Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer

The concept of mixed strategy is a fundamental component of game theory, and its normative importance is undisputed. However, its empirical relevance has sometimes been viewed with skepticism. The main concern over the practical usefulness of mixed strategies relates to the “indifference” property of a mixedstrategy equilibrium. In order to be willing to play a mixed strategy, an agent must be indifferent between each of the pure strategies that are played with positive probability in the mixed strategy, as well as any combination of those strategies. Given that the agent is indifferent across these many strategies, there is no beneŽ t to selecting precisely the strategy that induces the opponent to be indifferent, as required for equilibrium. Why an agent would, in the absence of communication between players, choose exactly one particular randomization is not clear. Of course, whether agents, in real life, actually play Nash equilibrium mixed strategies is ultimately an empirical question. The evidence to date on this issue is based almost exclusively on laboratory experiments (e.g., Barry O’Neill, 1987; Amnon Rapoport and Richard B. Boebel, 1992; Dilip Mookherjee and Barry Sopher, 1994; Jack Ochs, 1995; Kevin A. McCabe et al., 2000). The results of these experiments are mixed. O’Neill (1987) concludes that his experimental evidence is consistent with Nash mixed strategies, but that conclusion was contested by James N. Brown and Robert W. Rosenthal (1990). With the exception of McCabe et al. (2000), which looks at a three-person game, the other papers generally reject the Nash mixedstrategy equilibrium. While much has been learned in the laboratory, there are inherent limitations to such studies. It is sometimes argued that behavior in the simpliŽ ed, artiŽ cial setting of games played in such experiments need not mimic real-life behavior. In addition, even if individuals behave in ways that are inconsistent with optimizing behavior in the lab, market forces may discipline such behavior in the real world. Finally, interpretation of experiments rely on the assumption that the subjects are maximizing the monetary outcome of the game, whereas there may be other preferences at work among subjects (e.g., attempting to avoid looking foolish) that distort the results. Tests of mixed strategies in nonexperimental data are quite scarce. In real life, the games played are typically quite complex, with large strategy spaces that are not fully speciŽ ed ex ante. In addition, preferences of the actors may not be perfectly known. We are aware of only one paper in a similar spirit to our own research. Using data from classic tennis matches, Mark Walker and John Wooders (2001) test whether the probability the player who serves the ball wins the point is equal for serves to the right and * Chiappori: Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637; Levitt: Department of Economics, University of Chicago; Groseclose: Graduate School of Business, Stanford University, 518 Memorial Way, Stanford, CA 94305. The paper was presented at Games 2000 in Bilbao and at seminars in Paris and Chicago. We thank D. Braun, J. M. Conterno, R. Guesnerie, D. Heller, D. Mengual, P. Reny, B. Salanié, and especially J. L. Ettori for comments and suggestions, and M. Mazzocco and F. Bos for excellent research assistance. Any errors are ours. 1 The theoretical arguments given in defense of the concept of mixed-strategy equilibria relate either to puriŽ cation (John C. Harsanyi, 1973), or to the minimax property of the equilibrium strategy in zero-sum games. For recent elaborations on these ideas, see Authur J. Robson (1994) and Phil Reny and Robson (2001). 2 The ultimatum game is one instance in which experimental play of subjects diverges substantially from the predicted Nash equilibrium. Robert Slonim and Alvin E. Roth (1998) demonstrate that raising the monetary payoffs to experiment participants induces behavior closer to that predicted by theory, although some disparity persists.