The social rationality of footballers
Are footballers rational? It all depends on what their goals are (no pun intended). We will not be talking here about behavior outside the field, as it's not entirely clear what norms of rationality one should use in this case (as George Best put it: "I spent a lot of money on booze, birds and fast cars. The rest I just squandered."). However, when playing, footballers seem to have a very clear incentive: winning the game. After all, the indecent salaries of many professional footballers depend on their team winning as many games as possible. Nowhere is the situation as clear-cut as in penalty kicks. The kicker must put the ball into the nets while the goalkeeper must stop him from achieving his goal, period. Surely, the combination of huge stakes and intensive training should produce optimal behavior on both sides of a penalty kick. This is what Michael Bar-Eli and his colleagues have tried to find out in research reported here [1].
After having watched hundreds of games (or hundreds of penalty kicks at least), the team was able to compute what was the best strategy, both for the goalkeeper and for the kicker. Let's start with the goalkeeper. He has basically three choices: staying where he is, in the center, or diving to the left or to the right. In the sample of penalty kicks analyzed, his chances of stopping the ball were one out of three if he stayed put (very good odds indeed!), and below 15% if he chose to dive right or left. Is this how goalkeepers behave? Not at all. Even though the best bet is to stay in the center, the goalkeepers only did that in 6% of the penalty kicks. How is such an apparently irrational behavior to be explained?
According to the authors, who have conducted interviews of several goalkeepers, the explanation is to be found in the "action bias". Goalkeepers feel a pressure to act because they would feel guiltier missing a ball while staying in the center than missing it while trying to do something. From the perspective of someone watching the game, this is not surprising: everybody knows that stopping a penalty kick is very hard, so we would not think ill of a goalkeeper who fails while powerfully throwing himself to the side, whereas we are likely to think that one who hasn't budged didn't put much of an effort. If this is right, then the goalkeeper's behavior may very well be optimal, not in terms of stopping the ball, but in terms of avoiding blame from his coach, teammates, or other spectators.
What of the penalty kickers? Are they more ‘rational' in their choices than the goalies? According to the observation of the Bar-Eli team, the optimal behavior for a penalty kicker is to target the upper third of the goal. In the sample they analyzed, not a single kick in this part of the goal was stopped, as compared to 30% in the central third and 57% in the lower third. So, do the kickers consistently aim at the higher third? No. Why? Because the way of failing varies, and seems to be more important than the actual rate of failure. When the kicker targets the lower third, most failures will come from the goalkeeper having stopped the ball, an ‘honorable' defeat, brought about by the goalie's skills. On the other hand, when the kick is aimed at the upper third, most failures will come from kicks that miss the goal altogether. In such cases, the kicker only has himself to blame. And everybody else only has the kicker to blame. In this light, it makes sense that the kickers should act in a way that is going to minimize reproach rather than only the chances of missing.
In the end, I think that the behavior of the footballers in this case is quite rational. As long as no one knows about this study that is, because if these results were to become widely known, then goalkeepers who do not move and kickers who miss the goal might not be blamed anymore. But then their behavior would be likely to change and to align itself with these new norms.
[1] Bar-Eli M, Azar OH, Lurie Y. (2009) (Ir)rationality in action: do soccer players and goalkeepers fail to learn how to best perform during a penalty kick? Prog Brain Res. 2009;174:97-108.
Nicolas Baumard 29 March 2010 (14:33)
Very interesting! It looks like managers are irrational then. They should create monetary incentives or moral atmospheres that help goalkeepers and kickers to take the best decision for the team. In a way, football teams have a management problem!
Hugo Mercier 30 March 2010 (19:06)
You're right, but there's a twist in that it's far from clear what the equilibrium would be. For instance, if the goalkeepers started to always remain in the center, then the optimal strategy for the kicker would change (to a low kick on the side), and then the rational strategy for the goalie would change, etc. Some game theory anybody?
Christophe Heintz 11 April 2010 (17:00)
Hugo hints that the optimal strategies depends on the expectation of the opponent. Indeed, most games implies speculating on team members' and opponents' expectations. This is because winning strategies require to fulfil team members' expectations and thwart opponents' ones. Suppose that the motivation of kickers and goal-keepers is just to mark and prevent opponents to mark (i.e. forget about the motivation for easy post hoc justification) and suppose (for the sake of the modellisation) that Bar-Ali's paper include only advice for the goal keeper so that we have the following strategies: Goal keepers: stay put or throw yourself on the side. Kickers: aim at one side or aim at the centre of the gate. From reading Bar-Ali's paper, a goal keeper should infer that, caeteris paribus, one should stay put, even though this is not what they usually do. Imagine kicker K shoots against goal keeper G. K's decision will be dependent of whether (k0) he (i.e. K) has read Bar-Ali's paper [aim at the centre because goal keeper throw themselves on one side most of the time], (k1) but also whether he (i.e. K) thinks G has read the paper [aim at one side because G having read the paper will stand still] and (k2) whether he (i.e. K) thinks G think K has read the paper [aim at the centre because G will expect K to aim at one side and will therefore throw himself on one side], and (k3) whether he (i.e. K) thinks that G thinks that K thinks that G has read the paper [aim at one side because G will stay put since he expects K to aim at the centre because he (i.e. G) thinks that K believes he (i.e. G) will throw himself on one side]. (kn) whether K believes that G assumed that K made n-1 eductive inferences (= inferences about what the other think about what I think) [aim at a side if n is odd, aim at the centre if it is even] We have the same that goes with G: (g0) G has read Bar-Ali's paper [stay put] (g1) G has read Bar-Ali's paper and thinks that K has read it too [still stay put] (g2) G has read Bar-Ali's paper and think that K has read it too and that he knows that G has read it [throw himself on a side] (g3) .... [stay put] (gm) ... [stay put if m is odd and throw himself to the side if m is even] With all this, G wins when m is odd (stay put) and n is even (aim at the centre) or when m is even and n is odd, which is to say, G wins when m+n is odd. K wins when m+n is even. The series does not converge, so common knowledge (when m and n are infinite) is of no help here, which may frustrate some economists. But I guess it becomes fun only with repeated games with belief updating processes. Suppose K has a probabilistic belief p about m being odd or even. He updates p each time he can observe the behaviour of G. How does he do that? He can do some bayesian reasoning that confirm that m is odd when he observed G staying put. You can endow G with the same belief updating process. And at this is the point the game theorist should tell us if he manages to obtain a converging series or not (probably not). Unfortunately, the bayesian updating procedure could be taken to be quite different: e.g. if G stayed put, then increase the probability that he will throw himself on the side next time, since G will also change its own beliefs about K's beliefs ... and all these fun eductive thinking has not been side stepped. That was some amateur's thoughts towards penalty game theory. But for the students of cognition and culture, one should go beyond: the question is how do individuals actually manage to cope with this kind of complexity of our social world: We are pretty good at understanding others' expectations, but how good? And how many eductive steps means good? And are we good because we do lots of eductive steps or for some other reasons? Oh! And about the original topic: G and K could always say they thought the other did one more or one less eductive step than they thought. This may be an important characteristics of games to be based on our powerful but bounded social cognitive skills.