The Penalty-Kick Game under Incomplete Information

This paper presents a model of the penalty-kick game between a soccer goalkeeper and a kicker, in which there is uncertainty about the kicker’s type (and there are two possible types of kicker). Both the goalkeeper and the kicker can choose among three different strategies (right, left and center). To find a solution for this game we use the concept of Bayesian equilibrium. Comparing this equilibrium with the corresponding complete-information Nash equilibria, we find that in all cases the expected scoring probability increases (so that, on average, the goalkeeper is worse off under incomplete information). The model is also useful to explain why it could be optimal for a goalkeeper never to choose the center of the goal (although at the same time there are some kickers who always chose to shoot to the center).