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Representation Theory for Risk On Markowitz-Tversky-Kahneman Topology

G. Charles-Cadogan

posted on 11 June 2012

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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.

Discussion

This paper employs elementary differential geometry and Lie group theory to examine risk profiles in a neighbourhood topology for critical points in decision making under risk and uncertainty identified in seminal papers by Markowitz (1952), and Tversky and Kahneman (1979, 1992). It proves that utility is harmonic in a neighbourhood base around critical points so it is the solution to a Laplace equation. It provides some conjectures on how potential theory, i.e. Dirichlet problem, in tandem with skew Brownian motion could be used to estimate Tversky and Kahneman value functions.