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The Science of Penalties

One of my favourite pastimes is watching football (also called soccer in some parts of the world), there are few things more beautiful than seeing a ball kicked 40-50 yards and laying completely still at the feet of the intended recipient after a single deft touch.  The 2011/2012 season came to a close at the beginning of  May with dramatic finishes all over Europe.  The focus then shifted to Munich for the UEFA Champions League final, and now to Poland and the Ukraine for Euro 2012.

For the uninitiated, the UEFA Champions League sees the top 32 clubs in Europe taking part in the group stages, with two-legged knockout rounds later.  Euro 2012 is similar to the African Cup of Nations, the European continental competition. One only needs to look at this year’s ‘Group of Death’ containing Germany, Netherlands, Portugal and Denmark to see why it is said to be harder to win than the World Cup.

One thing that was interesting to note as this club season drew to a close was that penalties  became more and more important.  The Champions League final as well as the semi-final that pitted Bayern Munich against Real Madrid was decided by penalty shootouts.  After the final FIFA President Sepp Blatter called for an alternative to penalty shootouts, the general argument being that penalties is more a lottery than an accurate reflection of  skill.  I thought this was an interesting idea, so I decided to see whether any work has been done on what the best strategy is when you find yourself in that most lonely of spots where club or country depends on you for glory.

It turns out there has been and it brings us to the field of game theory.  Game theory originated from John von Neumann and Oskar Morgenstern’s work ‘The Theory of Games’  and attempts, very loosely, to explain human behaviour in cases where every participant has multiple action strategies, each with its own consequences or rewards. Game theory forms part of virtually every undergrad Economics course and has also found a home in some of the other social sciences.

It turns out that you can model a penalty as a two-person zero-sum game.  The two players are the kicker and goalkeeper, the actions available are kicking (diving) left or right and the payoffs are the probabilities of scoring (conceding) a goal, subject to the respective actions chosen. (Note: It has been shown that the option ‘Centre’ can be added to the options ‘Left’ and ‘Right’ without fundamentally changing the problem.)

Intuitively we would expect the kicker to act in such a way as to maximize the probability of scoring and the keeper to try to minimize it.  The question is now how often should you go, say, left in order to optimize your probability of success.

Well, if a player is following an optimal strategy the expected likelihood of scoring/conceding should be equal whether he kicks/dives left or right, otherwise he can easily do better choosing the action with the better expected payoff more often.  It is also necessary that an individual player’s choices are sufficiently unpredictable. The Minimax Theorem from game theory formalizes this.

Ignacio Palacios-Huerta from Brown University studied 1417 penalty kicks spread over five seasons to empirically determine the payoffs of the different action profiles.  Given these payoffs and our assumption in the previous paragraph, it was calculated that the kicker should go left 38.5% of the time and the goalkeeper 42% of the time.  What is interesting is that the actual frequencies over the five years showed the kicker went left 40% of the time and the goalkeeper 42.3% of the time.  These frequencies are close enough to optimal to be accepted at a 5% level.

Further tests focussed on individual players and for only 3 of the 42 players considered the null hypothesis that they are following optimal strategies could be rejected.

This leads to the interesting conclusion that, whether they know it or not, the vast majority of professional football players follow an optimal strategy when trying to score or save a penalty.

If we assume that all the players are playing an optimal strategy, it could suggest that the result of an individual penalty shootout is largely dependent on luck, leading some credibility to Blatter’s claim.  Football fans everywhere will be waiting to see whether FIFA manages to come up with a viable alternative within the next year or two, but while they concern themselves  with that, we’ll be enjoying Euro 2012 and all its potential penalty drama.  Germany all the way!

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