Bayesian Equilibria in the Penalty-Kick Game

Abstract

This paper presents a model of a game between a soccer kicker and a goalkeeper, in which each player is trying to score a goal or to avoid such goal. In this version of the penalty-kick game, there are two possible strategies for each player (related to the place that they choose to kick or to move themselves) and there is also uncertainty about the kicker’s type (with two possible types of kicker). To find a solution for this game we use the concept of Bayesian equilibrium, and we find that, typically, one the kicker’s types will play a mixed strategy while the other type will choose a pure strategy. Comparing this equilibrium with the corresponding Nash equilibria under complete information, we find that the expected scoring probability increases (so that, on average, the goalkeeper is worse off).