According To Game Theory, Germany And The US Should Collude To Get A Draw In Their Next World Cup Match

There's an argument based on game theory that both the United States and Germany should pretty much just stand around the ball during their match on Thursday.

How World Cup Points Work

In the World Cup tournament, the first round consists of eight groups of four teams each. Each team plays all the other teams in its group.

If a team wins a game, they get three points and the losing team gets nothing. If the teams tie, they both get one point.

The two teams with the highest number of points go on to the next round. After Sunday's heartbreaking tie with Portugal, the U.S. has four points, as does Germany. The other two members of our group, Ghana and Portugal, each have just one point.

So, if the U.S. and Germany draw on Thursday, both teams will have five points, and neither Ghana nor Portugal can have any more than four points. This would mean that both the U.S. and Germany would advance into the knockout round of the tournament. If the U.S. wins (which FiveThirtyEight only gives a 14% chance of happening), they will advance, and Germany will very likely also still advance, barring something very strange happening in the game between Ghana and Portugal leading to a blowout for one team or the other. If Germany wins then they advance, but things get rather complicated for the U.S. and their chances.

For more details on what the possibilities are for our group, check out our own Tony Manfred's write up here.

Neither Team Needs To Win And Neither Wants To Risk Injuring Players

This makes a win not particularly important for Germany. They should be perfectly happy to take the draw, and a draw is basically the best reasonably expectable outcome for the United States.

This has caused some people to speculate that Germany and the United States could collude and fix the game - neither team would try to do much of anything on Thursday, leading to a pre-ordained 0-0 tie. U.S. coach Jurgen Klinsmann has denied any intention of doing this, but it might be the best option for both teams, since both would advance, and without playing particularly hard, both teams could avoid any risk of injury or player suspensions going into the knockout round.

Making collusion work, however, would be tricky. While both teams would be best off not trying too hard and aiming for the draw, one team might betray the other, and launch a sneak attack against a team expecting a quiet day in the park. This would give the betraying team an advantage. However, collaborating and not really playing to win to get to the draw is still a better option, since now both teams are exposed to risk. Further, the collusion could completely collapse, with both teams trying to actually win, leading to a normal looking soccer game.

The Stag Hunt

This situation is actually an example of a classic problem in game theory: the stag hunt. The problem, originally formulated by Jean-Jacques Rousseau, involves two hunters who can choose between hunting stags or hunting rabbits. If the hunters team up, they can take down a stag, and eat like kings. If only one hunter tries to hunt a stag, and the other just goes for rabbits, the stag hunter is out of luck and goes home hungry, and the rabbit hunter gets a rabbit. If both hunters go for rabbits, they both get rabbits.

The issue is similar to whether or not the U.S. and Germany should play to win or just run ninety minutes of passing drills. While it would be better for both overall to cooperate, it's safer for a team or hunter to defect. Both hunters going for a stag will get more meat, but a hunter going for rabbits is guaranteed to get a rabbit.

It's helpful to put this situation into numerical terms. If both teams collude and don't try to win on Thursday, let's say each get 3 utility points, representing a safe path to the next round. If one team is not trying, and the other team defects and plays to win, the defecting team gets 2 points, and the now-betrayed cooperating team gets 0 points, representing the advantage the defecting team gets over the surprised cooperating team. If both teams defect and play to win, each gets 1 point, representing a normal soccer game.

In the analysis of game theory, this kind of situation is usually written out as a payout matrix like the one below. Each row shows the U.S. team's possible decisions, and each column shows Germany's possible decisions, with the respective U.S. utility points for each situation in blue and the German points in red:

This gives us an idea for what each team should do. If the USA cooperates and lounges on the field, they either get the best score of 3 utility points if Germany joins them, but with the risk of getting the worst score of 0 if Germany defects and plays seriously. Meanwhile, defecting is less risky - the U.S. gets at least one utility point - but there's no chance of getting the best possible outcome.

The resolution to this game relies on how much the two teams can trust each other. If Klinsmann wants to collude, and he's confident that Germany will go along with the plan, then both teams can reasonably safely aim for the best possible outcome, and safely and easily walk into the next round. However, if there's doubt in his mind and he expects the Germans to defect and play to win, then the U.S. team should do the same.

These Situations Happen Frequently

This puzzle and others like it, such as the famous Prisoner's Dilemma (which is very similar to this game, except that the payoff for betraying while the other player cooperates is higher than the reward for both cooperating, leading to the best strategy being to always betray the other player) are frequently used in economics and the social sciences as a highly simplified model for how interactions work in society.

The stag hunt game and the decision of whether or not to play to win on Thursday are small scale illustrations of situations where people collaborating and banding together can get much better outcomes for everyone involved than everyone working on their own, but if some people work together and some people work alone, the people working together get nothing.

One example given at Stanford's website explaining the stag hunt is pollution: everyone near a lake wants a clean lake, but if it's possible for one person to dump their garbage in the lake and ruin things for everyone, it's hard to coordinate everyone to keep the desirable outcome of a clean lake instead of moving to a nightmare scenario of everyone dumping their garbage.