Brazil v. Ivory Coast and World Cup Game Theory

In keeping with the World Cup series of posts, here’s a quick post today highlighting some interesting cases of game theory in World Cup play, historical and current.

Last World Cup in 2006, I read a Slate article describing how penalty kicks represent a perfect application of game theory in the real world. The situations are predictable and controlled, and the number of choices are extremely limited, providing for an easy model. In a penalty kick, a free kick is taken by one player twelve yards from the opposing team’s goal, with only the goalie of the defending team to protect the goal. There are only a handful of choices for the kicker to make, most importantly which direction to kick the ball–to the goalie’s right, left, or straight center–and to some extent whether to drive it to the high corner or low corner. For the goalie then, he must choose to go to his right, left, or stay in the center. According to the Slate writer, “Game theory, applied to the problem of penalties, says that if the striker and the keeper are behaving optimally, neither will have a predictable strategy. The striker might favor his stronger side, of course, but that does not mean that there will be a pattern to the bias.”

Professionals such as the French superstar Zinédine Zidane and Italy’s goalkeeper Gianluigi Buffon are apparently superb economists: Their strategies are absolutely unpredictable, and, as the theory demands, they are equally successful no matter what they do, indicating that they have found the perfect balance among the different options. These geniuses do not just think with their feet.

Freakonomics authors Levitt and Dubner also tackled the question in a recent piece for The Times (U.K.). They concluded that the best option was to kick the ball straight down center: “One strategy proved considerably more likely to score a goal than any other. It wasn’t shooting to the left. And it wasn’t shooting to the right. The wisest choice is to kick the ball straight down the centre. And yet that was the least-common choice.” This optimal strategy was the least common because it didn’t take into account the preferences of the kicker, who wanted not only to score, but to hedge against looking the utter fool. A kicker who kicks it straight down the middle and has his shot blocked will be the subject of scorn for making the seemingly ridiculous choice of kicking it straight at the goalkeeper.

In addition to penalty kicks, game theory in the group set-up can also come into play when deciding how to play an individual match. In the World Cup, the top two teams of each four-team group will advance to the knock-out rounds. In the Mind Your Decisions blog, Presh Talwalker recounts an incident in the 1982 FIFA World Cup:

West Germany played Austria in the last match of group B. A West German victory by 1 or 2 goals would result in both teams advancing; any less and Germany was out; any more and Austria was out (and replaced by Algeria, who had just beaten Chile). West Germany attacked hard and scored after 10 minutes. Afterwards, the players then proceeded to just kick the ball around aimlessly for the remainder of the match. Algerian supporters were so angered that they waved banknotes at the players, while a German fan burned his German flag in disgust. By the second half, the ARD commentator Eberhard Stanjek refused any further comment on the game, while the Austrian television commentator Robert Seeger advised viewers to switch off their sets.

He describes another perverse incentive in the 1994 Shell Caribbean Cup. Goals in overtime (sudden death) would count double, to reward teams in close matches.

Barbados needed to win by two goals. With less than ten minutes left in the match, Barbados led by exactly two goals and began to play very defensively. In the 83rd minute, Grenada finally scored, making the score 2-1. Barbados tried to answer but, with only three minutes remaining, was unable to score. Members of the Barbados team contemplated their options. To advance, they needed either to score one more goal in the last three minutes (winning by two), or force the game to extra time where a goal would count as if they won by two. Barbados scored on their own net, tying the game at 2-2.

This is not yet the odd part of the match. The Grenada players, initial shock abating, developed their own strategy. If they could score on Barbados in the waning minutes, they would win the match and advance. But, if they could score a goal on themselves, they would lose by one goal which was still enough to advance. For two minutes, Grenada tried to score on either goal, with Barbados players split between defending their own goal and that of their opponents!

Normal time ended in a tie and the game did go to overtime, in which Barbados scored a game winner and advanced (though was eliminated from the tournament in the next round). No penalties for the players’ actions in this game were handed down since both teams were earnestly trying to win their group, and the farce was the result of silly incentives.

Today in watching the Brazil v. Ivory Coast game, I witnessed some poor game theory on the part of the Ivory Coast team. Brazil had taken a commanding 3-0 lead, and were virtually guaranteed to win the game. In fact, according to the announcers, Brazil had never even lost a game where they led 1-0 at halftime. Didier Drogba scored for the Ivory Coast on a good header to make it 3-1, but still the game was likely out of reach for the underdogs. In the 87th minute near the conclusion of the game, Ivory Coast player Kader Keita ran into Kaka’s elbow, and dropped to the ground clutching his mouth. It was a clear dive, because nothing was near Keita’s mouth at all, and additionally he had run into Kaka. Kaka walked away amid some shouting and tension, and soon received a yellow card. It was his second yellow card of the tournament, which equaled a red card, forced him out of the remainder of the match, and suspending him for Brazil’s final Group game against Portugal. As the NYTimes reported:

The midfielder went off, improved from his opening showing but losing the chance to collect more precious minutes against Portugal, minutes he might need given any rustiness born of his injury-addled first season with Real Madrid that saw him play only 22 matches.

In fact, with Brazil in commanding lead of Group G with 6 points from two wins, guaranteed to advance (likely as the top seed), and Portugal and Ivory Coast now tied at only one point, it is actually the Ivory Coast that will miss Kaka’s “precious minutes against Portugal.” If Ivory Coast beats North Korea in their final Group game, and Brazil beats or ties Portugal, Ivory Coast can advance. If Ivory Coast only draws against North Korea, but Brazil beats Portugal, Ivory Coast will advance. Therefore, Ivory Coast should want Brazil playing at top form against their primary opponents for the second spot out of Group G. They should have preserved Kaka’s (ranked 4th best footballer in the World Cup by ESPN) eligibility for the next game. Thus, an inopportune red card for the Ivory Coast.


I made a mistake in reading the schedule and misrepresented the situations in which Ivory Coast can advance. Portugal currently at 1 pt still has 2 games left against North Korea, then Brazil, so the scenarios are more diverse than I implied. Either way, Ivory Coast should want Brazil with their best player when playing Portugal.

Ivory Coast player clutches mouth as he dives.

Ivory Coast player runs into Kaka's elbow.


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