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
This is a preview of subscription content, log in to check access.
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’.
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
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
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
Camerer, C., Ho, T., Chong, J., Weigelt, K. (2002). Strategic teaching and equilibrium models of repeated trust and entry games. Mimeo: CalTech.Google Scholar
Camerer, C., & Hua Ho, T. (1999). Experience-weighted attraction learning in normal form games. Econometrica,67(4), 827–874.Google Scholar
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
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
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
Fudenberg, D., & Levine, D. (1998). Learning in games. European Economic Review, 42(3), 631–639.CrossRefGoogle Scholar
Haruvy, E., Lahav, Y., & Noussair, C. (2007). Traders’ expectations in asset markets: Experimental evidence. The American Economic Review, 97(5), 1901–1920.CrossRefGoogle Scholar
Hoffrage, U., Lindsey, S., Hertwig, R., & Gigerenzer, G. (2000). Communicating statistical information. Science,290(5500), 2261–2262.Google Scholar
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
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
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
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
Rustichini, A., Satterthwaite, M., & Williams, S. (1994). Convergence to efficiency in a simple market with incomplete information. Econometrica, 62(5), 1041–1063.zbMATHCrossRefGoogle Scholar
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
Smith, V., Suchanek, G., & Williams, A. (1988). Bubbles, crashes, and endogenous expectations in experimental spot asset markets. Econometrica, 56(5), 1119–1151.CrossRefGoogle Scholar
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