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The penalty-kick game under incomplete information


  • Germán Coloma


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).

Suggested Citation

  • Germán Coloma, 2012. "The penalty-kick game under incomplete information," CEMA Working Papers: Serie Documentos de Trabajo. 487, Universidad del CEMA.
  • Handle: RePEc:cem:doctra:487

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    References listed on IDEAS

    1. Bar-Eli, Michael & Azar, Ofer H. & Ritov, Ilana & Keidar-Levin, Yael & Schein, Galit, 2007. "Action bias among elite soccer goalkeepers: The case of penalty kicks," Journal of Economic Psychology, Elsevier, vol. 28(5), pages 606-621, October.
    2. Florian Baumann & Tim Friehe & Michael Wedow, 2011. "General Ability and Specialization: Evidence From Penalty Kicks in Soccer," Journal of Sports Economics, , vol. 12(1), pages 81-105, February.
    3. Ignacio Palacios-Huerta, 2003. "Professionals Play Minimax," Review of Economic Studies, Oxford University Press, vol. 70(2), pages 395-415.
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    More about this item


    soccer penalty kicks; mixed strategies; Bayesian equilibrium; incomplete information;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • L83 - Industrial Organization - - Industry Studies: Services - - - Sports; Gambling; Restaurants; Recreation; Tourism

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