Neymar’s injury shows what happens when soccer enters a bad equilibrium.
It has not been the best week at the World Cup, especially for American fans. Striker Jozy Altidore failed to take the field for the United States, which lacked a spearpoint to its attack in its heartbreaking loss to Belgium. Meanwhile, after a goal glut in the group stage, the scoring taps have almost run dry. And worst of all, the tournament's poster boy has been knocked out by injury. All of these unfortunate events can be explained using game theory -- and at least one of them could have been avoided. Here's how.
First, consider Altidore, who'd strained a hamstring during his team's opening game against Ghana. Coach Jürgen Klinsmann had him in uniform for the clash with Belgium, but the big center forward didn't play. You could forgive American fans for their disappointment; chatter from the United States camp suggested throughout the run-up to the game that Altidore would be ready. In the end, though, it was a bluff. Altidore's injure was too severe for him to recover in time, and Klinsmann admitted as much after the game.
In game theory, this is called signaling. Talking up Altidore was an attempt to make Belgium waste time preparing for an opponent that they wouldn't even face. As long as there was a non-zero probability that Altidore would play, the Belgians' best strategy was to spend a non-zero amount of time considering how they would defend him.
The United States ended up losing to Belgium 2-1, which was a big score for the Round of 16 and the quarterfinals. Earlier, the tournament had offered 2.8 goals per game in the group stage, but the next 12 matches averaged only 1.9 even with several periods of extra time. Again, this was partly the result of strategy.
In the group stage, the rules of the tournament give teams an incentive to play offensive soccer and run up the score; after all, the United States only advanced because of goal difference. But in the knockout rounds, when teams are more evenly matched, the most important thing is not to concede. Once a team scores, it has little incentive to send players forward looking for more goals; the preferred way to finish the game is by bunkering the defense. Indeed, three of the 12 games in the Round of 16 and quarterfinals finished 1-0, compared with eight of the previous 48.
The events above can be explained by game theory, but they couldn't necessarily have been avoided. That's not the case with Neymar's injury. Towards the end of a rough-and-tumble match, the Brazilian star was kneed in the back by Colombia's Juan Zúñiga and had to leave the field on a stretcher. At the hospital, he was diagnosed with a broken back. Yet his World Cup did not have to end this way.
To understand why, imagine that Brazil and Colombia were playing without a referee. In this hypothetical situation, each team would have a choice of whether to play dirty or clean. If both play dirty, neither will have a clear advantage and both will leave the field with injuries. If both teams play clean, then they'll both avoid injuries as well. But if one team plays dirty and the other doesn't, the dirty team will have a clear advantage.
This is a classic Prisoner's Dilemma. With no cooperation between the teams, both will play dirty, as it's the best strategy regardless of how the other team plays. Clearly, it would be better if they could both commit to playing clean. On their own, however, they can't credibly make this commitment. That's where the referee comes in.
The referee's job is to impose the good equilibrium so that the teams avoid the bad equilibrium. By laying down the law early in the game, usually by calling fouls for any illegal contact and issuing cards for particularly dirty ones, he can give the teams a strong incentive to stick to the high road.
Unfortunately for Neymar, Spain's Carlos Velasco Carballo did not perform this vital function in Brazil's match against Colombia. As former World Cup referee Graham Poll put it, "Carballo wore the referee's kit, but he wasn't in charge." He let foul after foul go unpunished, sometimes not even stopping the play, until he finally pulled out a yellow card against James Rodríguez -- himself the victim of numerous fouls -- after 67 minutes. When Neymar went down with only a few minutes left to play, Carballo did nothing.
Game theory can be a tool to beat an opponent, but it can also make an entire tournament safer and more enjoyable. Hopefully FIFA will convey that message to its referees before the next round of the World Cup begins.
Eitan Abramovich / AFP / Getty Images