages, it is likely that at least one of the remaining matches will be tied after 120 minutes and then decided in soccer's peculiar endgame: Which team can make more penalty kicks in a shootout. In 2002, Korea advanced to the semifinals after knocking off Spain on PKs. In 1998, France could never have won the tournament had it not edged Italy in PKs in a quarterfinal match. And, of course, had Roberto Baggio not shanked his PK in 1994 (YouTube), the Italians could have been world champions themselves. So the ability to make penalty kicks (and stop them if you are a keeper) is tremendously important.

Ignacio Palacio-Huerta
Ignacio Palacio-Huerta
For Ignacio Palacio-Huerta, that ability is the focus of an exhaustive study [PDF] that reveals how keepers and shot-takers alike deal with penalty kicks. But Palacios-Huerta, who grew up in the soccer-mad Basque region of Northern Spain, doesn't analyze player tendencies in order to help a particular team. Rather, he's an economist at Brown University whose interest in soccer has led him to write several papers about how motivations, risk, and reward influence decision-making on the pitch.

Penalty kicks stood out to him because they are a rare real-life manifestation of two-person zero-sum games. A penalty kick is a situation in which the shot-taker either scores or doesn't score based on simultaneous actions taken by both the shot-taker and the keeper. Simplistically, both the shot-taker and keeper (the "players") must decide whether to aim right or left, without knowing the direction that the other will aim (the shot-taker aims with the ball, while the keeper aims with his body). Both players generally do better going to one of the two sides, and, logically, will choose to play their strong side more frequently than their weak side. However, neither player can always choose his strong side, because then the other player will know where to aim. Therefore, each player must decide how frequently to play each side, so as to maximize his expected payoff (for the shot-taker, the probability of scoring; for the keeper, the probability of preventing a score). According to Minimax, players should decide these frequencies such that their expected payoff will be the same whether they aim right or left; and each time a player aims right or left, the move should be random (unpredictable).

To see if professional soccer players were following classical game theory, Palacios-Huerta watched over one thousand penalty kicks taken in the highest professional leagues of England, Italy, and Spain. What he found was that, while different in their success levels, almost all frequent penalty-kick players were superb game theorists, choosing to aim right or left with appropriate frequencies.

He talked to Gelf over a series of emails about the implications of his research, why kicking to the center could be a good idea, and how he would advise teams to approach penalty kicks. The interview has been edited for clarity.

Gelf Magazine: How did you gather all the data? Did you personally watch all the tapes of highlight shows? Or did you have a grad student watch them?

Ignacio Palacios-Huerta: Yes, I did watch personally all the tapes. It is much easier than it seems. A three-to-four-minute summary of the Italian games can be watched on the web [at RAI, the Italian public service broadcaster], the Spanish games I got on tape (knowing which games had a PK and using the fast-forward on the VCR, it does not take that much time), and the English summaries I also watch every week.

GM: Does that mean you have a big bunch of tapes in your basement?

IPH: Yes, I do.

GM: Are these really zero-sum games? The outcome for the team could simplistically be +1 or -1 (for goal or no goal, if you don't consider the current score or the time left in the game), but these may not be the reward/punishment for the kicker and keeper specifically. They may be looking to increase their fame, pay, etc., and not specifically looking to get or prevent a goal. Also, depending on the score, the kicker could be a hero if he gets the goal and the team wins (+1), but the keeper only has a small punishment (-0.2) if he misses the small shot he had of preventing the goal.

IPH: This is very interesting. I think that:

For the most part they are zero-sum. The empirical evidence and all the statistical analysis are consistent with PK being zero-sum.

Let me explain. One advantage of zero-sum games is that everything depends ONLY on the fact that the interests of the players are exactly opposed to each other. In the case of PK, one definitely wants to score and the other wants to stop (no-score). This is the case regardless of whether there are other things associated with "score" or "no-score" (fame, embarrassment, etc.), and also regardless of the different degrees of importance of a penalty kick (i.e., whether it is a terribly important penalty or whether it will not affect the final outcome (win or lose) of the match), one still wants to score and the other wants no-score.

It is true that fame, embarrassment, etc., may affect the payoffs, but probably not too much. At least the evidence shows that there is not a huge variation (which might affect the results) in the vast majority of PKs.

GM: You write that the spin of the kick has no role in the outcome. But couldn't it be used to deceive the keeper?

IPH: I cannot prove this but I think it cannot really be used. The distance is too short for the spin to really matter. The closest I can come to prove it is that when, for instance, David Beckham kicks a free kick, he does not really like those that are too close to the goal (say only a couple of yards out of the big penalty area). He prefers some more distance, so that he can really bend it the way he wants, that is, so that he can use the spin. And he is the master of spinning the ball.

For penalty kicks that are much closer to the goal and take on average about 0.3 seconds, there is simply no room for spin to matter ... Also, if you look at the PKs by Beckham, he uses, as far as I can tell, no spin whatsoever.

With respect to deceiving, some players have told me that they "sometimes attempt" to use the run-up and the body language for attempting to deceive the goalie.

Courtesy Wikipedia
GM: Sports Jones suggested in a 1998 article on this topic that the reason more players don't shoot to the center on PKs is because it would be embarrassing to get such a kick blocked. Do you agree?

IPH: No, I do not. One can readily make exactly the opposite argument, namely that it is a great honor to score shooting to the middle, and not a big deal to have it stopped (rather than an embarrassment to have it stopped and not a big deal to score).

In fact, I think that in some sense it is a great honor. The most famous penalty shot (and I think the first one) to the middle was taken by Panenka in 1976 (YouTube). It is so famous that it has a name: when a penalty is shot softly to middle, say, 1 meter or 1.5 meters above the ground (like the second Ukrainian kicker did on Monday in the penalty shoot-out against Switzerland), it is said that the penalty was shot a la Panenka. [Editor's note: You can see the PK here on YouTube (it starts at the one-minute mark; the Supersport announcer describes the kick as "cheeky").]) Well, Panenka shot it like this in the last and decisive kick of the European final Germany-Czechoslovakia in 1976, and got totally famous for it. It is very risky but the fame payoffs is [great].

GM: In Slate's coverage of your study, Gianluigi Buffon and Zinedine Zidane are praised for being unpredictable and thus having a great understanding of game theory.

IPH: Actually, if you look at my paper, the vast majority are unpredictable.

GM: But as the New York Times points out, neither Zidane nor Buffon are particularly effective compared to other keepers or kickers. Do you think it's possible that their focus on being unpredictable has made them less effective? (On a related note, do you think that any players actually worry about being random?)

IPH: I have talked to many players, and my sense is that they are not really thinking of being unpredictable. At least, I do not think that Zidane or Buffon are really focusing more than others...

In general, some players have some gut feelings about where to shoot or where to move, some do not really know where to shoot or move, some change their mind in the last milisecond. If they start thinking a lot about it they will probably stop being random.

You may be interested in reading one of the chapters of The Best American Sports Writing 2004 on Mia Hamm (published originally in Sports Illustrated in 2003), arguably the best female soccer player ever, about her PK in the penalty shoot-out in the World Cup final in the Rose Bowl a few years back. When asked about that PK, she says that she did not want to take a penalty kick because she was way too nervous, that she was "forced" by teammates and the coach, and that she did not know what she was doing and does not remember anything about it ... that she just "woke up" after seeing the ball in the net.

GM: You mention that you've spoken to several players and teams. How did you end up being in contact with them?

IPH: I played semi-professional soccer in my youth (when I was 21 to 22 while in college, before going to the University of Chicago for my PhD in economics) and I still have some contacts from those years. Also, my brother is an agent for some soccer players. And lastly, there is such a high density of professional soccer players in the north of Spain that it is relatively easy to approach them in practice.

GM: You write that it would make sense for players to be unpredictable in their PK patterns.

IPH: Yes, this is part of the equilibrium strategy. In equilibrium, (1) the scoring rate should be the same across the different choices that they have (in the simplest case, across left and right); and (2) they should be unpredictable, the same way it is unpredictable which side a coin is going to land.

GM: But do you think that, practically, any teams analyze the patterns of players they may face when it comes to PKs (besides obvious giveaways like always going right)?

IPH: In my experience, they analyze very little. It is more at the level of individual players (goalkeepers and designated kickers) than at the level of teams. Some players do keep written records, but by and large many, and I think most, do not.

Interestingly enough, though, what players have is a terrific PK memory. If you ask them, they remember very well what they did and what their opponents did, in many, many penalties even far back in time, sometimes going back years. Somehow, and probably unconsciously, they have those records in their brain.

Courtesy Wikipedia
GM: Do you think there are differences between the standalone PKs that you mostly studied, and the shootout situation we'll see now in the Cup? Perhaps in the latter, players and keepers start thinking more about randomness, because the kicks happen back-to-back, rather than many games apart against different opponents and keepers?

IPH: Yes, I think there may be big differences. One reason is that a standalone PK pretty much is always shot by the same player (the "specialist"), whereas in a shoot-out there are five different kickers, some of which rarely shoot for their teams.

Second, I also think you are right: Players may be thinking more about randomness than in standalone PKs, especially the non-specialists. It may be argued that the fact that standalone PKs happen somewhat infrequently does help kickers and goalkeepers to the extent that they may tend to forget and/or not think that much about what they did in the past (and forgetting helps a lot to generate random sequences—think for example of a coin, which we know has absolutely no memory and does not think on which side to land).

Lastly, standalone PKs are sometimes important and sometimes are not important at all (although still the kicker wants to score and the goalkeeper wants to stop the ball). But a PK in a penalty shootout is always very important. In such important penalties there is a lot of, say, psychological pressure. If you look at table 1 in my paper you will see that the scoring rate in PKs shot in the last 10 minutes of the game (when a goal will tend to be more decisive) is much lower (73.3 percent) than the average (80.1 percent). Although this is only an average, more sophisticated tests (regression analysis taking into account the score, etc.) do show that the scoring rate is lower under pressure (e.g., close games late in the match).

(This, by, the way is not inconsistent with the Nash equilibrium behavior, since Nash says nothing about the level, it only says that the scoring rate should be statistically identical across left and right choices, and the sequence should be random—and both of these hypotheses are still satisfied.)

GM: Often, the keeper chooses the correct side but still gets beaten. How does game theory apply in these cases?

IPH: No problem at all. On the contrary, this is part of the game. As I mentioned above, the "payoffs" are the probabilities that a goal will be scored or not for each combination of strategies. These probabilities are simply constructed using the observed frequencies (e.g., of all the Left-Left [keeper and player both play to the left] 59 percent were goals and 41 weere no-goal).

GM: But are there cases in which a player is so skilled to one side that he should not try to be random?

IPH: This is a great question. I have never seen any one like this, and I do not think any like this has ever existed. Perhaps the closest was Alan Shearer, who shot a lot to the right and little to the left (but, as predicted, he had—statistically speaking—the same success rate on either side).

I think that no player like this ever existed because when one gets too close to ALWAYS using one strategy all the time, the goalkeeper will tend to ALWAYS go that way, in which case the kicker will always have the other side open and should then begin shooting to the other side ... I think something like this may have happened to Alan Shearer in his career ...

GM: On a related note, an article in the Daily Telegraph covers a study done in Amsterdam that suggests that late changes adversely affect the kick efficacy, thus you should "decide where you are going to kick the ball beforehand—and never change your mind." Do you agree?

IPH: I am not a soccer player, although I played semiprofessionally for a couple of years when I was in college, but talking to players some (perhaps even many) say that they think about it beforehand but that somehow, unconsciously, they change at the last instant. Thus it is not clear they have control over never changing their mind. Moreover, some will tell you that they do not really know where they are going to shoot, and they do not really decide ...They just simply do it. See also my comment on Mia Hamm above.

GM: Are you happy with how the media has covered your study?

IPH: Yes, I have seen The NY Times article and Tim Harford´s articles in the Financial Times and in Slate. I thought that these articles describe very well for the general public, in a way that can be well understood, the game theoretical aspects of a PK as well as what the empirical evidence says. This is not an easy thing to do and I think they did it very well.

GM: If you were an advisor to a team, how would you prepare your side for a penalty kick situation?

IPH: I would prepare them especially for shootouts as these are 100 percent decisive (like in the current World Cup) and involve players that do not typically kick PKs very often, or at all, in their teams. Take for instance David Villa for Spain. The Spanish team had been preparing a lot in practice these days and you can tell that Villa was very confident, I think, because of the preparation. His shot was incredibly close to the post, totally unstoppable by Barthez (YouTube). Most of the effect of practicing, I think, would go towards making more confident each and every player.

Also, and interestingly enough, there are various kickers who are somehow predictable ... but let me keep that information secret for now. I will only mention that a few days before the European Cup final (Barcelona-Arsenal) last month, I emailed some data to one of the teams indicating how one of the usual penalty-kick takers of the other team was very predictable. I think that, also, the information that some kicker are unpredictable is valuable (since perhaps one might think that there is a pattern when in fact there is none) ...

GM: Before the Barcelona-Arsenal game, had you ever contacted a team about its opponent before?

IPH: No, it was the first time.

GM: Did you let your rooting interests decide which team to tell?

IPH: No. The reason why I contacted one of the sides is that I knew the chief financial manager of one of the clubs.

GM: Did you hear back?

IPH: Yes. He said that he appreciated the information and data analysis I was giving him, and that he would pass it on to the coach. (It turned out that there were no penalty kicks in the game.)

Related in Gelf

Blogging referee Aaron Corman tells Gelf why the World Cup refs haven't been so bad after all, Zooming In rounds up press coverage of the Cup from local papers around the world, while Gelflog looks at 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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- Sports
- posted on Apr 28, 12

None of the youtube links work.

Article by David Goldenberg and Rachel Bialik

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