The Game Theory of Soccer Penalty Kicks
With the World Cup starting in two months, I think now is a great time to dig deeper into the game theory behind the soccer penalty kicks. Grew up watching soccer, I was always playing prediction games with my dad about where the kicker will shoot and where the goalie will dive. Now that I am taking Networks, I can model penalty kicks in soccer as a two-player game and solve it easily using Game Theory.
Before digging deeper, there are some assumptions that we have to make. The kicker can only aim the ball to the left or the right of the goal, and the goalie can dive to either the left or right as well. The ball moves to the goal fast enough that the decisions of the kicker and goalie are effectively being made simultaneously; based on these decisions the kicker is likely to score or not. We simply ignore the possibility that the kicker can score even though the goalie dives to the side that the kicker aims. Below is the payoff matrix where +1 represents a win and -1 represents a loss.
Observe that to maximize its payoff, the kicker will try to shoot to the side opposite the side that the goalie dives. In this case, kickers get a payoff of +1 and the goalie gets -1. Similarly, to maximize its payoff, the goalie tries to dive to the side that the kicker shoots. In this case, the goalie gets a payoff of +1 and the kicker gets -1.
If the kicker is going to kick right and the goalie knows it, the goalie will defend right. This is not good for the kicker since the kicker is likely to lose and not score the penalty kick. So, having both of them choose right cannot be a Nash Equilibrium. Intuitively, having both of them choose right also cannot be a Nash Equilibrium because the kicker would want to kick right if the goalie is going to defend left. There are no pair of strategies that are best responses to each other and there is no Nash equilibrium in the pure strategies of simply going left or right. So both the kicker and the goalie should be unpredictable. Each should randomize their choice so their opponents cannot predict what they are going to do. The solution to the Nash equilibrium in mixed strategies is that each player should choose each side with the same probability to not get exploited by their opponent.
Sources:
- Spaniel, W. (2015, May 28). The Game Theory of Soccer Penalty Kicks. William Spaniel. Retrieved September 21, 2022, from https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/
- Easley, D., & Kleinberg, J. (2010). Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press.