Do soccer players play the mixed-strategy Nash equilibrium?

Azar, Ofer H. and Bar-Eli, Michael (2009): Do soccer players play the mixed-strategy Nash equilibrium? Forthcoming in: Applied Economics

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, 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 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 analyze, its Nash equilibrium is in mixed strategies, and players' actions can be observed. We collected and analyzed 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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Soccer; Football; MSNE; Mixed-strategies; Mixed-strategy Nash equilibrium; Sports; Penalty kicks
• Subjects: L - Industrial Organization > L8 - Industry Studies: Services > L83 - Sports ; Gambling ; Restaurants ; Recreation ; Tourism; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C70 - General; D - Microeconomics > D0 - General > D00 - General; C - Mathematical and Quantitative Methods > C9 - Design of Experiments > C93 - Field Experiments; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Bar-Eli, Michael, Ofer H. Azar, Ilana Ritov, Yael Keidar-Levin and Galit Schein, 2007. Action bias among elite soccer goalkeepers: The case of penalty kicks. Journal of Economic Psychology 28(5): 606-621.

Brown, James N. and Robert W. Rosenthal, 1990. Testing the minimax hypothesis: A re-examination of O'Neill's game experiment. Econometrica 58(5): 1065-1081.

Budescu, David V. and Amnon Rapoport, 1994. Subjective randomization in one- and two-person games. Journal of Behavioral Decision Making 7(4): 261-278.

Chiappori, P.A., S. Levitt and T. Groseclose, 2002. Testing mixed-strategy equilibria when players are heterogeneous: the case of penalty kicks in soccer. American Economic Review 92(4): 1138-1151.

Erev, Ido and Alvin E. Roth, 1998. Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria. American Economic Review 88(4): 848-881.

Fudenberg, Drew and Jean Tirole, 1991. Game Theory. The MIT Press, Cambridge, MA.

Harsanyi, John C., 1973. Games with randomly disturbed payoffs: a new rationale for mixed strategy equilibrium points. International Journal of Game Theory 2, 1-23.

McCabe, Kevin A, Arijit Mukherji and David E. Runkle, 2000. An experimental study of information and mixed strategy play in the three person matching pennies game. Economic Theory 15(2): 421-462.

Mookherjee, Dilip and Barry Sopher, 1994. Learning behavior in an experimental matching pennies game. Games and Economic Behavior 7(1): 62-91.

Mookherjee, Dilip and Barry Sopher, 1997. Learning and decision costs in experimental constant sum games. Games and Economic Behavior 19(1): 97-132.

Moschini, GianCarlo, 2004. Nash Equilibrium in strictly competitive games: live play in soccer. Economics Letters 85(3): 365-371.

Nash, John, 1950. Equilibrium points in n person games. Proceedings of the National Academy of Sciences 36: 48-49.

Ochs, Jack, 1995. Games with unique, mixed strategy equilibria: an experimental study. Games and Economic Behavior 10(1): 202-217.

Osborne, Martin J. and Ariel Rubinstein, 1994. A Course in Game Theory. The MIT Press, Cambridge, MA.

Palacios-Huerta, Ignacio, 2003. Professionals play minimax. Review of Economic Studies 70(2): 395-415.

Rapoport, Amnon and Richard B. Boebel, 1992. Mixed strategies in strictly competitive games: a further test of the minimax hypothesis. Games and Economic Behavior 4(2): 261-83.

Rapoport, Amnon and David V. Budescu, 1992. Generation of random series in two-person strictly competitive games, Journal of Experimental Psychology: General 121(3): 352-363.

Rapoport, Amnon and Wilfred Amaldoss, 2000. Mixed strategies and iterative elimination of strongly dominated strategies: an experimental investigation of states of knowledge. Journal of Economic Behavior and Organization 42(4): 483-521.

Rapoport, Amnon and Wilfred Amaldoss, 2004. Mixed-strategy play in single-stage first-price all-pay auctions with symmetric players. Journal of Economic Behavior and Organization 54(4), 585-607.

Roth, Alvin E. and Ido Erev, 1995. Learning in extensive-form games: experimental data and simple dynamic models in the intermediate term. Games and Economic Behavior 8(1): 164-212.

Shachat, Jason M., 2002. Mixed strategy play and the minimax hypothesis. Journal of Economic Theory 104(1): 189-226.

Vulkan, Nir, 2000. An economist's perspective on probability matching. Journal of Economic Surveys 14(1): 101-118.

Walker, Mark and John Wooders, 2001. Minimax play at Wimbledon. American Economic Review 91(5): 1521-1538.