Experimental Economics

, Volume 13, Issue 4, pp 461–475 | Cite as

A finite mixture analysis of beauty-contest data using generalized beta distributions

  • Antoni Bosch-Domènech
  • José G. Montalvo
  • Rosemarie Nagel
  • Albert SatorraEmail author


This paper introduces a mixture model based on the beta distribution, without pre-established means and variances, to analyze a large set of Beauty-Contest data obtained from diverse groups of experiments (Bosch-Domènech et al. 2002). This model gives a better fit of the experimental data, and more precision to the hypothesis that a large proportion of individuals follow a common pattern of reasoning, described as Iterated Best Reply (degenerate), than mixture models based on the normal distribution. The analysis shows that the means of the distributions across the groups of experiments are pretty stable, while the proportions of choices at different levels of reasoning vary across groups.


Beauty-contest experiments Decision theory Reasoning hierarchy Finite mixture distribution Beta distribution EM algorithm 

JEL Classification

C14 C16 C72 C91 D03 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andersen, S., Harrison, G. W., Hole, A. R., & Rutström, E. E. (2009). Non-linear mixed logit and the characterization of individual heterogeneity (Working Paper 09-02). Department of Economics, College of Business Administration, University of Central Florida. Google Scholar
  2. Bognanno, M. L. (2001). Corporate tournaments. Journal of Labor Economics, 19(2), 290–315. CrossRefGoogle Scholar
  3. Bosch-Domènech, A., Montalvo, J. G., Nagel, R., & Satorra, A. (2002). One, two, (three), infinity, …: Newspaper and lab beauty-contest experiments. American Economic Review, 92(5), 1687–1701. CrossRefGoogle Scholar
  4. Bozdogan, H. (1987). Model selection and Akaike’s information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370. CrossRefGoogle Scholar
  5. Breitmoser, Y. (2010). Hierarchical reasoning versus iterated reasoning in p-beauty contest guessing games (MPRA Paper No. 19893). Munich Personal RePEc Archive. Google Scholar
  6. Burchardi, K. B., & Penczynski, S. P. (2010). Out of your mind: Eliciting individual reasoning in one shot games. Mimeo, London School of Economics. Google Scholar
  7. Camerer, C., Ho, T., & Chong, J. (2004). A cognitive hierarchy theory of one-shot games and experimental analysis. Quarterly Journal of Economics, 119, 861. CrossRefGoogle Scholar
  8. Costa-Gomes, M., & Crawford, V. (2006). Cognition and behavior in guessing games: An experimental study. American Economic Review, 96, 1737–1768. CrossRefGoogle Scholar
  9. Costa-Gomes, M., Crawford, V., & Broseta, B. (2001). Cognition and behavior in normal-form games: An experimental study. Econometrica, 69, 1193–1235. CrossRefGoogle Scholar
  10. Crawford, V., & Iriberri, N. (2007). Level-k auctions: Can boundedly rational strategic thinking explain the winner’s curse and overbidding in privatevalue auctions? Econometrica, 75, 1721–1770. CrossRefGoogle Scholar
  11. Dastrup, S. R., Hartshorn, R. Y., & Mcdonald, J. B. (2007). The impact of taxes and transfer payments on the distribution of income: A parametric comparison. Journal of Economic Inequality, 5, 3, 1569–1721. Google Scholar
  12. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via de EM algorithm (with discussion). Journal of the Royal Statistical Society B, 39, 1–38. Google Scholar
  13. Efron, B., & Tibshirani, R. (1986). Bootstrap measures for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, 1, 54–77. CrossRefGoogle Scholar
  14. Goeree, J. K., & Holt, C. A. (2004). A model of noisy introspection. Games and Economic Behavior, 46(2), 365–382. CrossRefGoogle Scholar
  15. Harrison, G., & List, J. A. (2004). Field experiments. Journal of Economic Literature, XLII, 1013–1059. Google Scholar
  16. Harrison, G. W., & Rutström, E. E. (2009). Expected utility and prospect theory: One wedding and decent funeral. Experimental Economics, 12(2), 133–158. CrossRefGoogle Scholar
  17. Heckman, J. J., & Willis, R. J. (1977). A betalogistic model for the analysis of sequential labor force participation by married women. Journal of Political Economy, 85(1), 27–58. CrossRefGoogle Scholar
  18. Ho, T., Camerer, C., & Weigelt, K. (1998). Iterated dominance and iterated best-response in experimental ‘p-beauty-contests’. American Economic Review, 88(4), 947–969. Google Scholar
  19. Keynes, J. M. (1936). The general theory of employment, interest and money. London: Macmillan. Google Scholar
  20. Kübler, D., & Weizacker, G. (2004). Limited depth of reasoning and failure of cascade formation in the laboratory. Review of Economic Studies, 71, 425–441. CrossRefGoogle Scholar
  21. Lesaffre, E., Rizopoulos, D., & Tsonaka, R. (2007). The logistic transform for bounded outcome scores. Biostatistics, 8(1), 72–85. CrossRefGoogle Scholar
  22. Mcdonald, J. B., & Xu, Y. J. (1995). A generalization of the beta distribution with applications. Journal of Econometrics, 66, 133–152. CrossRefGoogle Scholar
  23. McKelvey, R. D., & Palfrey, T. R. (1992). An experimental study of the centipede game. Econometrica, 60(4), 803–836. CrossRefGoogle Scholar
  24. McLachlan, G., & Peel, D. (2000). Finite mixture models. New York: Wiley. CrossRefGoogle Scholar
  25. Nagel, R. (1995). Unraveling in guessing games: An experimental study. American Economic Review, 85(5), 1313–1326. Google Scholar
  26. Nagel, R. (1998). A survey on experimental beauty-contest games: Bounded rationality and learning. In D. Budescu, I. Erev, & R. Zwick (Eds.), Games and human behavior, Essays in honor of Amnon Rapoport (pp. 105–142). Hillsdale: Lawrence Erlbaum Associates, Inc. Google Scholar
  27. Nagel, R. (2004). How to improve reasoning in experimental beauty-contest games: A survey. In D. Friedman, & A. Cassar (Eds.), Economics Lab: An intensive course on experimental economics. Lessons from the Trento summer school in experimental economics. London: Routledge. Google Scholar
  28. Nagel, R., Bosch-Domènech, A., Satorra, A., & Montalvo, J. G. (1999). One, two, (three), infinity… . Newspaper and lab beauty-contest experiments (Economics and Business WPS, Paper # 438). Barcelona: Universitat Pompeu Fabra. Google Scholar
  29. Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308–313. Google Scholar
  30. Stahl, D. O. (1996). Boundedly rational rule learning in a guessing game. Games and Economic Behavior, 16, 303–330. CrossRefGoogle Scholar
  31. Stahl, D. O. (1998). Is step-j thinking an arbitrary modelling restriction or a fact of human nature? Journal of Economic Behavior and Organization, 37, 33–51. CrossRefGoogle Scholar
  32. Stahl, D. O., & Wilson, P. W. (1994). ‘Experimental evidence on players’ models of other players. Economic Behavior & Organization, 25, 309–327. CrossRefGoogle Scholar
  33. Stahl, D. O., & Wilson, P. W. (1995). ‘On players’ models of other players: Theory and experimental evidence. Games and Economic Behavior, 10, 213–254. CrossRefGoogle Scholar
  34. Titterington, D., Smith, A., & Makov, U. (1992). Statistical analysis of finite mixture distributions. New York: Wiley. Google Scholar

Copyright information

© Economic Science Association 2010

Authors and Affiliations

  • Antoni Bosch-Domènech
    • 1
  • José G. Montalvo
    • 1
  • Rosemarie Nagel
    • 1
  • Albert Satorra
    • 1
    Email author
  1. 1.Department of Economics and BusinessUniversitat Pompeu FabraBarcelonaSpain

Personalised recommendations