Eliciting Probabilities, Means, Medians, Variances and Covariances without Assuming Risk Neutrality

Affiliation(s)

University Vienna, Vienna, Austria.

Department of Economics, J. W. Goethe University, Frankfurt am Main, Germany.

University Vienna, Vienna, Austria.

Department of Economics, J. W. Goethe University, Frankfurt am Main, Germany.

ABSTRACT

We are interested in incentivizing experimental subjects to report their beliefs truthfully, without imposing assumptions on their risk preferences. We prove that if subjects are not risk neutral, it is not possible to elicit subjective probabilities or the mean of a subjective probability distribution truthfully using deterministic payments schemes, which are predominant in the literature. We present a simple randomization trick that transforms deterministic rewards into randomized rewards, such that agents with arbitrary risk preferences report as if they were risk neutral. Using this trick, we show how to elicit probabilities, means, medians, variances and covariances of the underlying distribution without assuming risk neutrality.

Cite this paper

K. H. Schlag and J. J. van der Weele, "Eliciting Probabilities, Means, Medians, Variances and Covariances without Assuming Risk Neutrality,"*Theoretical Economics Letters*, Vol. 3 No. 1, 2013, pp. 38-42. doi: 10.4236/tel.2013.31006.

K. H. Schlag and J. J. van der Weele, "Eliciting Probabilities, Means, Medians, Variances and Covariances without Assuming Risk Neutrality,"

References

[1] G. W. Brier, “Verification of Forecasts Expressed in Terms of Probability,” Monthly Weather Review, Vol. 78, No. 1, 1950, pp. 1-3. doi:10.1175/1520-0493(1950)078<0001:VOFEIT>2.0.CO;2

[2] R. Winkler and A. Murphy, “Nonlinear Utility and the Probability Score,” Journal of Applied Meteorology, Vol. 9, 1970, pp. 143-148. doi:10.1175/1520-0450(1970)009<0143:NUATPS>2.0.CO;2

[3] C. A. Holt and S. Laury, “Risk Aversion and Incentive Effects,” The American Economic Review, Vol. 92, No. 5, 2002, p. 1644. doi:10.1257/000282802762024700

[4] O. Armantier and N. Treich, “Eliciting Beliefs: Proper Scoring Rules, Incentives, Stakes and Hedging,” IDEI Working Paper, Vol. 643, 2010.

[5] T. Offerman, J. Sonnemans, G. Van de Kuilen and P. P. Wakker, “A Truth Serum for Non-Bayesians,” Review of Economic Studies, Vol. 76, No. 4, 2009, pp. 1461-1489. doi:10.1111/j.1467-937X.2009.00557.x

[6] C. Smith, “Consistency in Statistical Inference and Decision,” Journal of the Royal Statistical Society. Series B, Vol. 23, No. 1, 1961, pp. 1-37.

[7] L. J. Savage, “The Foundation of Statistics,” Wiley, New York, 1954.

[8] L. Savage, “Elicitation of Personal Probabilities and Expectations,” Journal of the American Statistical Association, Vol. 66, No. 336, 1971, pp. 783-801. doi:10.1080/01621459.1971.10482346

[9] A. Roth and M. Malouf, “Game-Theoretic Models and the Role of Information in Bargaining,” Psychological review, Vol. 86, No. 6, 1979, pp. 574-594. doi:10.1037/0033-295X.86.6.574

[10] D. Grether, “Financial Incentive Effects and Individual Decision-Making,” Working Paper 401, California Institute of Technology, 1981.

[11] F. Allen, “Discovering Personal Probabilities When Utility Functions Are Unknown,” Management Science, Vol. 33, No. 4, 1987, pp. 542-544. doi:10.1287/mnsc.33.4.542

[12] G. Becker, M. DeGroot and J. Marschak, “Measuring Utility by a Single-Response Sequential Method,” Behavioral Science, Vol. 9, No. 3, 1964, p. 226. doi:10.1002/bs.3830090304

[13] T. Hossain and R. Okui, “The Binarized Scoring Rule,” Manuscript, 2009.

[14] G. W. Harrison, J. Martffnez-Correa and J. Swarthout, “Eliciting Subjective Probabilities with Binary Lotteries,” Manuscript, Georgia State University, Atlanta, 2012.

[15] C. Holt, “Markets, Games and Strategic Behavior,” Pearson/Addison-Wesley, Boston, 2006.

[16] E. Karni, “A Mechanism for Eliciting Probabilities,” Econometrica, Vol. 77, No. 2, 2009, pp. 603-606. doi:10.3982/ECTA7833

[17] R. McKelvey and T. Page, “Public and Private Information: An Experimental Study of Information Pooling,” Econometrica: Journal of the Econometric Society, Vol. 58, No. 6, 1990, pp. 1321-1339. doi:10.2307/2938318

[18] J. L. Cervera and J. Munoz, “Proper Scoring Rules for Fractiles,” In: J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics 5, Oxford University Press, Oxford, 1996, pp. 513-519.

[19] R. Selten, A. Sadrieh and K. Abbink, “Money Does Not Induce Risk Neutral Behavior, but Binary Lotteries Do even Worse,” Theory and Decision, Vol. 46, 1999, pp. 211-249. doi:10.1023/A:1005038628305

[20] G. W. Harrison, J. Martinez-Correa and T. Swarthout, “Inducing Risk Neutral Preferences with Binary Lotteries: A Reconsideration,” Journal of Economic Behavior and Organization, 2012, in Press. doi:10.1016/j.jebo.2012.09.008

[1] G. W. Brier, “Verification of Forecasts Expressed in Terms of Probability,” Monthly Weather Review, Vol. 78, No. 1, 1950, pp. 1-3. doi:10.1175/1520-0493(1950)078<0001:VOFEIT>2.0.CO;2

[2] R. Winkler and A. Murphy, “Nonlinear Utility and the Probability Score,” Journal of Applied Meteorology, Vol. 9, 1970, pp. 143-148. doi:10.1175/1520-0450(1970)009<0143:NUATPS>2.0.CO;2

[3] C. A. Holt and S. Laury, “Risk Aversion and Incentive Effects,” The American Economic Review, Vol. 92, No. 5, 2002, p. 1644. doi:10.1257/000282802762024700

[4] O. Armantier and N. Treich, “Eliciting Beliefs: Proper Scoring Rules, Incentives, Stakes and Hedging,” IDEI Working Paper, Vol. 643, 2010.

[5] T. Offerman, J. Sonnemans, G. Van de Kuilen and P. P. Wakker, “A Truth Serum for Non-Bayesians,” Review of Economic Studies, Vol. 76, No. 4, 2009, pp. 1461-1489. doi:10.1111/j.1467-937X.2009.00557.x

[6] C. Smith, “Consistency in Statistical Inference and Decision,” Journal of the Royal Statistical Society. Series B, Vol. 23, No. 1, 1961, pp. 1-37.

[7] L. J. Savage, “The Foundation of Statistics,” Wiley, New York, 1954.

[8] L. Savage, “Elicitation of Personal Probabilities and Expectations,” Journal of the American Statistical Association, Vol. 66, No. 336, 1971, pp. 783-801. doi:10.1080/01621459.1971.10482346

[9] A. Roth and M. Malouf, “Game-Theoretic Models and the Role of Information in Bargaining,” Psychological review, Vol. 86, No. 6, 1979, pp. 574-594. doi:10.1037/0033-295X.86.6.574

[10] D. Grether, “Financial Incentive Effects and Individual Decision-Making,” Working Paper 401, California Institute of Technology, 1981.

[11] F. Allen, “Discovering Personal Probabilities When Utility Functions Are Unknown,” Management Science, Vol. 33, No. 4, 1987, pp. 542-544. doi:10.1287/mnsc.33.4.542

[12] G. Becker, M. DeGroot and J. Marschak, “Measuring Utility by a Single-Response Sequential Method,” Behavioral Science, Vol. 9, No. 3, 1964, p. 226. doi:10.1002/bs.3830090304

[13] T. Hossain and R. Okui, “The Binarized Scoring Rule,” Manuscript, 2009.

[14] G. W. Harrison, J. Martffnez-Correa and J. Swarthout, “Eliciting Subjective Probabilities with Binary Lotteries,” Manuscript, Georgia State University, Atlanta, 2012.

[15] C. Holt, “Markets, Games and Strategic Behavior,” Pearson/Addison-Wesley, Boston, 2006.

[16] E. Karni, “A Mechanism for Eliciting Probabilities,” Econometrica, Vol. 77, No. 2, 2009, pp. 603-606. doi:10.3982/ECTA7833

[17] R. McKelvey and T. Page, “Public and Private Information: An Experimental Study of Information Pooling,” Econometrica: Journal of the Econometric Society, Vol. 58, No. 6, 1990, pp. 1321-1339. doi:10.2307/2938318

[18] J. L. Cervera and J. Munoz, “Proper Scoring Rules for Fractiles,” In: J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics 5, Oxford University Press, Oxford, 1996, pp. 513-519.

[19] R. Selten, A. Sadrieh and K. Abbink, “Money Does Not Induce Risk Neutral Behavior, but Binary Lotteries Do even Worse,” Theory and Decision, Vol. 46, 1999, pp. 211-249. doi:10.1023/A:1005038628305

[20] G. W. Harrison, J. Martinez-Correa and T. Swarthout, “Inducing Risk Neutral Preferences with Binary Lotteries: A Reconsideration,” Journal of Economic Behavior and Organization, 2012, in Press. doi:10.1016/j.jebo.2012.09.008