(Team) Game theory: why Arsenal is top, and what Man Utd can do
Posted by GTG on February 29, 2008
Check out this article applying game theory to the NBA. Having established a correlation between better teamwork (more passing) and win percentage through the sheer force of mathematics, it goes on to analyse why some teams don’t pass as much as they should, even when it leads to better chances of winning.
Enter game theory, analysing a two-player team presented with two options (pass or shoot):
| Look to pass to open man | Be a chucker | |
|---|---|---|
| Look to pass to open man | win-win | lose much-win much |
| Be a chucker | win much-lose much | lose-lose |
Given these payoffs, the Nash equilibrium for both players will be to “go for glory” and make the shot, even when passing is the best-case scenario. This effect is more pronounced in teams with more players, since the win-win outcome requires co-operation from all players, making the temptation to be the one who chucks the ball in too great to bear. So much for “there’s no I in team”.
I wonder: can this concept be applied to other team games, such as football? Looking at the EPL, sitting pretty at the top of the table is Arsenal, whose passing ratio far surpasses any other team in the league. Even after correcting for differences in the style of play, perhaps we should give credit to the fact that their players value the team over personal glory. Or should we?
Perhaps Arsene Wenger may have managed to set the payoffs for each player such that it is beneficial to pass rather than shoot. (Some fans may claim he has overdone it, seeing how the team tends to overpass the ball - they aren’t complaining this season though.)
Which moves me to my next point: how do we overcome the problem of selfish play dictated by the Nash equilibrium?
I believe it’s all a matter of incentives. Make passing more attractive and shooting less so. A direct way will be to alter goal and assist bonuses - extra cash paid for assists made and goals scored - so that assists are equally, if not more lucrative, than goalscoring. Another way would be to threaten selfish players with the bench. The point is to try to adjust the payoff scheme such that:
| Look to pass to open man | Go for goal | |
|---|---|---|
| Look to pass to open man | win-win | draw-draw |
| Go for goal | draw-draw | lose-lose |
In this case, players are indifferent between shooting and passing, having adjusted for the risk of missing and earning the ire of the manager (or worse, a place on the bench). The best action for players, clearly, is to keep passing the ball, until players find themselves in a situation when it’s harder to miss than to score (note: you may be surprised by the number of times that some players staunchly continue to miss).
If anything, players become adverse to shooting, which may lead to the problem of overpassing: last season, Arsenal players even preferred passing the ball into goal than shooting.
A look at the pay scheme of Arsenal players versus those of other teams should prove interesting. Too bad your author is just a lousy, low-down student without even the money to buy an ordinary ticket to Arsenal games.
