Friday, June 18, 2010

Game Theory in the World Cup

The classic Game Theory puzzle is the "Prisoner's Dilemma." In short, two criminals are caught and separated (so they can't collaborate on their respective stories), if neither rats on the other, they'll both get one year. If one confesses and the other does not, the confessor goes free, and the non-confessor gets 5 years. If both fess up, they both get 3 years. The numbers can vary - and, if you want to be purely mathematical about it, the numbers can make a big difference - but the core of the problem is always the same. Each criminal has to make a choice, either denying everything and getting either 1 or 3 years, or trying to rat out the other and getting either 0 or 5 years.

There are infinite variations of this problem, but since we're in the midst of a World Cup, I think it's informative to look at Group A as an exemplar of the classic prisoner scenario. Consider the current standings in Group A:

1 Uruguay 2 1 1 0 3 0 3 4
2 Mexico 2 1 1 0 3 1 2 4
3 France 2 0 1 1 0 2 -2 1
4 South Africa 2 0 1 1 1 4 -3 1

Uruguay and Mexico are sitting pretty, with 4 points each, while France and South Africa are both on the outside looking in. As a result, this is a kind of reverse-dilemma for Uruguay and Mexico because they play each other in their next match while France plays South Africa. Since a win nets three points and a draw one point, Uruguay and Mexico are in a position to both advance to the next round if they tie each other. They could both, in theory, simply let the ball sit at the midfield and play Duck-Duck-Goose for 90 minutes, and move on to the knockout round. They're not likely to do that, but there isn't a lot of impetus to play hard.

Let's assume that France dominates South Africa. Granted, France has been far from impressive in its first two games, but South Africa has also been quite poor, and they simply can't match France's skill. With a 4-0 scoreline, France would be in a position to make the next round if either Uruguay or France were to beat the other 1-0.

Now, assuming France can pull off a 4-0 thrashing of the host nation, what should Uruguay and Mexico do? If they tie 0-0, they both move on, whereas a win for either team means one of them goes home. Like the prisoners, their best mutual option is for neither team to try, and to therefore both advance. Like the prisoners, however, each individual team will benefit most from playing hard while the other team doesn't. It is true that, unlike the prisoners, both teams playing hard doesn't have any significant disadvantages, except the risk of picking up suspensions or injuries moving into the next round.

The analogy fails slightly on one point, which is that the order of finish in Group A has serious repercussions. The winner of Group A is likely to play either Greece or South Korea in their first knockout game. Neither of those teams is bad, mind you, but the Group A runner-up faces (barring some shocking results in Group B's final match) Argentina, the odds-on favorite to win the entire tournament. Uruguay, by virtue of their goal differential after their thumping of South Africa, are in a position to win the group with a draw.

The result is a twist on the "Prisoner's Dilemma," wherein Uruguay's position should be that a draw is fine, and that Mexico clearly shouldn't play hard because, hey, we're both moving on. Mexico, on the other hand, might say that we need to play hard because the difference between playing Argentina and South Korea is potentially the difference between winning and losing in the round of 16. Conversely, however, Mexico might say that Uruguay need not worry as much about getting caught by France, because they already have a better goal differential. A 2-1 final for Mexico, for example, would mean that France would need a 5-1 (or 5-0) victory instead of 4-0 to catch up.

The strange logical result here is that Uruguay and Mexico should intentionally allow each other to score one goal, in order to increase their mutual likelihood of moving on. At that point, playing to win, but doing so cautiously, seems the best option.

It is unlikely either team will intentionally concede, but I do expect Mexico and Uruguay to play each other somewhat tentatively. Mexico has more to play for, but not by much, and while Argentina is a terrifying foe, no one gets through the knockout stages of the World Cup without playing terrifying foes.

Undoubtedly there will be other examples of the Prisoner's Dilemma in the World Cup, as it goes with the territory of the three-team group format. In all likelihood, we'll see something even more true to the Game Theory form. Indeed, within any particular match there are dozens of Game Theory situations at play in terms of tactical approach. I think that's why soccer - or football - is such an engaging sport. It is both tactical and strategic, and it affords not only opportunity for appreciation of physical feats, but also mental ones.

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