A Note on Sandroni-Shmaya Belief Elicitation Mechanism

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Arthur Carvalho

Abstract

Incentive-compatible methods for eliciting beliefs, such as proper scoring rules, often rely on strong assumptions about how humans behave when making decisions under risk and uncertainty. For example, standard proper scoring rules assume that humans are risk neutral, an assumption that is often violated in practice. Under such an assumption, proper scoring rules induce honest reporting of beliefs, in a sense that experts maximize their expected scores from a proper scoring rule by honestly reporting their beliefs.Sandroni and Shmaya [Economic Theory Bulletin, volume 1, issue 1, 2013] suggested a remarkable mechanism based on proper scoring rules that induces honest reporting of beliefs without any assumptions on experts’ risk attitudes. In particular, the authors claimed that the mechanism relies only on the natural assumptions of probabilistic sophistication and dominance. We suggest in this paper that the reduction of compound lotteries axiom is another assumption required for Sandroni and Shmaya’s mechanism to induce honest reporting of beliefs. We further elaborate on the implications of such an extra assumption in light of recent findings regarding the reduction of compound lotteries axiom.

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References

Allen, F., 1987. Discovering Personal Probabilities when Utility Functions are Unknown. Management Science, 33(4), pp.542–544.

Armantier, O. & Treich, N., 2013. Eliciting Beliefs: Proper Scoring Rules, Incentives, Stakes and Hedging. European Economic Review, 62, pp.17–40.

Becker, G.M., Degroot, M.H. & Marschak, J., 1964. Measuring Utility by a Single-Response Sequential Method. Behavioral Science, 9(3), pp.226–232.

Carvalho, A., 2015. Tailored Proper Scoring Rules Elicit Decision Weights. Judgment and Decision Making, 10(1), pp.86–96.

Cason, T. & Plott, C., 2014. Misconceptions and Game Form Recognition: Challenges to Theories of Revealed Preference and Framing. Journal of Political Economy, 122(6), pp.1235–1270.

Gneiting, T. & Raftery, A.E., 2007. Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association, 102(477), pp.359–378.

Harrison, G.W., Martínez-Correa, J. & Swarthout, J.T., 2015. Reduction of Compound Lotteries with Objective Probabilities: Theory and Evidence. Journal of Economic Behavior and Organization, 119, pp.32–55.

Holt, C.A. & Laury, S.K., 2002. Risk Aversion and Incentive Effects. American Economic Review, 92(5), pp.1644–1655.

Karni, E., 2009. A Mechanism for Eliciting Probabilities. Econometrica, 77(2), pp.603–606.

Kothiyal, A., Spinu, V. & Wakker, P., 2011. Comonotonic Proper Scoring Rules to Measure Ambiguity and Subjective Beliefs. Journal of Multi-Criteria Decision Analysis, 17(3-4), pp.101–113.

Offerman, T., Sonnemans, J., Van De Kuilen, G., Wakker, P.P., 2009. A Truth Serum for Non-Bayesians: Correcting Proper Scoring Rules for Risk Attitudes. Review of Economic Studies, 76(4), pp.1461–1489.

Plott, C.R. & Zeiler, K., 2005. The Willingness to Pay-Willingness to Accept Gap, the “Endowment Effect,” Subject Misconceptions, and Experimental Procedures for Eliciting Valuations. American Economic Review, 95(3), pp.530–545.

Rutström, E.E., 1998. Home-Grown Values and Incentive Compatible Auction Design. International Journal of Game Theory, 27(3), pp.427–441.

Sandroni, A. & Shmaya, E., 2013. Eliciting Beliefs by Paying in Chance. Economic Theory Bulletin, 1(1), pp.33–37.

Schlag, K.H. & van der Weele, J.J., 2013. Eliciting Probabilities, Means, Medians, Variances and Covariances without Assuming Risk Neutrality. Theoretical Economics Letters, 3(1), pp.38–42.

Wakker, P.P., 2010. Prospect theory: For Risk and Ambiguity, Cambridge University Press.

Weber, B.J. & Chapman, G.B., 2005. Playing for Peanuts: Why is Risk Seeking More Common for Low-Stakes Gambles? Organizational Behavior and Human Decision Processes, 97(1), pp.31–46.

Winkler, R.L., 1969. Scoring Rules and the Evaluation of Probability Assessors. Journal of the American Statistical Association, 64(327), pp.1073–1078.

Winkler, R.L. & Murphy, A.H., 1968. “Good” Probability Assessors. Journal of Applied Meteorology, 7(5), pp.751–758.

Winkler, R.L. & Murphy, A.H., 1970. Nonlinear Utility and the Probability Score. Journal of Applied Meteorology, 9, pp.143–148.