## Game Theory in the World Cup

This article from Business Insider shows an interesting example of how game theory can be used to model real life scenarios. The article is about a scenario that occurred during the group stage of the 2014 World Cup. The first round of the world cup involves eight groups of four teams who will each play the other teams in their group once, and based on points earned from the results of these games, the top two teams in each group will move onto the next round of the tournament. In the group consisting of Germany, the U.S., Portugal, and Ghana, the U.S. and Germany were tied for the most points in the group at four, while Ghana and Portugal had one point each. The U.S. team and the German team both had their last game of the group stage against each other. The winning team would earn three points, while the loser would earn none, and they would both get one point for a tie. If the U.S. and German teams were to tie, they would both end with five points and be guaranteed to advance to the next round, while not tying would potentially allow Ghana or Portugal to take the second place spot away from the loser. This created an interesting situation that could easily be modeled using game theory.

Because of this situation, both teams could actually benefit the most through collusion. If the teams were to agree to intentionally tie the game, they could both be sure that they would move on to the next round, and they would avoid any risks during the game that could potentially affect the rest of the tournament, such as player injuries or suspensions. This especially would benefit the U.S. team because the German team, who went on to win the World Cup, were by far the favorites for the match. However, there was a risk to this collusion. If one team were to betray the other and actually play to win after agreeing to tie, then the team that was cooperating would be at a significant disadvantage after being surprised by the betrayal. And of course, both teams could not collude and play normally, resulting in a standard game. In this scenario, the optimal result would be for the teams to collude and agree to tie, but this would not be done without complete and total trust of the other team. If there was any expectation by either team that the other would betray them, then they would decide to play normally as well, ending in a normal game. In the end, the two teams did not cooperate, and Germany won the game 1-0, although the U.S. still did end up advancing past the group round.

This scenario showed exactly how similar real life situations can be to problems modeled in class by game theory. While real life examples of game theory are usually much more complex than this, the U.S. vs Germany game could be easily modeled by a simple two by two payoff matrix with two players and two strategies each. Its result is also a great example of how cooperation in games can often result in a better outcome for all its players, yet typically the players still do not cooperate out of fear that the other players may betray them and leave them even worse off than if they hadn’t tried to cooperate.

Payoff matrix modeling the scenario