Argument

The Game Theory of the Game

The Game Theory of the Game

Sports are supposed to be simple. Two teams, one goal — may the best men (or women) win. Tournaments can complicate matters though.

All of a sudden, instead of the single goal of winning a game, teams have a bigger objective: take home the trophy. And that’s where things start to get complicated, as they are for Germany and the United States in Brazil, where they face off in a third and final group match on Thursday, June 26.

In single-elimination tournaments like the NCAA’s March Madness or the NFL playoffs, the strategy is still straightforward enough — win and advance, or lose and go home. But in the World Cup and other round-robin competitions, especially when there’s a transition from group stages to knockout rounds, the intrigue begins. In Brazil, with four teams vying for two spots, the third round of the group stage offers a host of competing incentives, and the aims of the protagonists can be far from clear. Luckily, we have game theory to help us understand what is and isn’t at stake for the two sides.

You remember game theory — you probably had a junior high teacher who taught you about the Prisoner’s Dilemma, or you went and saw A Beautiful Mind (though you’d probably understand Nash equilibrium better if you hadn’t, but I digress). In general, the idea is that if you examine what decision-makers have to gain by making different choices, you can predict what they will do. So, let’s take a look at what the United States and Germany will be considering.

The first thing that stands out is how different this game is from a normal match for Germany. Given the respective point totals and tiebreakers in the group, Germany is all but assured of going through. Win, lose, or draw — Germany will play in the Round of 16. However, with a win or a draw they will also win the group; with a loss they will come in second. Germany is playing only to determine what their path to the final looks like.

Come first, and Germany will face a path that consists first of the runner-up in Group H (a decidedly lower-ranked team like Algeria seems likely), followed by France and possibly Brazil before the finals. Again, this will happen whether Germany wins or draws against the United States; unlike in most soccer matches, there is literally zero difference between the results here. But if Germany comes second, it will likely face Belgium, possibly Argentina, and then whichever victor emerges from the Netherlands, Mexico, Greece, and Costa Rica.

For the United States, the story is slightly different. Win, and they’ll top the group with the path that entails; draw, and they come in second. If they lose, their future rests on what happens between Portugal and Ghana in the other game, and they could be eliminated. So, while there may be slight differences for the United States between winning and drawing, there’s a huge difference between drawing and losing. Just like Germany, the United States faces a very different set of incentives from what you’d normally see in a soccer match.

That’s how this one game lines up, but remember: Thursday’s match is only one part of the bigger picture. If the result makes no difference for Germany’s chances to win the tournament — that is, if the two paths are roughly equivalent — then coach Joachim Löw should also consider other things. Does resting his stars, giving them time to recover while offering his reserves a chance to stay sharp, increase his team’s chances of winning the entire tournament? Or is the opposite is true — that playing his best team together for another match will increase their cohesion and generate even bigger benefits?

Such considerations could extend to tactics within the game as well. Might Germany decide not to use its energetic pressing defense, if there was even the possibility it might hurt them down the line? If the sole purpose of the tactic is to expend energy and win the game, it’s difficult to see why they would press. But again, if Löw thinks that pressing might help to train his players, or that deviating from his usual gameplan might do more harm to his players’ psyches than good to their fitness, perhaps he will continue as before. If the outcome of the match truly makes little difference to Germany’s chances in the tournament, then considerations like these — which might seem marginal in a knockout match — might be important enough to sway him.

The United States has no such luxury. For coach Jürgen Klinsmann and his squad, the goal is clear: they must do everything in their power not to lose. There will be no room for adventurous forays that could leave the defense exposed. Counterattacking will be the order of the day. If Löw’s decisions help them to keep the game under control, so much the better.

If it sounds like a draw might suit both teams, then you’ve been paying attention. And it’s not the first time such a thing has happened. In the 2004 European Championship, Sweden and Denmark played a 2-2 draw that saw both advance from the group stage only by having superior goal difference to Italy. They even managed to save the last goal for the 89th minute — quite a risk if the fix was in.

Of course, all of this assumes that the primary goal of both teams is to advance as far as possible in the tournament. But there’s one more dynamic at work. Löw was Klinsmann’s assistant when he took his home country of Germany to the semifinals of the World Cup in 2006. Afterwards, some said Löw was the real brains of the operation. Löw took over the national team and Klinsmann moved to California. The two rivals might just send their players out hell bent for leather, even if it means running them into the ground, just to prove a point.

It’s also important to consider exactly whom the players will be playing for in Recife. If it’s for themselves, they may strive to score audacious goals even if the coaches tell them to play it safe. If it’s for the people who bought tickets from FIFA, then they’ll try to play entertaining soccer even if its means risking elimination. Only if it’s for their own fans will they stick to the smart strategies above.

When fans arrive at the stadium for a World Cup match, they usually expect both teams to do everything possible to win. Yet thanks to the structure of the tournament, Thursday’s match will be between a team trying not to lose, and one to whom the result is largely incidental. German and American fans may nod their heads solemnly as their teams play out the corresponding strategies. For the neutral, it’s likely to be a snooze.