Where do fouls come from? Listening to Arsene Wenger or reading analyses of the World Cup final, they reflect an inferior team's strategic decision to disrupt the play of superior teams. That may well be, but I bet that's not the end of the story. Anyone who's ever played the game knows that on some days the match just turns out to be a little nastier, tougher, more competitive than on others.
Why would this be? Game theory may offer one possible answer. Game theory is a branch of applied mathematics social scientists use to capture behavior in strategic situations or games, in which an individual's success in making choices depends on the choices of others (Wikipedia). It has famously been applied to study the interaction of goalkeepers and shooters during penalty kicks (for a nice piece on that, see this article in Slate Magazine). But beyond penalty kicks, it has rarely been applied to soccer analyses.
So on the heels of my recent posts about fouls committed and suffered during the course of a match, I started to wonder if game theory could also provide insights into other situations in football. In particular, I remembered there is a famous concept in game theory called "tit for tat", which refers to a particular way of playing the game (a strategy, if you will). Quoting Wikipedia: "an agent using this strategy will initially cooperate, then respond in kind to an opponent's previous action. If the opponent previously was cooperative, the agent is cooperative. If not, the agent is not." The key thing about tit for tat is that is is the optimal (winning) strategy in so-called repeated prisoner's dilemma games.
So where would we look for tit for tat in football matches? Very simply put, more fouls by one team in the course of a single match should beget more fouls by the opposite team. If all the mention of game theory has your eyes glazing over by now, just think of it as sort of an "eye for an eye" strategy. Thus, contrary to the idea that bad teams foul more, both good and bad teams should foul more when the other team fouls more, and they should foul less when the other team fouls less.
To get a quick and dirty handle on this, I collected data on fouls committed and suffered by each side during matches in the Big Four leagues (Bundesliga, EPL, La Liga, and Serie A) for the 2005/06 to 2009/10 seasons (I was able to get these data for 7,219 matches). I then graphed fouls committed against fouls suffered - remember, we would expect to see a positive correlation: as one goes up, so should the other. Here's what we get when we do that:
The scatterplot reveals systematic evidence of tit for tat. As the number of fouls committed goes up in a match, the number of fouls suffered goes up as well (and vice versa, of course). Moreover, this pattern exists in every one of the four leagues. Sure, there is variation around this central tendency - plenty of matches deviate in one way or another - but the fact that we can see it, given such a large sample of matches, and the fact that we do not see the opposite pattern or no pattern are worth remembering.
We can see this pattern most clearly when we use the data to extract the linear combination by calculating regressions (where we make outcome - fouls committed - a linear function of the other - fouls suffered). Here's what the data show when we do that (these graphs strip out the variation around the average in the data):
Clearly, these regressions show that fouls by one team go hand in hand with fouls by the other. Put another way: there is no league in which this is not the case. To put numbers on these linear correlations, the strongest connection between fouls committed suffered appears to exist in the Bundesliga, La Liga, and Serie A (where Pearson's r is a remarkably similar .19 across the board) and somewhat weaker in the Premiership (r = .14). Not sure why this would be, except to say that play in the EPL is slightly less tit for tat oriented than in the other three leagues because there are fewer opportunities to play tit for tat, given the lower foul totals in the Premiership I reported in a previous post (yet another difference between the EPL and the rest of the top leagues).
