As a Brazilian person, I can not forget the most popular sport in Brazil: soccer. It is not that I care much for it, but at least all Brazilian male do it.
The six dimensions of Game Theory classification for Penalty
- Simultaneous Game
- One-shot
- Zero-sum
- Full information / partial
- Rules Fixed
- Cooperatives Agreements possible? No!
Depending of the championship's rules, the game can not end equal, and each team has an equal number of chances to make penalty kicks (PK) in a shootout. In 2002 World Cup, 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, the Italians could have been world champions themselves, but then Brazil won the World Cup for the fourth time. So, the ability to make penalty kicks (and stop them if you are a keeper) is tremendously important. For Ignacio Palacio-Huerta, that ability is the focus of an exhaustive study that reveals how keepers and shot-takers alike deal with penalty kicks. But Palacios-Huerta doesn't analyze player tendencies in order to help a particular team, his 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). 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.
Why kicking to the center could be a good idea? How he would advise teams to approach penalty kicks? The interview has been edited for clarity:
Gelf Magazine: 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.
Ignacio Palacios-Huerta: 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. 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.
GM: 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. 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, it is said that the penalty was shot a la Panenka. It is very risky but the fame payoffs is great.
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. 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, it is said that the penalty was shot a la Panenka. 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 millisecond. If they start thinking a lot about it they will probably stop 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 millisecond. If they start thinking a lot about it they will probably stop being random.
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.
GM: Often, the keeper chooses the correct side but still gets beaten. How does game theory apply in these cases?
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.
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: 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...
Even if I am not a big fan of soccer, who doesn't remember this moment in the 1994 World Cup, when Baggio lost his penalty, and gave Brazil the World Cup leadership?
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