Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 27, 2020

Identification of Non-Equilibrium Beliefs in Games of Incomplete Information Using Experimental Data

  • Victor Aguirregabiria ORCID logo EMAIL logo and Erhao Xie


This paper studies the identification of players’ preferences and beliefs in discrete choice games using experimental data. The experiment comprises a set of games that differ in their matrices of monetary payoffs. The researcher is interested in the identification of preferences (utility of money) and beliefs on the opponents’ expected behavior, without imposing equilibrium restrictions or parametric assumptions on utility and belief functions. We show that the hypothesis of unbiased/rational beliefs is testable as long as the set of games in the experiment imply variation in monetary payoffs of other players, keeping the own monetary payoff constant. We present conditions for the full identification of utility and belief functions at the individual level – without restrictions on players’ heterogeneity in preferences or beliefs. We apply our method to data from two experiments: a matching pennies game, and a public good game.

JEL Classification: C57; C72


We would like to thank the editor and two anonymous referees for their helpful and constructive feedback. We have benefited from comments and helpful discussions with Martin Burda, Evan Calford, Robert Gazzale, Tanjim Hossain, Arvind Magesan, Yesim Orhun, David Pacini, Eduardo Souza-Rodrigues, Colin Stewart, Junichi Suzuki, Ruli Xiao, and especially Ryan Webb. We also thank comments from seminar participants at Bristol, Cornell, and Warwick, and at the Behavioral Industrial Organization & Marketing Symposium (in Ann Arbor, in 2019), the Canadian Economic Association conference (in Banff, in 2019), and the Economics Science Association North American meeting (in Tucson, in 2016). This research has been supported by the Social Sciences and Humanities Research Council of Canada.


Aguirregabiria, V., and C.-Y. Ho. 2012. “A Dynamic Oligopoly Game of the US Airline Industry: Estimation and Policy Experiments.” Journal of Econometrics 168 (1): 156–173.10.1016/j.jeconom.2011.09.013Search in Google Scholar

Aguirregabiria, V., and A. Magesan. 2019. “Identification and Estimation of Dynamic Games when Players’ Beliefs Are not in Equilibrium.” Review of Economic Studies Forthcoming. DOI: Search in Google Scholar

Anderson, S., J. Goeree, and C. Holt. 2001. “Minimum-Effort Coordination Games: Stochastic Potential and Logit Equilibrium.” Games and Economic Behavior 34 (2): 177–199.10.1006/game.2000.0800Search in Google Scholar

Bajari, P., H. Hong, J. Krainer, and D. Nekipelov. 2010. “Estimating Static Models of Strategic Interactions.” Journal of Business & Economic Statistics 28 (4): 469–482.10.3386/w12013Search in Google Scholar

Bajari, P., H. Hong, and D. Nekipelov. 2013. “Game Theory and Econometrics: A Survey of Some Recent Research.” Advances in Economics and Econometrics, 10th World Congress of the Econometric Society. Vol. 3.10.1017/CBO9781139060035.002Search in Google Scholar

Berry, S., and E. Tamer. 2006. “Identification in Models of Oligopoly Entry.” In Advances in Economics and Econometrics, edited by R. Blundell, W. Newey and T. Persson, 46–85. Cambridge: Cambridge University Press.10.1017/CBO9781139052276.004Search in Google Scholar

Brown, A., C. Camerer, and D. Lovallo. 2013. “Estimating Structural Models of Equilibrium and Cognitive Hierarchy Thinking in the Field: The Case of Withheld Movie Critic Reviews.” Management Science 59: 733–747.10.1287/mnsc.1120.1563Search in Google Scholar

Camerer, C., T. Ho, and J. Chong. 2004. “A Cognitive Hierarchy Model of One-Shot Games.” Quarterly Journal of Economics 119: 861–898.10.1162/0033553041502225Search in Google Scholar

Cheung, Y. W., and D. Friedman. 1997. “Individual Learning in Normal Form Games: Some Laboratory Result.” Games and Economic Behavior 19 (1): 46–76.10.1006/game.1997.0544Search in Google Scholar

Chmura, T., S. Goerg, and R. Selten. 2012. “Learning in Experimental 2×2 Games.” Games and Economic Behavior 76: 44–73.10.1016/j.geb.2012.06.007Search in Google Scholar

Costa-Gomes, M., and V. Crawford. 2006. “Cognition and Behavior in Two-Person Guessing Games: An Experimental Study.” American Economic Review 96: 1737–1768.10.1257/aer.96.5.1737Search in Google Scholar

Costa-Gomes, M., and G. Weizsacker. 2008. “Stated Beliefs and Play in Normal-Form Games.” Review of Economic Studies 75: 729–762.10.1111/j.1467-937X.2008.00498.xSearch in Google Scholar

Crawford, V., and H. Haller. 1990. “Learning How to Cooperate: Optimal Play in Repeated Coordination Games.” Econometrica 58 (3): 571–595.10.2307/2938191Search in Google Scholar

Crawford, V., and N. Iriberri. 2007. “Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner’s Curse and Overbidding in Private-Value Auctions?” Econometrica 75: 1721–1770.10.1111/j.1468-0262.2007.00810.xSearch in Google Scholar

Crawford, V., M. Costa-Gomes, and N. Iriberri. 2013. “Structural Models of Nonequilibrium Strategic Thinking: Theory, Evidence, and Applications.” Journal of Economic Literature 51 (1): 5–62.10.1257/jel.51.1.5Search in Google Scholar

Doraszelski, U., G. Lewis, and A. Pakes. 2018. “Just Starting Out: Learning and Equilibrium in a New Market.” American Economic Review 108: 565–615.10.3386/w21996Search in Google Scholar

Ellickson, P., and S. Misra. 2008. “Supermarket Pricing Strategies.” Marketing Science 27 (5): 811–828.10.1287/mksc.1080.0398Search in Google Scholar

Fehr E., and K. Schmidt. 1999. “A Theory of Fairness, Competition and Cooperation.” Quarterly Journal of Economics 114: 817–868.10.2307/j.ctvcm4j8j.14Search in Google Scholar

Feltovich, N. 2000. “Reinforcement-based vs. Belief-based Learning Models in Experimental Asymmetric-information Games.” Econometrica 68: 605–641.10.1111/1468-0262.00125Search in Google Scholar

Goeree J., and C. Holt. 2001. “Ten Little Treasures of Game Theory and Ten Intuitive Contradictions.” American Economic Review 91: 1402–1422.10.1257/aer.91.5.1402Search in Google Scholar

Goeree J., C. Holt, and T. Palfrey. 2003. “Risk Averse Behavior in Generalized Matching Pennies Games.” Games and Economic Behavior 45: 97–113.10.1016/S0899-8256(03)00052-6Search in Google Scholar

Goeree J., C. Holt, and T. Palfrey. 2005. “Regular Quantal Response Equilibrium.” Experimental Economics 8: 347–367.10.1007/s10683-005-5374-7Search in Google Scholar

Goldfarb, A., and M. Xiao. 2011. “Who Thinks About the Competition? Managerial Ability and Strategic Entry in US Local Telephone Markets.” American Economic Review 101: 3130–3161.10.1257/aer.101.7.3130Search in Google Scholar

Grether, D. M. 1992. “Testing Bayes Rule and the Representativeness Heuristic: Some Experimental Evidence.” Journal of Economic Behavior & Organization 17 (1): 31–57.10.1016/0167-2681(92)90078-PSearch in Google Scholar

Guerra, G., and D. Zizzo. 2004. “Trust Responsiveness and Beliefs.” Journal of Economic Behavior & Organization 55 (1): 25–30.10.1016/j.jebo.2003.03.003Search in Google Scholar

Harrison, G., and E. Rutström. 2008. “Risk Aversion in the Laboratory.” In Risk Aversion in Experiments (Research in Experimental Economics, Volume 12), edited by James C. Cox, Glenn W. Harrison, 41–196. Emerald Group Publishing Limited.10.1016/S0193-2306(08)00003-3Search in Google Scholar

Heinemann F., R. Nagel, and P. Ockenfels. 2009. “Measuring Strategic Uncertainty in Coordination Games.” Review of Economic Studies 76: 181–221.10.1111/j.1467-937X.2008.00512.xSearch in Google Scholar

Hossain T., and R. Okui. 2013. “The Binarized Scoring Rule.” Review of Economic Studies 80: 984–1001.10.1093/restud/rdt006Search in Google Scholar

Kahneman, D., and A. Tversky. 1979. “Prospect Theory: An Analysis of Decision under Risk.” Econometrica 47: 263–293.10.21236/ADA045771Search in Google Scholar

Karni, E. 2009. “A Mechanism for Eliciting Probabilities.” Econometrica 77: 603–606.10.3982/ECTA7833Search in Google Scholar

Kline, B. 2018. “An Empirical Model of Non-Equilibrium Behavior in Games.” Quantitative Economics 9 (1): 141–181.10.3982/QE647Search in Google Scholar

Mckelvey, R., and T. Palfrey. 1995. “Quantal Response Equilibria for Normal Form Games.” Games and Economic Behavior 10: 6–38.10.1006/game.1995.1023Search in Google Scholar

Mckelvey, R., and T. Palfrey. 1998. “Quantal Response Equilibria for Extensive Form Games.” Experimental Economics 1: 9–41.10.1007/s10683-015-9471-ySearch in Google Scholar

Mckelvey, R., T. Palfrey, and R. Weber. 2000. “The Effects of Payoff Magnitude and Heterogeneity on Behavior in 2× 2 Games with Unique Mixed Strategy Equilibria.” Journal of Economic Behavior & Organization 42: 523–548.10.1016/S0167-2681(00)00102-5Search in Google Scholar

Melo, E., K. Pogorelskiy, and M. Shum. 2019. “Testing the Quantal Response Hypothesis.” International Economic Review 60 (1): 53–74.10.1111/iere.12344Search in Google Scholar

Morris, S., and H. Shin. 2002. “Social Value of Public Information.” American Economic Review 92: 1521–1534.10.1257/000282802762024610Search in Google Scholar

Morris, S., and H. Shin. 2004. “Coordination Risk and the Price of Debt.” European Economic Review 48: 133–153.10.1016/S0014-2921(02)00239-8Search in Google Scholar

Nyarko Y., and A. Schotter. 2002. “An Experimental Study of Belief Learning Using Elicited Beliefs.” Econometrica 70: 971–1005.10.1111/1468-0262.00316Search in Google Scholar

Ochs, J. 1995. “Games with Unique, Mixed Strategy Equilibria: An Experimental Study.” Games and Economic Behavior 10: 202–217.10.1006/game.1995.1030Search in Google Scholar

Offerman, T., J. Sonnemans, G. Van De Kuilen, and P. P. Wakker. 2009. “A Truth Serum for Non-Bayesians: Correcting Proper Scoring Rules for Risk Attitudes.” Review of Economic Studies 76 (4): 1461–1489.10.1111/j.1467-937X.2009.00557.xSearch in Google Scholar

Palfrey, T., and S. Wang. 2009. “On Eliciting Beliefs in Strategic Games.” Journal of Economic Behavior and Organization 71: 98–109.10.1016/j.jebo.2009.03.025Search in Google Scholar

Rogers, B., T. Palfrey, and C. Camerer. 2009. “Heterogeneous Quantal Response Equilibrium and Cognitive Hierarchies.” Journal of Economic Theory 144 (4): 1440–1467.10.1016/j.jet.2008.11.010Search in Google Scholar

Roth, A., and M. Malouf. 1979. “Game-Theoretic Models and the Role of Information in Bargaining.” Psychological Review 86: 574–594.10.1037/0033-295X.86.6.574Search in Google Scholar

Rutström, E., and N. Wilcox. 2009. “Stated Beliefs versus Inferred Beliefs: A Methodological Inquiry and Experimental Test.” Games and Economic Behavior 67: 616–632.10.1016/j.geb.2009.04.001Search in Google Scholar

Schlag, K. H., J. Tremewan, and J. J. Van Der Weele. 2015. “A Penny for Your Thoughts: A Survey of Methods for Eliciting Beliefs.” Experimental Economics 18 (3): 457–490.10.1007/s10683-014-9416-xSearch in Google Scholar

Schotter, A., and I. Trevino. 2014. “Belief Elicitation in the Laboratory.” Annual Review of Economics 6 (1): 103–128.10.1146/annurev-economics-080213-040927Search in Google Scholar

Selten, R., and T. Chmura. 2008. “Stationary Concepts for Experimental 2×2 Games.” American Economic Review 98: 938–966.10.1257/aer.101.2.1041Search in Google Scholar

Selten, R., A. Sadrieh, and K. Abbink. 1999. “Money Does Not Induce Risk Neutral Behavior, But Binary Lotteries do Even Worse.” Theory and Decision 46: 211–249.10.1023/A:1005038628305Search in Google Scholar

Smith, C. 1961. “Consistency in Statistical Inference and Decision.” Journal of the Royal Statistical Society. Series B 23 (1): 1–37.10.1111/j.2517-6161.1961.tb00388.xSearch in Google Scholar

Sweeting, A. 2009. “The Strategic Timing Incentives of Commercial Radio Stations: An Empirical Analysis Using Multiple Equilibria.” The RAND Journal of Economics 40 (4): 710–742.10.3386/w14506Search in Google Scholar

Van Huyck, R. Battalio, and R. Beil. 1990. “Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure.” American Economic Review 80 (1): 234–248.Search in Google Scholar

Published Online: 2020-03-27

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 22.2.2024 from
Scroll to top button