Penalty kicks: an adversarial risk analysis (ARA) perspective

Penalty kicks are critical to game outcomes in soccer. The typical quantitative strategy is as follows. First, model such an interaction as a two or three strategy game between the kicker and the goalkeeper. Second, find the mixed-strategy Nash equilibrium (MSNE) to determine the “optimal” probabilities of each player choosing to kick or dive to either side or to the center of the goal. While this is the usual path to a solution, it is also fraught with many assumptions due to the nature of penalty kick data as well as due to implicit assumptions within the game theory model. In this paper, we introduce an alternative set of strategies, known as adversarial risk analysis (ARA), to determine optimal decisions for the penalty kick game. ARA, which is grounded in the principles of Bayesian decision analysis, allows the decision maker to avoid many of the assumptions and limitations encountered when using the game theory approach. By examining 2018–2019 Major League Soccer (MLS) penalty kicks on an individual basis, we show that our most accurate ARA model predicts the correct goalkeeper decision 65 % of the time, while the game-theoretic model predicts correctly 51 % of the time, a statistically significant difference.