Ghossoub, Mario (2011): Towards a Purely Behavioral Definition of Loss Aversion.

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Abstract
This paper suggests a behavioral, preferencebased definition of loss aversion for decision under risk. This definition is based on the initial intuition of Markowitz [30] and Kahneman and Tversky [19] that most individuals dislike symmetric bets, and that the aversion to such bets increases with the size of the stake. A natural interpretation of this intuition leads to defining loss aversion as a particular kind of risk aversion. The notions of weak loss aversion and strong loss aversion are introduced, by analogy to the notions of weak and strong risk aversion. I then show how the proposed definitions naturally extend those of Kahneman and Tversky [19], Schmidt and Zank [48], and Zank [54]. The implications of these definitions under Cumulative Prospect Theory (PT) and ExpectedUtility Theory (EUT) are examined. In particular, I show that in EUT loss aversion is not equivalent to the utility function having an S shape: loss aversion in EUT holds for a class of utility functions that includes Sshaped functions, but is strictly larger than the collection of these functions. This class also includes utility functions that are of the FriedmanSavage [14] type over both gains and losses, and utility functions such as the one postulated by Markowitz [30]. Finally, I discuss possible ways in which one can define an index of loss aversion for preferences that satisfy certain conditions. These conditions are satisfied by preferences having a PTrepresentation or an EUTrepresentation. Under PT, the proposed index is shown to coincide with Kobberling and Wakker’s [22] index of loss aversion only when the probability weights for gains and losses are equal. In Appendix B, I consider some extensions of the study done in this paper, one of which is an extension to situations of decision under uncertainty with probabilistically sophisticated preferences, in the sense of Machina and Schmeidler [27].
Item Type:  MPRA Paper 

Original Title:  Towards a Purely Behavioral Definition of Loss Aversion 
Language:  English 
Keywords:  Loss Aversion, Risk Aversion, MeanPreserving Increase in Risk, Prospect Theory, Probability Weights, SShaped Utility 
Subjects:  D  Microeconomics > D0  General > D03  Behavioral Economics; Underlying Principles D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D81  Criteria for DecisionMaking under Risk and Uncertainty 
Item ID:  37628 
Depositing User:  Mario Ghossoub 
Date Deposited:  26. Mar 2012 02:26 
Last Modified:  18. Feb 2013 05:57 
References:  [1] C.D. Aliprantis and K.C. Border. Infinite Dimensional Analysis  3rd edition. SpringerVerlag, 2006. [2] J. Apesteguıa and M.A. Ballester. A Theory of ReferenceDependent Behavior. Economic Theory, 40(3):427–455, 2009. [3] K.J. Arrow. Essays in the Theory of RiskBearing. Chicago: Markham Publishing Company, 1971. [4] N. Barberis and M. Huang. Stocks as Lotteries: The Implications of ProbabilityWeighting for Security Prices. American Economic Review, 98(5):2066–2100, 2008. [5] N. Barberis, M. Huang, and T. Santos. Prospect Theory and Asset Prices. Quarterly Journal of Economics, 116(1):1–53, 2001. [6] S. Benartzi and R.H. Thaler. Myopic Loss Aversion and the Equity Premium Puzzle. The Quarterly Journal of Economics, 110(1):73–92, 1995. [7] C. Bernard and M. Ghossoub. Static Portfolio Choice under Cumulative Prospect Theory. Mathematics and Financial Economics, 2(4):277–306, 2010. [8] P.R. Blavatskyy. Loss Aversion. Economic Theory, 46(1):127–148, 2011. [9] D. Bowman, D. Minehart, and M. Rabin. Loss Aversion in a ConsumptionSavings Model. Journal of Economic Behavior and Organization, 38(2):155–178, 1999. [10] C.F. Camerer. Prospect Theory in the Wild: Evidence from the Field. In D. Kahneman and A. Tversky (eds.), Choices, Values, and Frames. Cambridge: Cambridge University Press, 2000. [11] G. Carlier and R.A. Dana. Optimal Demand for Contingent Claims when Agents Have Law Invariant Utilities. Mathematical Finance, 21(2):169–201, 2011. [12] J. Diestel and J.J. Uhl. Vector Measures. AMS Mathematical Surverys, 1977. [13] N. Dunford and J.T. Schwartz. Linear Operators, Part 1: General Theory. WileyInterscience, 1958. [14] M. Friedman and L.J. Savage. The Utility Analysis of Choices Involving Risk. The Journal of Political Economy, 56(4):279, 1948. [15] R.M. Griffith. Odds Adjustments by American HorseRace Bettors. The American Journal of Psychology, 62(2):290–294, 1949. [16] J. Hadar and W. Russell. Rules for Ordering Uncertain Prospects. American Economic Review, 59(1):25–34, 1969. [17] X.D. He and X.Y. Zhou. Portfolio Choice under Cumulative Prospect Theory: An Analytical Treatment. Management Science, 57(2):315–331, 2011. [18] H. Jin and X.Y. Zhou. Behavioral Portfolio Selection in Continous Time. Mathematical Finance, 18(3):385–426, 2008. [19] D. Kahneman and A. Tversky. Prospect Theory: An Analysis of Decision Under Risk. Econometrica, 47(2):263–291, 1979. [20] J.L. Kelley. General Topology. Springer, 1975. [21] J.L. Knetsch and J.A. Sinden. Willingness to Pay and Compensation Demanded: Experimental Evidence of an Unexpected Disparity in Measures of Value. Quarterly Journal of Economics, 99(3):507–521, 1984. [22] V. Kobberling and P.P. Wakker. An Index of Loss Aversion. Journal of Economic Theory, 122(1):119–131, 2005. [23] B. Koszegi and M. Rabin. A Model of ReferenceDependent Preferences. The Quarterly Journal of Economics, 121(4):1133–1165, 2006. [24] B. Koszegi and M. Rabin. ReferenceDependent Risk Attitudes. American Economic Review, 97(4):1047–1073, 2007. [25] B. Koszegi and M. Rabin. ReferenceDependent Consumption Plans. The American Economic Review, 99(3):909–936, 2009. [26] S.K. Kundu and B.K. Lahiri. Integration of VectorValued Functions. Indian Journal of Pure and Applied Mathematics, 10(5):617–628, 1979. [27] M.J. Machina and D. Schmeidler. A More Robust Definition of Subjective Probability. Econometrica, 60(4):745–780, 1992. [28] M.J. Machina and D. Schmeidler. Bayes without Bernoulli: Simple Conditions for Probabilistically Sophisticated Choice. Journal of Economic Theory, 67(1):106–128, 1995. [29] M. Marinacci and L. Montrucchio. Introduction to the Mathematics of Ambiguity. In I. Gilboa (ed.), Uncertainty in Economic Theory. Routledge, London, 2004. [30] H. Markowitz. The Utility of Wealth. The Journal of Political Economy, 60(2):151–158, 1952. [31] Y. Masatlioglu and E.A. Ok. A Canonical Model of Choice with Initial Endowments. mimeo (2009). [32] Y. Masatlioglu and E.A. Ok. Rational Choice with Status Quo Bias. Journal of Economic Theory, 121(1):1–29, 2005. [33] R. Mehra and E.C. Prescott. The Equity Premium: A Puzzle. Journal of Monetary Economics, 15(2):145–161, 1985. [34] F. Mosteller and P. Nogee. An Experimental Measurement of Utility. The Journal of Political Economy, 59(5):371–404, 1951. [35] A. Muller. Another Tale of Two Tails: On Characterizations of Comparative Risk. Journal of Risk and Uncertainty, 16(2):187–197, 1998. [36] A. Muller. Comparing Risks with Unbounded Distributions. Journal of Mathematical Economics, 30(2):229–239, 1998. [37] L. Nachbin. Topology and Order. D. Van Nostrand, 1965. [38] W.S. Neilson. Comparative Risk Sensitivity with ReferenceDependent Preferences. The Journal of Risk and Uncertainty, 24(2):131–142, 2002. [39] E.A. Ok, P. Ortoleva, and G. Riella. Revealed (P)Reference Theory. mimeo (2009). [40] P. Ortoleva. Status Quo Bias, Multiple Priors and Uncertainty Aversion. Games and Economic Behavior, 69(2):411–424, 2010. [41] A.L. Peressini. Ordered Topological Vector Spaces. Harper & Row, 1967. [42] R.S. Phillips. Integration in a Convex Linear Topological Space. Transactions of the American Mathematical Society, 47(1):114–145, 1940. [43] J.W. Pratt. Risk Aversion in the Small and in the Large. Econometrica, 32(1/2):122–136, 1964. [44] M.G. Preston and P. Baratta. An Experimental Study of the AuctionValue of an Uncertain Outcome. The American Journal of Psychology, 61(2):183–193, 1948. [45] W. Roth. OperatorValued Measures and Integrals for ConeValued Functions. Springer, 2009. [46] M. Rothschild and J.E. Stiglitz. Increasing Risk: I. A Definition. Journal of Economic Theory, 2(3):225–243, 1970. [47] W. Samuelson and R. Zeckhauser. Status quo Bias in Decision Making. Journal of Risk and Uncertainty, 1(1):7–59, 1988. [48] U. Schmidt and H. Zank. What is Loss Aversion? The Journal of Risk and Uncertainty, 30(2):157–167, 2005. [49] M. Talagrand. Pettis Integral and Measure Theory. Memoirs of the AMS – number 307, 1984. [50] R. Thaler. Toward a Positive Theory of Consumer Choice. Journal of Economic Behavior and Organization, 1(1):39–60, 1980. [51] A. Tversky and D. Kahneman. Advances in Prospect Theory: Cumulative Representation of Uncertainty. The Journal of Risk and Uncertainty, 5(4):297–323, 1992. [52] P.P. Wakker. Prospect Theory for Risk and Ambiguity. Cambridge University Press, 2010. [53] P.P. Wakker and A. Tversky. An Axiomatization of Cumulative Prospect Theory. The Journal of Risk and Uncertainty, 7(7):147–176, 1993. [54] H. Zank. On Probabilities and Loss Aversion. Theory and Decision, 68(3):243–261, 2010. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/37628 