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). To find a solution for this game we use the concept of Bayesian equilibrium, and we find that, typically, one of the kicker’s types will play a mixed strategy while the other type will choose a pure strategy (or, sometimes, a “restricted mixed strategy”). The model has a simpler version in which the players can only choose between two strategies (right and left), and a more complex version in which they can also choose a third strategy (the center of the goal). Comparing the incomplete-information Bayesian equilibria 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 three-strategy model is also useful to explain why it could be optimal for a goal keeper never to choose the center of the goal (although at the same time there were some kickers who always chose to shoot to the center).