Cadogan, Godfrey (2010): Commutative Prospect Theory and Stopped Behavioral Processes for Fair Gambles.
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We augment Tversky and Khaneman (1992) (“TK92”) Cumulative Prospect Theory (“CPT”) function space with a sample space for “states of nature”, and depict a commutative map of behavior on the augmented space. In particular, we use a homotopy lifting property to mimic behavioral stochastic processes arising from deformation of stochastic choice into outcome. A psychological distance metric (in the class of Dudley-Talagrand inequalities) popularized by Norman (1968); Nosofsky and Palmeri (1997), for stochastic learning, was used to characterize stopping times for behavioral processes. In which case, for a class of nonseparable space-time probability density functions, based on psychological distance, and independently proposed by Baucells and Heukamp (2009), we find that behavioral processes are uniformly stopped before the goal of fair gamble is attained. Further, we find that when faced with a fair gamble, agents exhibit submartingale [supermartingale] behavior, subjectively, under CPT probability weighting scheme. We show that even when agents’ have classic von Neuman-Morgenstern preferences over probability distribution, and know that the gamble is a martingale, they exhibit probability weighting to compensate for probability leakage arising from the their stopped behavioral process.
|Item Type:||MPRA Paper|
|Original Title:||Commutative Prospect Theory and Stopped Behavioral Processes for Fair Gambles|
|Keywords:||commutative prospect theory; homotopy; stopping time; behavioral stochastic process|
|Subjects:||D - Microeconomics > D0 - General > D03 - Behavioral Economics; Underlying Principles
D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty
D - Microeconomics > D7 - Analysis of Collective Decision-Making > D70 - General
C - Mathematical and Quantitative Methods > C0 - General
C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods
|Depositing User:||godfrey cadogan|
|Date Deposited:||28. Apr 2010 00:11|
|Last Modified:||22. Feb 2013 23:52|
Allgower, E. and K. Georg (1994, August). Numerical path following. mimeo. Dep’t. Math., Colorado State U.
Baucells, M. and F. H. Heukamp (2009, June). Probability and time trade-off. http://web.iese.edu/mbaucells/downloads/PTT˙06˙22.pdf. Working Paper, IESE Business School, Spain.
Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis (2nd ed.). Springer Series in Statistics. New York, N.Y.: Springer-Verlag.
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.
Dawes, R. M. (1979, July). The robust beauty of improper linear models. American Psychologist, 571–582.
Debreu, G. (1958). Stochastic choice and cardinal utility. Econometrica 26, 440–444.
DeGroot, M. (1970). Optimal Statistical Decisions. New York, N.Y.: McGraw-Hill, Inc.
Dellacherie, C. and P. Meyer (1982). Probabilities and Potential B:Theory of Martingales. Number 72 in North-Holland Mathematical Studies. Amsterdam: North-Holland Publishing, Co.
Doob, J. L. (1953). Stochastic Processes. New York, N. Y.: John Wiley & Sons.
Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian process. Journal Functional Analysis 1, 290–330.
Gray, B. (1975). Homotopy Theory: An Introduction to Algebraic Topology. New York: Academic Press.
Grimmett, G. R. and D. R. Stirzaker (2001). Probability and Random Processes(3rd ed.). Oxford Univ. Press.
Guggenheimer, H. W. (1977). Differential Geometry. Mineola, New York: Dover Publications, Inc.
Ingersoll, J. E. (2008, June). Non-monotonicity of the Khaneman-Tversky probabiility weighting function: A cautionary note. European Financial Management 14(3), 385–390.
Karatzas, I. and S. E. Shreve (1991). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. New York, N.Y.: Springer-Verlag.
Karlin, S. and H. M. Taylor (1975). A First Course in Stochastic Processes (2nd ed.). Academic Press.
Lefshetz, S. (1942). Algebraic Topology, Volume 27 of Colloquium Publications. Providence, R.I.: Amer. Math. Soc.
Lindquist, M. A. and I.W. McKeague (2009). Logistic regression with Brownianlike predictors. Journal of the American Statistical Association 104(488), 1575– 1585.
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. (2001). Reduction invariance and Prelec’s weighting functions. Journal of Mathematical Psychology 45, 167–179.
Massa, M. and A. Simonov (2005). Is learing a dimension of risk? Journal of Banking and Finance 29, 2605–2632.
Massart, P. (1998, November). About the constant in Talagrand’s concentration inequalities for empirical processes. mimeo. Dep’t. Math., Univ. Paris-Sud.
McFadden, D. P. (1974). Frontiers in Econometrics, Chapter IV. Conditional Logit Analysis of Qualitative Choice Behavior, pp. 105–142. New York: Academic Press.
Norman, M. F. (1968). Some convergence theorems for stochastic learning models with distance diminishing operators. Journal of Mathematical Psychology 5, 61–101.
Nosofsky, R. M. (1997). An exemplar based random walk model of speeded categorization and absolute judgment. In A. A. J. Marley (Ed.), Choices, Decisions, and Measurement, pp. 347–365. New Jersey: Lawrence Erlbaum Associates.
Nosofsky, R. M. and T. J. Palmeri (1997). An exemplar based random walk model of speeded classification. Psychological Review 104(2), 266–300.
Prelec, D. (1998). The probability weighting function. Econometrica 60, 497–528.
Shao, J. (2007). Mathematical Statistics (2nd ed.). Springer Texts in Statistics. New York, N.Y.: Springer-Verlag.
Steinbacher, M. (2009). Stochastic processes in finance and behavioral finance. http://mpra.ub.uni-muenchen.de/13603/.
Talagrand, M. (2005). The Generic Chaining: Upper and Lower Bounds for Empirical Processes. New York,N.Y.: Springer-Verlag.
Tversky, A. and D. Khaneman (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, 297–323.
Wickens, T. (1982). Models for Behavior: Stochastic Processes in Psychology. San Francisco, CA: W. H. Freeman & Sons, Inc.
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