Do soccer players play the mixed-strategy Nash equilibrium?

Abstract

Mixed-Strategy Nash Equilibrium (MSNE) is a commonly used solution concept in game-theoretic models in various fields in economics, management and other disciplines, but the experimental results whether the MSNE predicts well actual play in games is mixed. Consequently, the evidence for naturally occurring games in which the MSNE predicts the outcome well is of great importance, as it can justify the vast use of MSNE in models. The game between the kicker and the goalkeeper in soccer penalty kicks is a real-world game that can be used to examine the application of the MSNE concept or its accuracy, because payoffs are a common knowledge, the players have huge incentives to play correctly, the game is simple enough to analyse, its Nash equilibrium is in mixed strategies, and players’ actions can be observed. We collected and analysed the data on the direction of kicks and jumps in penalty kicks in various top leagues and tournaments. Our analysis suggests that the MSNE predictions are the closest to the actual sample data, even though some other prediction methods use information on the marginal distribution of kicks or jumps, whereas the MSNE does not.