## Game Theory in Penalty Kicks

Reference: https://williamspaniel.com/2014/06/12/the-game-theory-of-soccer-penalty-kicks/

Soccer penalty kicks is one of the most intriguing and suspenseful parts of the soccer game after a long, tiring 120 minutes of battle on the soccer pitch. While other sports, at crucial moments, usually employ complex, elaborate tactics to win over opposing teams, soccer is rather straightforward. Each penalty shot is a one-on-one game between the kicker and the goalie, one of which must win or lose. Penalty kicks, thus, is a “zero-sum game” in that as one participant wins the other participants must lose. The game procedure is simple: A kicker lines up with the ball in front of him, and he runs forward and kicks the ball in the net while the goalie tries to prevent the ball from going in. Because it takes less than a second for a ball to hit the back of the net, both the kicker and the goalie should move simultaneously. A goalie does not have enough time to dive after he discerns the trajectory of the ball. Therefore, the goalie must make a decision where to dive before recognizing any pertinent information from the kicker.

Game theory is essential in explaining the decision-making process with regards to Penalty kicks as represented by the 2 x 2 playoff matrix between the kicker and the goalie matchup. In this matrix, for convenience, I assumed that both kicker and goalie are excellent players in that the kicker will always score if the goalie fails guessing and the goalie will always block if he/she succeeds guessing.

Kicker/ Goalie | Goalie jumps left | Goalie jumps right |

Kicker kicks left | -1, +1 | +1, -1 |

Kicker kicks right | +1, -1 | -1, +1 |

(+1 = win, -1 = loss)

Both players are fundamentally inclined to maximize their utility (happiness) by garnering highest possible payoffs. For a goalie to maximize his payoff, he/she would like to jump on the same side that the kicker kicks (In this case, the goalie always blocks, thus gaining +1 for himself and -1 for the kicker). For a kicker to maximize his payoff, he/she would like to kick in the opposite side that the goalie jumps (In this case, the kicker always scores, thus gaining +1 for himself -1 for the goalie). However, this game has neither dominant strategy nor pure strategy Nash equilibrium. Since all finite games have at least one Nash equilibrium, this game must have a mixed strategy Nash equilibrium. And the solution is that both players should choose each side with the same probability. If a kicker is very confident in the direction where he can put more power, the goalie will anticipate this choice for the same reason and vice versa. To avoid getting exploited by each other, both players should randomly perform (less calculable manner). As the game theory shows us, soccer penalty shots is not all about physical abilities. It is about being random, unpredictable, and smartly disguising the opposing player.