## Game Theory: Penalty Kicks

In soccer games penalty kicks can drastically change the games trajectory. There are two different cases when a player would take a kick: if someone is fouled in the box during the game or if the game is still tied at the end of overtime (after 120 minutes).  The process of taking a penalty shot is as follows: the kicker places the ball at the top of the box and takes the shot when given the okay from the referee, simultaneously as the player is taking the shot the goalie must decide which way they want to dive to attempt to save the ball. Since the shots are at such a close range they are very quick, so it is necessary for both players to make their move at the same time. Penalty shots are relatively simple for something that holds so much weight in the game. They are a one-on-one game between a player and the goalie where one player will always win and the other will always lose making them a “zero-sum game”.

The game theory behind the decisions that go into penalty kicks can be shown as a 2×2 matrix. The goalie and the kicker are the two players and for the sake of this example we will assume that if the goalie dives correctly they will always save the ball (goalie wins) and if the dive incorrectly the kicker will always make the shot (kicker wins).

 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)

The two players both want to maximizes their payoff in the game. For the kicker this means kicking the opposite way that the goalie dives which would result in +1 for the kicker and -1 for the goalie, and the kicking always winning. Maximizing payoff for the goalie would be if they always dived in the same direction as the ball was shot, resulting in +1 for the goalie and -1 for the kicker and the goalie always winning. Since we know that all finite games have at least one Nash Equilibrium and additionally that penalty kicks don’t have a pure strategy Nash Equilibrium or a dominant strategy it must have a mixed strategy Nash equilibrium. The solution for the mixed strategy NE is that each player will choose each direction with equal probability indicating that they should play to their strengths and anticipate their opponents. If a kicker has a strong side they should kick there and the goalie should anticipate that they will and if the goalie has a weak side they should kick there and the goalie should anticipate that. But since this strategy is predictable both players should attempt to randomize their actions to make them less predictable. This game theory connection indicates that penalty kicks are more than just playing to ones strengths as a player, instead the involve randomization and strategy to trick the opponent.