Game Theory in Soccer

As we shall soon see, game theory has a number of applications in sports. This is not all that surprising because, after all, sports are games that have implicit costs, utilities and strategies. In this discussion, the game theoretic underpinnings of soccer will be considered. Likely the two most common invocations of game theory with regard to soccer are the semifinalists’ dilemma and the decision theory of penalty kicks.

The semifinalists’ dilemma arises from the distribution of a finite level of stamina between the semifinal round and final round. We assume there are four equally matched semifinalists (representative of opposing soccer teams) A, B, C, and D. In the first round—the semifinal round—A will contest B, and C will contest D. The two prevailing teams from the semifinals will advance to the final round, in which both players will vie for the championship title. The two losers from the semifinal round will return in the final round to compete for third place. The players’ payoffs will be given by the results of the tournament. Let’s denote these different payoffs i , j, k, and l, such that i < j < k < l. This implies that l represents the payoff of the winning player, while i represents the loser’s payoff.

Figure 1. How do you distribute stamina in the Soccer World Cup?

We can assume the outcome of a game can be modeled by a probability density function, which gives a player’s probability as an increasing function of stamina investment. This suggests that a player’s probability of winning the semifinal game, p, depends on the amount of stamina he is willing to expend. Because the level of stamina is assumed to be finite, the player has a probability 1-p of winning in the final round.

Recall that we stipulated that our players A, B, C, and D were evenly matched and had equal amounts of stamina. Each player’s strategy—and resulting payoff—will depend on how much stamina he is willing to expend in the semifinal round. Therein lies the dilemma; how much stamina should be spent in the semifinal round? Is it better for a player to fully exert himself ensuring a victory in the semifinal, or is it better to reserve some stamina for the subsequent championship round?

The answer of course depends on the value of i, j, k, l. There are three possible Nash equilibria. First, any pure strategy (i.e. exert or reserve) will be a Nash equilibrium when the winner receives a much greater payoff than the other players.  Second, if first and second place have much higher payoffs than third or fourth place, and the payoffs are not significantly different, then the only Nash equilibrium is such that all players employ a pure strategy and expend all of their stamina in the semifinal round. Third, if the payoff for last place is significantly less than any other finish, a Nash equilibrium is formed when all players adopt a mixed strategy whereby each player uses all or none of his stamina in the semifinal game. In essence, the players’ strategies will depend on their value of the payoffs for each finish in the tournament.   

                My explanation here is a little ambiguous. If you are curious about the onerous math behind these generalizations, check out this article.

                Now, let’s turn our attention to the decision theory behind penalty kicks. Penalty kicks are awarded to the team whose opponent has committed a foul. In a penalty kick, a kicker attempts to score on the opponent’s goal, defended only by the goalkeeper. The goalkeeper must stand on the goal line until the ball has been kicked, and then he must make a dive left or right to try and block the incoming ball.

                For the sake of a game theoretic analysis of penalty kicks, we must make the following assumptions:

1. The kicker kicks precisely at the same time the goalie dives. This isn’t an entirely unreasonable assumption, as it takes an average of 0.3 seconds for the ball to reach the goal. So the goalie definitely doesn’t have time to deliberate which way he needs to dive.
2. The kicker and goalie employ mixed strategies. That is, the kicker may kick to the left or the right of the goal with probabilities p and 1-p, and the goalie may dive left or right with probabilities q and 1-q.
3.  If the kicker kicks the ball in the same direction that the goalie dives, the ball will be successfully blocked. If the kicker kicks the ball in the opposite direction that the goalie dives, the kicker will successfully score a goal.

This model closely resembles the game of matching pennies. In matching pennies, two competitors, A and B, simultaneously display pennies. If the pennies do not match (heads, tails), then player A collects one dollar from player B. On the other hand, if the pennies match ((heads, heads) or (tails, tails)) then player B receives one dollar from player A. The payoff matrix is as follows:

Figure 2. A payoff matrix for the matching pennies game. 

The example of matching pennies can be modified to model penalty kicks. If the direction of the goalie’s dive "matches" the direction of the kick, then he successfully defends the goal and both players receive a payoff of zero; the score remains the same. However, if the direction of the goalie’s dive does not match the direction of the kick, he misses the ball and receives a payoff of -1, and the kicker scores a payoff of 1. The results are summarized in the payoff matrix below. 

  

Figure 3. A payoff matrix for the penalty kick. 

      No doubt, this model is flawed. Here, we assume there’s a 50/50 chance that a player will choose one direction over another. In truth, there is a much higher probability that a right-footed player will kick the ball left; a left-footed player is also more likely to kick the ball right (and surely ambipedal soccer players are few and far between). We can also presume that goalies don’t just dive willy-nilly. Professional goalies probably make a point of checking players’ footedness. The kicker’s approach to the ball is also pretty telling. If he is right-footed, he would actually have to approach the ball on an arc and angle his body away from the goal to make a right shot.

               Goalie be like…

Figure 4. Goalies aren't quite as clueless as our model makes them out to be. 

Well, that’s all folks. Looking forward to our discussion of game theory in sports!  

References:

Baek, Seung Ki, Seung-Woo Son, and Jeong Hyeong-Chai. "Nash Equilibrium and Evolutionary Dynamics in                                Semifinalists' Dilemma." American Physical Society, 30 Apr. 2015. Web. 15 Nov. 2015.

Grant, Andrew. "Here's What Game Theory Says about How to Win in Semifinals." Science News. N.p., 22 May                              2015. Web. 15 Nov. 2015.

Harford, Tim. "What Economics Teaches about Penalty Kicks." Slate. N.p., 24 June 2006. Web. 15 Nov. 2015.

Spaniel, William. "The Game Theory of Soccer Penalty Kicks." William Spaniel. N.p., 12 June 2014. Web. 15 Nov.                          2015.