Greco, Salvatore and Rindone, Fabio (2011): The bipolar Choquet integral representation.

PDF
MPRA_paper_38957.pdf Download (604kB)  Preview 
Abstract
Cumulative Prospect Theory of Tversky and Kahneman (1992) is the modern version of Prospect Theory (Kahneman and Tversky (1979)) and is nowadays considered a valid alternative to the classical Expected Utility Theory. Cumulative Prospect theory implies GainLoss Separability, i.e. the separate evaluation of losses and gains within a mixed gamble. Recently, some authors have questioned this assumption of the theory, proposing new paradoxes where the GainLoss Separability is violated. We present a generalization of Cumulative Prospect Theory which does not imply GainLoss Separability and is able to explain the cited paradoxes. On the other hand, the new model, which we call the bipolar Cumulative Prospect Theory, genuinely generalizes the original Prospect Theory of Kahneman and Tversky (1979), preserving the main features of the theory. We present also a characterization of the bipolar Choquet Integral with respect to a bicapacity in a discrete setting.
Item Type:  MPRA Paper 

Original Title:  The bipolar Choquet integral representation 
English Title:  The bipolar Choquet integral representation 
Language:  English 
Keywords:  Cumulative Prospect Theory; GainsLoss Separability; bi Weighting Function; Bipolar Choquet Integral 
Subjects:  D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D81  Criteria for DecisionMaking under Risk and Uncertainty C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60  General 
Item ID:  38957 
Depositing User:  Fabio Rindone 
Date Deposited:  22 May 2012 20:54 
Last Modified:  18 Nov 2016 04:15 
References:  Abdellaoui, M. (2000). Parameterfree elicitation of utility and probability weighting functions. Management Science, 46(11), 1497–1512. Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’´ecole Am´ericaine. Econometrica, 21(4), 503–546. Baltussen, G., Post, T., and van Vliet, P. (2006). Violations of cumulative prospect theory in mixed gambles with moderate probabilities. Management Science, 52(8), 1288. Barberis, N., Huang, M., and Santos, T. (2001). Prospect Theory and Asset Prices. Quarterly Journal of Economics, 116(1), 1–53. Benartzi, S. and Thaler, R. (1995). Myopic loss aversion and the equity premium puzzle. The Quarterly Journal of Economics, 110(1), 73–92. Birnbaum, M. (2005). Three new tests of independence that differentiate models of risky decision making. Management Science, 51(9), 1346–1358. Birnbaum, M. and Bahra, J. (2007). Gainloss separability and coalescing in risky decision making. Management Science, 53(6), 1016–1028. Blavatskyy, P. (2005). Back to the St. Petersburg paradox? Management Science, 51(4), 677–678. Bleichrodt, H. and Pinto, J. (2000). A parameterfree elicitation of the probability weighting function in medical decision analysis. Management Science, 46(11), 1485–1496. Camerer, C. (2000). Prospect theory in the wild: Evidence from the field. Advances in Behavioral Economics. Choquet, G. (1953). Theory of capacities. Ann. Inst. Fourier, 5(131295), 54. Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 75(4), 643–669. Gilboa, I. (1987). Expected utility with purely subjective nonadditive probabilities. Journal of Mathematical Economics, 16(1), 65–88. Goldstein, W. and Einhorn, H. (1987). Expression theory and the preference reversal phenomena. Psychological Review, 94(2), 236. Gonzalez, R. and Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38(1), 129–166. Grabisch, M. and Labreuche, C. (2005a). Bicapacities–I: definition, Mobius transform and interaction. Fuzzy Sets and Systems, 151(2), 211–236. Grabisch, M. and Labreuche, C. (2005b). Bicapacities–II: the Choquet integral. Fuzzy Sets and Systems, 151(2), 237–259. Greco, S., Matarazzo, B., and Slowinski, R. (2002). Bipolar Sugeno and Choquet integrals. In EUROFUSE Workshop on Informations Systems, pages 191–196. Ingersoll, J. (2008). Nonmonotonicity of the TverskyKahneman probability weighting function: a cautionary note. European Financial Management, 14(3), 385–390. Jolls, C., Sunstein, C., and Thaler, R. (1998). A behavioral approach to law and economics. Stanford Law Review, 50(5), 1471–1550. Kahneman, D. and Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, 47(2), 263–291. Lattimore, P., Baker, J., and Witte, A. (1992). The influence of probability on risky choice: A parametric examination. Journal of Economic Behavior & Organization, 17(3), 377–400. Levy, M. and Levy, H. (2002). Prospect Theory: Much ado about nothing? Management Science, 48(10), 1334–1349. Luce, R. (1999). Binary Gambles of a Gain and a Loss: an Understudied Domain. Mathematical utility theory: utility functions, models, and applicaitons in the social sciences, 8, 181–202. Luce, R. (2000). Utility of Gains and Losses:: MeasurementTheoretical, and Experimental Approaches. McNeil, B., Pauker, S., Sox Jr, H., and Tversky, A. (1982). On the elicitation of preferences for alternative therapies. New England journal of medicine, 306(21), 1259–1262. Prelec, D. (1998). The probability weighting function. Econometrica, 66(3), 497–527. Quattrone, G. and Tversky, A. (1988). Contrasting rational and psychological analyses of political choice. The American political science review, 82(3), 719–736. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior & Organization, 3(4), 323–343. Rieger, M. and Wang, M. (2006). Cumulative prospect theory and the St. Petersburg paradox. Economic Theory, 28(3), 665–679. Schade, C., Schroeder, A., and Krause, K. (2010). Coordination after gains and losses: Is prospect theory’s value function predictive for games? Journal of Mathematical Psychology. Schmeidler, D. (1986). Integral representation without additivity. Proceedings of the American Mathematical Society, 97(2), 255–261. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica: Journal of the Econometric Society, 57(3), 571–587. Tversky, A. and Fox, C. (1995). Weighing risk and uncertainty. Psychological review, 102(2), 269–283. Tversky, A. and Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty, 5(4), 297–323. Von Neumann, J. and Morgenstern, O. (1944). Theories of games and economic behavior. Princeton University Press Princeton, NJ. Wu, G. and Gonzalez, R. (1996). Curvature of the probability weighting function. Management Science, 42(12), 1676–1690. Wu, G. and Markle, A. (2008). An empirical test of gainloss separability in prospect theory. Management Science, 54(7), 1322–1335. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/38957 