Experimental Economics

, Volume 18, Issue 4, pp 569–608 | Cite as

Eliciting beliefs in continuous-choice games: a double auction experiment

  • Claudia Neri
Original Paper


This paper proposes a methodology to implement probabilistic belief elicitation in continuous-choice games. Representing subjective probabilistic beliefs about a continuous variable as a continuous subjective probability distribution, the methodology involves eliciting partial information about the subjective distribution and fitting a parametric distribution on the elicited data. As an illustration, the methodology is applied to a double auction experiment, where traders’ beliefs about the bidding choices of other market participants are elicited. Elicited subjective beliefs are found to differ from proxies such as Bayesian Nash equilibrium beliefs and empirical beliefs, both in terms of the forecasts of other traders’ bidding choices and in terms of the best-response bidding choices prescribed by beliefs. Elicited subjective beliefs help explain observed bidding choices better than BNE beliefs and empirical beliefs. By extending probabilistic belief elicitation beyond discrete-choice games to continuous-choice games, the proposed methodology enables to investigate the role of beliefs in a wider range of applications.


Probabilistic beliefs Belief elicitation Private information Experiments  



I had the opportunity to present earlier versions of this work at the 2010 International Meeting of the Economic Science Association, and at Northwestern University, University of St.Gallen, University of Zurich, LUISS, Universidade Nova de Lisboa, CSEF, Bank of Italy. I am grateful to seminar participants and to Charles Manski, Marco Ottaviani, Brian Rogers, Charles Bellemare, and Jacob Goeree for their comments. A previous version of this work circulated as ‘Strategic Thinking and Subjective Expectations in a Double Auction Experiment’.

Supplementary material

10683_2014_9420_MOESM1_ESM.pdf (107 kb)
Supplementary material 1 (PDF 107 kb)


  1. Armantier, O., & Treich, N. (2009). Subjective probabilities in games: An application to the overbidding puzzle. International Economic Review, 50(4), 1079–1102.MathSciNetCrossRefGoogle Scholar
  2. Armantier, O., & Treich, N. (2013). Eliciting beliefs: Proper scoring rules, incentives, stakes and hedging. European Economic Review, 62, 17–40.CrossRefGoogle Scholar
  3. Bellemare, C., Sebald, A., & Strobel, M. (2010). Measuring the willingness to pay to avoid guilt: Estimation using equilibrium and stated belief models. Journal of Applied Econometrics.
  4. Bellemare, C., Sebald, A., & Strobel, M. (2011). Measuring the willingness to pay to avoid guilt: Estimation using equilibrium and stated belief models. Journal of Applied Econometrics, 26, 437–453.MathSciNetCrossRefGoogle Scholar
  5. Bellemare, C., Bissonnette, L., & Kröger, S. (2012). Flexible approximation of subjective expectations using probability questions. Journal of Business and Economic Statistics, 30(1), 213–245.Google Scholar
  6. Blanco, M., Engelmann, D., Normann, H., & Koch, A. (2010). Belief elicitation in experiments: Is there a hedging problem? Experimental Economics, 13(4), 412–438.zbMATHCrossRefGoogle Scholar
  7. Camerer, C., Ho, T., Chong, J., Weigelt, K. (2002). Strategic teaching and equilibrium models of repeated trust and entry games. Mimeo: CalTech.Google Scholar
  8. Camerer, C., & Hua Ho, T. (1999). Experience-weighted attraction learning in normal form games. Econometrica, 67(4), 827–874.Google Scholar
  9. Cason, T., & Friedman, D. (1997). Price formation in single call markets. Econometrica, 65(2), 311–345.zbMATHCrossRefGoogle Scholar
  10. Cheung, Y. W., & Friedman, D. (1997). Individual learning in normal form games: Some laboratory results. Games and Economic Behavior, 19(1), 46–76.zbMATHMathSciNetCrossRefGoogle Scholar
  11. Costa-Gomes, M., & Weizsäcker, G. (2008). Stated beliefs and play in normal-form games. Review of Economic Studies, 75(3), 729–762.zbMATHMathSciNetCrossRefGoogle Scholar
  12. Danz, D. N., Fehr, D., & Kübler, D. (2012). Information and beliefs in a repeated normal-form game. Experimental Economics, 15(4), 622–640.CrossRefGoogle Scholar
  13. Engelberg, J., Manski, C., & Williams, J. (2009). Comparing the point predictions and subjective probability distributions of professional forecasters. Journal of Business and Economic Statistics, 27(1), 30–41.MathSciNetCrossRefGoogle Scholar
  14. Erev, I., & Roth, A. (1998). Predicting how people play games: Reinforcement learning in experimental games with unique, mixed strategy equilibria. The American Economic Review, 88(4), 848–881.Google Scholar
  15. Fudenberg, D., & Levine, D. (1998). Learning in games. European Economic Review, 42(3), 631–639.CrossRefGoogle Scholar
  16. Haruvy, E., Lahav, Y., & Noussair, C. (2007). Traders’ expectations in asset markets: Experimental evidence. The American Economic Review, 97(5), 1901–1920.CrossRefGoogle Scholar
  17. Hoffrage, U., Lindsey, S., Hertwig, R., & Gigerenzer, G. (2000). Communicating statistical information. Science, 290(5500), 2261–2262.Google Scholar
  18. Hommes, C., Sonnemans, J., Tuinstra, J., & Velden, H. V. D. (2005). Coordination of expectations in asset pricing experiments. Review of Financial Studies, 18(3), 955.CrossRefGoogle Scholar
  19. Hyndman, K., Özbay, E., Schotter, A., & Ehrblatt, W. Z. (2012). Convergence: An experimental study of teaching and learning in repeated games. Journal of the European Economic Association, 10(3), 573–604.CrossRefGoogle Scholar
  20. Kirchkamp, O., & Reiß, J. P. (2011). Out-of-equilibrium bids in first-price auctions: Wrong expectations or wrong bids. The Economic Journal, 121(557), 1361–1397.CrossRefGoogle Scholar
  21. Manski, C. (2002). Identification of decision rules in experiments on simple games of proposal and response. European Economic Review, 46(4–5), 880–891.CrossRefGoogle Scholar
  22. Manski, C. (2004). Measuring expectations. Econometrica, 72(5), 1329–1376.zbMATHCrossRefGoogle Scholar
  23. Manski, C., & Neri, C. (2013). First-and second-order subjective expectations in strategic decision-making: Experimental evidence. Games and Economic Behavior, 81(C), 232–254.zbMATHMathSciNetCrossRefGoogle Scholar
  24. Nyarko, Y., & Schotter, A. (2002). An experimental study of belief learning using elicited beliefs. Econometrica, 70(3), 971–1005.zbMATHMathSciNetCrossRefGoogle Scholar
  25. Palfrey, T., & Wang, S. (2009). On eliciting beliefs in strategic games. Journal of Economic Behavior & Organization, 71(2), 98–109.CrossRefGoogle Scholar
  26. Rey-Biel, P. (2009). Equilibrium play and best response to (stated) beliefs in normal form games. Games and Economic Behavior, 65(2), 572–585.zbMATHMathSciNetCrossRefGoogle Scholar
  27. Rustichini, A., Satterthwaite, M., & Williams, S. (1994). Convergence to efficiency in a simple market with incomplete information. Econometrica, 62(5), 1041–1063.zbMATHCrossRefGoogle Scholar
  28. Rutström, E., & Wilcox, N. (2009). Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test. Games and Economic Behavior, 67(2), 616–632.zbMATHMathSciNetCrossRefGoogle Scholar
  29. Seidenfeld, T. (1985). Calibration, coherence, and scoring rules. Philosophy of Science, 52, 274–294.MathSciNetCrossRefGoogle Scholar
  30. Smith, V., Suchanek, G., & Williams, A. (1988). Bubbles, crashes, and endogenous expectations in experimental spot asset markets. Econometrica, 56(5), 1119–1151.CrossRefGoogle Scholar
  31. Sonnemans, J., Hommes, C., Tuinstra, J., & Velden, H. V. D. (2004). The instability of a heterogeneous cobweb economy: A strategy experiment on expectation formation. Journal of Economic Behavior & Organization, 54(4), 453–481.CrossRefGoogle Scholar

Copyright information

© Economic Science Association 2015

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of St. GallenSt. GallenSwitzerland

Personalised recommendations