Random Expected Utility and Certainty Equivalents: Mimicry of Probability Weighting Functions

Wilcox, Nathaniel (2016): Random Expected Utility and Certainty Equivalents: Mimicry of Probability Weighting Functions.

Abstract

For simple prospects routinely used for certainty equivalent elicitation, random expected utility preferences imply a conditional expectation function that can mimic deterministic rank dependent preferences. That is, an agent with random expected utility preferences can have expected certainty equivalents exactly like those predicted by rank dependent probability weighting functions of the inverse-s shape discussed by Quiggin (1982) and advocated by Tversky and Kahneman (1992), Prelec (1998) and other scholars. Certainty equivalents may not nonparametrically identify preferences: Their conditional expectation (and critically, their interpretation) depends on assumptions concerning the source of their variability.

• Item Type: MPRA Paper
• Language: English
• Keywords: Random Expected Utility, Certainty Equivalents, Money Equivalents, Probability Weighting, Probability Weighting Function, Weighting Function
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C9 - Design of Experiments > C91 - Laboratory, Individual Behavior; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty

References

Abdellaoui, M., 2000, Parameter-free elicitation of utility and probability weighting functions. Management Science 46, 1497-1512.

Abdellaoui, M., H. Bleichrodt and C. Paraschiv, 2007, Loss aversion under prospect theory: A parameter-free measurement. Management Science 53, 1659-1674.

Ahn, D. S. and T. Sarver, 2013, Preference for flexibility and random choice. Econometrica 81:341-361.

Apesteguia, and M. Ballester, 2016, Monotone stochastic choice models: The case of risk and time preferences. Journal of Political Economy (forthcoming).

Becker, G. M., M. H. DeGroot and J. Marschak, 1963, Stochastic models of choice behaviour. Behavioral Science 8, 41–55.

Blavatskyy, P. and G. Pogrebna, 2010, Models of stochastic choice and decision theories: Why both are important for analyzing decisions. Journal of Applied Econometrics 25, 963-986.

Bleichrodt, H. and J. L. Pinto, 2000, A parameter-free estimation of the probability weighting function in medical decision analysis. Management Science 46:1485-1496.

Bruhin, A., H. Fehr-Duda and T. Epper, 2010, Risk and rationality: Uncovering heterogeneity in probability distortion. Econometrica 78:1375-1412.

Butler, D. and G. Loomes, 2007, Imprecision as an account of the preference reversal phenomenon. American Economic Review 97, 277-297.

Eliashberg, J. and J.R. Hauser, 1985, A measurement error approach for modeling consumer risk preference. Management Science 31, 1-25.

Feller, W., 1971, An Introduction to Probability Theory and Its Applications Vol. 2, 2nd ed. New York: Wiley.

Fox, C. R. and R. A. Poldrack, 2009, Prospect theory and the brain. In P. Glimcher, C. Camerer, E. Fehr and R. Poldrack, eds., Neuroeconomics: Decision Making and the Brain. London, UK: Academic Press.

González-Velasco, E. A., 1995, Fourier analysis and boundary value problems. San Diego: Academic Press.

Gonzalez, R. and G. Wu, 1999, On the shape of the probability weighting function. Cognitive Psychology 38, 129-166.

Gul, F. and W. Pesendorfer, 2006, Random expected utility. Econometrica 74,121-146.

Halevy, Y., 2007, Ellsberg revisited: An experimental study. Econometrica 75, pp. 503-536.

Hendry, D. F. and M. S. Morgan, 2005, The foundations of econometric analysis. Cambridge: Cambridge University Press.

Hilton, R.W., 1989. Risk attitude under random utility. Journal of Mathematical Psychology 33, 206–222.

Hougaard, P., 1986, Survival models for heterogeneous populations derived from stable distributions. Biometrika 73:387-396.

Karni, E. and Z. Safra, 2016, A theory of stochastic choice under uncertainty. Journal of Mathematical Economics 63:164–173.

Krahnen, J. P., C. Rieck and E. Theissen, 1997, Inferring risk attitudes from certainty equivalents: Some lessons from an experimental study. Journal of Economic Psychology 18:469-486.

Loomes, G., P. Moffatt, R. Sugden, 2002, A microeconometric test of alternative stochastic theories of risky choice. Journal of Risk and Uncertainty 24, 103-130.

Loomes, G. and G. Pogrebna, 2014, Measuring individual risk attitudes when preferences are imprecise. Economic Journal 124, 569-593.

Loomes, G. and R. Sugden, 1995, Incorporating a stochastic element into decision theories. European Economic Review 39, 641-648.

Loomes, G. and R. Sugden, 1998, Testing different stochastic specifications of risky choice. Economica 65, 581-598.

Luce, R. D., 1997, Some unresolved conceptual problems in mathematical psychology. Journal of Mathematical Psychology 41:79-87.

________, 2000, Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches. London, UK: Erlbaum.

Navarro-Martinez, D., G. Loomes, A. Isoni and D. Butler, 2014, Boundedly rational expected utility theory. University of Warwick working paper.

Nelder, J. A. and R. Mead, 1965, “A simplex method for function minimization.” Computer Journal 7, 308–313.

Nielson, W. S., 2003, Probability transformations in the study of behavior toward risk. Synthese 135:171–192. Nolan, J. P., 2017 (forthcoming), Stable Distributions—Models for Heavy Tailed Data. Boston: Birkhauser.

Pennings, J. M. E. and A. Smidts, 2000, Assessing the construct validity of risk attitude. Management Science 46:1337-1348.

Powell, M. J. D., 1992, A direct search optimization method that models the objective and constraint functions by linear interpolation. Technical Report DAMTP 1992/NA5, Department of Applied Mathematics and Theoretical Physics, University of Cambridge.

Prelec, D., 1998, The probability weighting function. Econometrica 66, 497-527.

Quiggin, J., 1982, A theory of anticipated utility. Journal of Economic Behavior and Organization 3, 323-343.

Regenwetter, M., J. Dana and C. P. Davis-Stober, 2011, Transitivity of preferences. Psychological Review 118:42–56.

Regenwetter, M. and A. A. J. Marley, 2001, Random relations, random utilities and random functions. Journal of Mathematical Psychology 45:864-912.

Seshadri, V., 1993, The inverse Gaussian distribution—A case study in exponential families. Oxford: Clarendon Press.

Tversky, A. and C. R. Fox, 1995, Weighing risk and uncertainty. Psychological Review 102, 269-283.

Tversky, A. and D. Kahneman, 1992, Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, 297–323.

Vieider, F. M., M. Lefebvre, R. Bouchouicha, T. Chmura, R. Hakimov, M. Krawczyk and P. Martinsson, 2015, Common components of risk and uncertainty attitudes across contexts and domains: Evidence from 30 countries. Journal of the European Economic Association 13, 421-452.

von Winterfeldt, D., N.-K. Chung, R. D. Luce, and Y.-H. Cho, 1997, Tests of consequence monotonicity in decision making under uncertainty. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 406–426.

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

Wilcox, N., 2008, Stochastic models for binary discrete choice under risk: A critical primer and econometric comparison. In J. C. Cox and G. W. Harrison, eds., Research in Experimental Economics Vol. 12: Risk Aversion in Experiments pp. 197-292. Bingley, UK: Emerald.

Wilcox, N., 2011, ‘Stochastically more risk averse:’ A contextual theory of stochastic discrete choice under risk. Journal of Econometrics 162, 89-104.

Wilcox, N., 2015. Unusual estimates of probability weighting functions. Chapman University, Economic Science Institute Working Paper #15-10.

Wu, G. and R. Gonzalez, 1999, Nonlinear decision weights in choice under uncertainty. Management Science 45, 74-85.