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Representation Theory for Risk On Markowitz-Tversky-Kahneman Topology

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  • Godfrey Charles-Cadogan

Abstract

We introduce a representation theory for risk operations on locally compact groups in a partition of unity on a topological manifold for Markowitz-Tversky-Kahneman (MTK) reference points. We identify (1) risk torsion induced by the flip rate for risk averse and risk seeking behaviour, and (2) a structure constant or coupling of that torsion in the paracompact manifold. The risk torsion operator extends by continuity to prudence and maxmin expected utility (MEU) operators, as well as other behavioural operators introduced by the Italian school. In our erstwhile chaotic dynamical system, induced by behavioural rotations of probability domains, the loss aversion index is an unobserved gauge transformation; and reference points are hyperbolic on the utility hypersurface characterized by the special unitary group SU(n). We identify conditions for existence of harmonic utility functions on paracompact MTK manifolds induced by transformation groups. And we use those mathematical objects to estimate: (1) loss aversion index from infinitesimal tangent vectors; and (2) value function from a classic Dirichlet problem for first exit time of Brownian motion from regular points on the boundary of MTK base topology.

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File URL: http://arxiv.org/pdf/1206.2665
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Paper provided by arXiv.org in its series Papers with number 1206.2665.

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Date of creation: Jun 2012
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Handle: RePEc:arx:papers:1206.2665

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  2. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, Econometric Society, vol. 74(6), pages 1447-1498, November.
  3. Ariane Lambert Mogiliansky & Shmuel Zamir & Herve Zwirn, 2003. "Type Indeterminacy: A Model of the KT(Kahneman-Tversky)-man," Discussion Paper Series, The Center for the Study of Rationality, Hebrew University, Jerusalem dp343, The Center for the Study of Rationality, Hebrew University, Jerusalem.
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  8. Mohammed Abdellaoui & Han Bleichrodt & Corina Paraschiv, 2007. "Loss Aversion Under Prospect Theory: A Parameter-Free Measurement," Management Science, INFORMS, INFORMS, vol. 53(10), pages 1659-1674, October.
  9. Tversky, Amos & Kahneman, Daniel, 1992. " Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, Springer, vol. 5(4), pages 297-323, October.
  10. Bernard, Carole & Ghossoub, Mario, 2009. "Static Portfolio Choice under Cumulative Prospect Theory," MPRA Paper 15446, University Library of Munich, Germany.
  11. Drazen Prelec, 1998. "The Probability Weighting Function," Econometrica, Econometric Society, Econometric Society, vol. 66(3), pages 497-528, May.
  12. Milton Friedman & L. J. Savage, 1948. "The Utility Analysis of Choices Involving Risk," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 56, pages 279.
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