The Game Theory of Penalty Kicks : A framework for approximating Nash equilibria
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- Master Thesis 
This paper presents a game-theoretic analysis of penalty kicks in football. There must exist a Nash equilibrium in penalty kicks, but the nature of it depends on the unique abilities of the penalty taker and the goalkeeper. Thus, this study introduces a flexible framework that approximates a Nash equilibrium, based on a user-inputted set of player-dependent assumptions. The study also seeks to improve our general understanding of the characteristics of Nash equilibria in penalty kicks. This is achieved by investigating a diverse set of playerdependent assumptions, as well as gradually adding new elements of complexity to the models, and observing the shifts in the equilibrium. In the most basic model, the players make a simultaneous choice, where the penalty taker decides where to aim, and the goalkeeper decides which area to cover. The penalty taker isn’t able to shoot with perfect accuracy, so the hit-coordinate will likely differ from the aimcoordinate. In later models, the penalty taker is also allowed to choose the velocity of the ball, which in turn impacts the area coverage of the goalkeeper. In the final model, an element of sequential choice is added, such that the penalty taker may pretend to shoot, and observe if the goalkeeper starts to move. The Counterfactual Regret Minimization algorithm is employed to locate the Nash equilibrium, while an enhanced Coordinate Search algorithm is developed for determining the optimal aimcoordinates for the penalty taker. The most complex and realistic model indicates that the penalty taker should abstain from aiming at the middle region of the goal, and rather either pretend to shoot, or shoot at one of the sides. This is because the goalkeeper needs to stay in the middle fairly often, to avoid revealing their intention in the case where the penalty taker pretends to shoot. It’s also viable for the goalkeeper to commit to diving to either side without waiting to observe the trajectory of the ball.