Group Representations for Decision Making under Risk and Uncertainty

Abstract

This paper extends decision making under risk and uncertainty to group theory via representations of invariant behavioural space for prospect theory. First, we predict that canonical specifications for value functions, probability weighting functions, and stochastic choice maps are homomorphic. Second, we derive a continuous singular matrix operator T for affine transformation of a vector space of skewed S-shape value functions V isomorphic to a vector spaceW of inverted S-shaped probability weighting functions. To characterize the transformation, we decompose the operator into shear, scale and translation components. In that milieu, Moore-Penrose psuedoinverse transformation recovers value functions from probability weighting functionals. Removal of 0 from the point spectrum induces nonsingular operators that support group representation of stochastic choice maps in an invariant subspace of the general linear group GL(V). Third, we demonstrate how group theoretic operations on a gamble provide mathematical foundations of probability weighting functions that subsume the Prelec class. Fourth, we predict that a gamble is isomorphic to an invariant cyclic sub-group in weighted probablity space. This result implies that probability weighting functions [and value functions] fluctuate near their extremes, and explain violation of transitivity axioms in decision theory. Moreover, representations include the special unitary group SU(2) and orthogonal group Θ*3. The former includes Pauli’s spin matrices and accounts for skewness. It also provides microfoundations for construction of (1) behavioural stochastic processes from group character in the frequency domain; and (2) Schrodinger-Pauli Hamiltonian to compute, inter alia, time dependent probabilities in decision field theory.