Cadogan, Godfrey (2010): Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures.

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Abstract
We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly comonotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under BrouwerSchauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the HoffmanJorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gainloss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.
Item Type:  MPRA Paper 

Original Title:  Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures 
Language:  English 
Keywords:  monotone class theorem; stochastic choice functional; embedded probability; comonotonic probability; isomorphism 
Subjects:  D  Microeconomics > D0  General > D03  Behavioral Microeconomics: Underlying Principles D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D81  Criteria for DecisionMaking under Risk and Uncertainty C  Mathematical and Quantitative Methods > C4  Econometric and Statistical Methods: Special Topics > C44  Operations Research ; Statistical Decision Theory C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods 
Item ID:  22380 
Depositing User:  g charlescadogan 
Date Deposited:  29 Apr 2010 00:38 
Last Modified:  26 Sep 2019 12:21 
References:  Acerbi, C. (2001). Risk aversion and coherent risk measures: A spectral representation theorem. http://arxiv.org/abs/condmat/0107190. Abaxbank Working Paper, Milano, Italy. Akheizer, N. and I. Glazman (1961). Theory of Linear Operators in Hilbert Space. Frederick Ungar Publ. Co.:New York. Dover reprint 1993. al Nowaihi, A. and S. Dhami (2005). A simple derivation of Prelec’s probability weighting functions. http://ssrn.com/abstract=1130811. Working Paper No. 05 20, Dept. Economics, Univ. Leicester, UK. Ash, R. B. (2000). Probability and Measure Theory. HarcourtAcademic Press. With contributions from Catherine Dol´eansDade. Athreya, K. B. and S. N. Lahiri (2006). Measure Theory and Probability Theory. New York, N.Y.: SpringerVerlag. Billingsley, P. (1968). Convergence in Probability Measures. New York, N.Y.: John Wiley & Sons, Inc. Billingsley, P. (1995). Probability and Measure (3rd ed.). New York: John Wiley & Sons, Inc. Birchby, J., G. Gigliotti, and B. Sopher (2008, July). Consistency and aggregation in individual choice under uncertainty. pp. 1–21. Foundations and Applications of Utility, Risk and Decision Theory–Barcelona. Dep’t. Economics, Rutgers Univ. New Brunswick, New Jersey. Bliechrodt, H. and J. L. Pinto (2000). A parameter free eliicitation of the probability weightingg function in medical decisionns. Management Science 46(11), 1485–1496. Blumenthal, R. M. and R. K. Geetor (1968). Markov Processes and Potential. New York: Academic Press. Bochner, S. (1955). Harmonic Analysis and the Theory of Probability. Berkeley, CA: Univ. California Press. Brieman, L. (1968). Probability and Measure. Reading,MA. AddisonWesley. SIAM reprint 1992. Burns, Z., A. Chiu, and G. Wu (Forthcoming, Fall 2010). Encyclopedia of Operations Research and Management, Chapter Overweighting of Small Probabilities. John Wiley & Sons, Inc. Cadogan, G. (2010). On spectral analysis of risk operators with applications to prospect theory. WorkinProgress. C´ıˇzek, P. (2007). Robust and efficient adaptative estimation of binary choice regression models. http://arno.uvt.nl/show.cgi?fid=57456. CentER Discussion Paper No. 200712, Tilburg University, Netherlands. Curtain, R. F. and A. J. Pritchard (1977). Functional Analysis in Modern Appplied Mathematics, Volume 132 of Mathematics in Science and Engineering. Academic Press, New York. Dagsvik, J. (2006, July). Axiomatization of stochastic choice models under uncertainty. Discussion Paper No. 465, Statistics Norway, Research Department. Davidson, D. and J. Marschak (1958, July). Experimental tests of stochastic decision theory. Technical Report No. 17, Behavioral Sciences Division, Applied Math and Statistical Laboratory, Stanford Univ. de Palma, A., M. BenAkiva, D. Brownstone, C. Holt, T. Magnac, D. McFadden, P. Moffatt, N. Picard, K. Train, P. Wakker, and J. Walker (2008). Risk, Uncertainty and Discrete Choice. Marketing Letters 19, 269–285. Debreu, G. (1953, May). Representation of a preference ordering by a numerical function. mimeo, Cowles Comm. Discussion Paper No. 2083, Yale Univ. Debreu, G. (1958). Stochastic choice and cardinal utility. Econometrica 26, 440–444. Dugundji, J. (1966). Topology. Boston, MA: Allyn and Bacon, Inc. Fabiyi, M. E. (2008). Advances in Decision Making Under Risk and Uncertainty, Chapter II, pp. 109–118. Number 42 in Series C: Mathematical Programming and Operations Research. Berlin: SpringerVerlag. Gonzalez, R. and G. Wu (1999). On the shape of the probability weighting function. Cognitive Psychology 38, 129–166. Gonzalez, R. and G. Wu (2003, August). Composition rules in original and cumulative prospect theory. http://faculty.chicagobooth.edu/george.wu/research/papers/CompositionRules.pdf. Working Paper, Dep’t. Psychology, Univ. Michigan. H¨ardle, W. and L. Simar (2003). Applied Multivariate Statistical Analysis. SpringerVerlag. Galley proof copy. Hewitt, E. and K. Stromberg (1965). Real and Abstract Analysis. Number 25 in Graduate Texts in Mathematics. SpringerVerlag. HoffmanJorgenson, J. (1971). Existence of conditional probability. Math. Scand. 28, 257–264. Horton, G. A. (2004). Defining risk. Financial Analyst Journal 60(6), 19–25. Hsu, M., I. Krajbich, and C. F. Camerer (2009, February). Neural response to reward anticipation under risk is nonlinear in probabilities. Journal of Neuroscience 29(7), 2231–2237. Hunt, E. (2007). The Mathematics of Behavior. New York: Cambridge University Press. Ingersoll, J. E. (2008, June). Nonmonotonicity of the KhanemanTversky probabiility weighting function: A cautionary note. European Financial Management 14(3), 385–390. Jacobson, N. (1951). Lectures in Abstract Algebra. SpringerVerlag, New York. Reprint 1975. Loewenstein, G. and D. Prelec (1991). Negative time prefence. In AEA Papers and Proceedings, Volume 81, pp. 347–352. Amer. Econ. Rev. Luce, D. and L. Narens (2008). Theory of measurement. In L. Blume and S. N. Durlauf (Eds.), Palgrave Dictionary of Economics (2nd ed.). Palgrave Macmillan. Preprint. Luce, R. D. (1959). Individual Choice Behavior. New York,: John Wiley & Sons, Inc. Luce, R. D. (2001). Reduction invariance and Prelec’s weighting functions. Journal of Mathematical Psychology 45, 167–179. McFadden, D. P. (1974). Frontiers in Econometrics, Chapter IV. Conditional Logit Analysis of Qualitative Choice Behavior, pp. 105–142. New York: Academic Press. McFadden, D. P. (1980, July). Econometric models for probabilistic choice among products. Journal of Business 53(3), S13–S29. McShane, E. (1963). Integrals devised for special purposes. Bull. Amer. Math. Soc. 69, 597–627. Moy, S.T. C. (1954). Characterizations of conditional expectation as a transformation on function spaces. Pacific Journal of Mathematics 4, 47–63. Munkres, J. (2000). General Topology. New Jersey. PrenticeHall, Inc. Musial, K., W. Strauss, and N. D. Macheras (2009). Liftings for topological products of measures. http://www.math.uni.wroc.pl/ musial/mms22.pdf. mimeo, Wroclaw Univ., Poland. Musial, K. Strauss,W. and N. D. Macheras (2002). Handbook of Measure Theory, Chapter 28. Liftings, pp. 1131–1184. Elsevier NorthHolland. Narens, L. (2007). Theories of Probability: An Examination of Logical and Qualitatiive Foundations, Volume 2 of Advanced Series on Mathematical Psychology. Singapore: World Scientific. Prelec, D. (1998). The probability weighting function. Econometrica 60, 497–528. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behaviour and Organization 3, 323–343. Reed, M. and B. Simon (1980). Modern Methods of Mathematical Physics (Revised and Enlarged Edidion ed.), Volume I:Functional Analysis. San Diego, CA: Academic Press, Inc. Richards, I. (1959). A note on the Daniell integral. Rendicotti del Semenario Mathematico della Univ. di Padova 29, 401–410. Rota, G.C. (1960). On the representation of averaging operators. Rendicotti del Semenario Mathematico della Univ. di Padova 30, 52–64. Royden, H. L. (1988). Real Analysis (3rd ed.). New York: Macmillan Publishing Co. Rudin, W. (1973). Functional Analysis. New York, N.Y.: McGrawHill, Inc. Saxe, K. (2002). Beginning Functional Alalysis. SpringerVerlag, New York, Inc. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica 57(3), 571–587. Shilov, G. (1973). Elementary Real and Complex Analysis. MA:MIT Press. Dover Publications, Inc. reprint 1996. Stott, H. P. (2003). Cumulative prospect theory’s functional menagerie. mimeo, Dept. Psychology, Univ. Warwick, UK, Journal of Risk and Uncertainty, 32, 101130 (2006). Takaheshi, T. (2006). A mathematical framework for probabilistic choice based on information theory and psychophysics. Med. Hypotheses 67(1), 183–186. Thomas, E. G. F. and A. Vol˘ci˘c (1989). Daniell integral represented by Radon measures. Rendicotti del Circolo Matematico di Palermo 38, 39–59. Train, K. (2003). Discrete Choice Methods With Simulations. New York: Cambridge University Press. Tversky, A. and D. Khaneman (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, 297–323. Wu, G. and R. Gonzalez (1996). Curvature of the probability weighting function. Management Science 42, 1676–90. Wu, G. and R. Gonzalez (1999). Nonlinear decision weights in choice under uncertainty. Management Science 45, 74–85. Wu, G., J. Zhang, and R. Gonzalez (2004). Handbook of Judgment and Decision Making, Chapter :Decision Under Risk, pp. 399–423. Oxford: Blackwell. Yosida, K. (1980). Functional Analysis (6th ed.). New York: Springerverlag. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/22380 