Abstract
The Arrow–Pratt index, a gold standard in studies of risk attitudes, is not directly observable from choice data. Existing methods to measure it rely on parametric assumptions. We introduce a discrete Arrow–Pratt index, and its relative counterpart, that can be directly obtained from choices. Our approach is general: it is (i) non-parametric, (ii) applicable to both risk and uncertainty, (iii) and robust to probability transformation, non-additive beliefs and multiple priors. Our index can also be used to characterize various decision models through various simple consistency requirements. We analyze its properties and demonstrate how it can be measured.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abdellaoui, M., Bleichrodt, H., Paraschiv, C.: Loss aversion under prospect theory: a parameter-free measurement. Manag. Sci. 53(10), 1659–1674 (2007)
Allais, M.: Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’école americaine. Econometrica 21(4), 503–546 (1953)
Apesteguia, J., Ballester, M.A.: Monotone stochastic choice models: the case of risk and time preferences. J. Politi. Econ. 126(1), 74–106 (2018)
Arrow, K.J.: Aspects of the Theory of Risk-Bearing. Yrjö Jahnssonin Säätiö, Helsinki (1965)
Baillon, A., Driesen, B., Wakker, P.P.: Relative concave utility for risk and ambiguity. Games Econ. Behav. 75(2), 481–489 (2012)
Barsky, R.B., Juster, F.T., Kimball, M.S., Shapiro, M.D.: Preference parameters and behavioral heterogeneity: an experimental approach in the health and retirement study. Q. J. Econ. 112(2), 537–579 (1997)
Bouchouicha, R., Vieider, F.M.: Accommodating stake effects under prospect theory. J. Risk Uncertainty 55(1), 1–28 (2017)
Bruhin, A., Fehr-Duda, H., Epper, T.: Risk and rationality: uncovering heterogeneity in probability distortion. Econometrica 78(4), 1375–1412 (2010)
Chateauneuf, A., Wakker, P.: From local to global additive representation. J. Math. Econ. 22(6), 523–545 (1993)
Cohen, M., Jaffray, J.-Y., Said, T.: Experimental comparison of individual behavior under risk and under uncertainty for gains and for losses. Organ. Behav. Hum. Decis. Process. 39(1), 1–22 (1987)
de Finetti, B.: Sulla preferibilitá. Giornale degli Economisti e Annali di Economia 11(11/12), 685–709 (1952)
Dean, M., Ortoleva, P.: Allais, Ellsberg, and preferences for hedging. Theor. Econ. 12(1), 377–424 (2017)
Dohmen, T., Falk, A., Huffman, D., Sunde, U., Schupp, J., Wagner, G.G.: Individual risk attitudes: measurement, determinants, and behavioral consequences. J. Eur. Econ. Assoc. 9(3), 522–550 (2011)
Eeckhoudt, L., Liu, L., Meyer, J.: Restricted increases in risk aversion and their application. Econ. Theor. 64(1), 161–181 (2017)
Ellsberg, D.: Risk, ambiguity, and the savage axioms. Q. J. Econ. 75(4), 643–669 (1961)
Ghirardato, P., Marinacci, M.: Risk, ambiguity, and the separation of utility and beliefs. Math. Oper. Res. 26(4), 864–890 (2001)
Ghirardato, P., Pennesi, D.: Mixing without randomness. Working Paper, Collegio Carlo Alberto, University of Torino (2019)
Ghirardato, P., Maccheroni, F., Marinacci, M., Siniscalchi, M.: A subjective spin on roulette wheels. Econometrica 71(6), 1897–1908 (2003)
Ghirardato, P., Maccheroni, F., Marinacci, M.: Differentiating ambiguity and ambiguity attitude. J. Econ. Theory 118(2), 133–173 (2004)
Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18(2), 141–153 (1989)
Gul, F., et al.: Savage’s theorem with a finite number of states. J. Econ. Theory 57(1), 99–110 (1992)
Hardy, G., Littlewood, J., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Klibanoff, P., Marinacci, M., Mukerji, S.: A smooth model of decision making under ambiguity. Econometrica 73(6), 1849–1892 (2005)
Köbberling, V., Wakker, P.P.: Preference foundations for nonexpected utility: a generalized and simplified technique. Math. Oper. Res. 28(3), 395–423 (2003)
Lajeri, F., Nielsen, L.T.: Parametric characterizations of risk aversion and prudence. Econ. Theor. 15(2), 469–476 (2000)
Luce, R.D.: Rank-and sign-dependent linear utility models for binary gambles. J. Econ. Theory 53(1), 75–100 (1991)
Machina, M.J.: “Expected utility” analysis without the independence axiom. Econometrica 50(2), 277–323 (1982)
Meyer, J.: Representing risk preferences in expected utility based decision models. Ann. Oper. Res. 176(1), 179–190 (2010)
Montesano, A.: De Finetti and the Arrow–Pratt measure of risk aversion. In: Gavalotti, M. (ed.) Bruno de Finetti Radical Probabilist. College Publications, London (2009)
Pratt, J.W.: Risk aversion in the small and in the large. Econometrica 32(1), 122–136 (1964)
Prelec, D.: Decreasing impatience: a criterion for non-stationary time preference and “hyperbolic” discounting. Scand. J. Econ. 106(3), 511–532 (2004)
Quiggin, J.: Risk perception and risk aversion among Australian farmers. Aust. J. Agric. Econ. 25(2), 160–169 (1981)
Rieger, M.O., Wang, M., Hens, T.: Risk preferences around the world. Manag. Sci. 61(3), 637–648 (2014)
Rohde, K.I.: Measuring decreasing and increasing impatience. Manag. Sci. 65(4), 1700–1716 (2019)
Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)
Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57(3), 571–587 (1989)
Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertainty 5(4), 297–323 (1992)
Van de Kuilen, G., Wakker, P.P.: The midweight method to measure attitudes toward risk and ambiguity. Manag. Sci. 57(3), 582–598 (2011)
Vind, K.: Independent preferences. J. Math. Econ. 20(1), 119–135 (1991)
Wakker, P.P.: Additive Representations of Preferences: A New Foundation of Decision Analysis. Kluwer Academic Publishers, Dordrecht (1989)
Wakker, P.P.: Prospect Theory: For Risk and Ambiguity. Cambridge University Press, Cambridge (2010)
Wakker, P., Deneffe, D.: Eliciting von Neumann–Morgenstern utilities when probabilities are distorted or unknown. Manag. Sci. 42(8), 1131–1150 (1996)
Werner, K.M., Zank, H.: A revealed reference point for prospect theory. Econ. Theor. 67(4), 731–773 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Baillon, A., L’Haridon, O. Discrete Arrow–Pratt indexes for risk and uncertainty. Econ Theory 72, 1375–1393 (2021). https://doi.org/10.1007/s00199-020-01315-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-020-01315-8