The Source of Uncertainty in Probabilistic Preferences Over Gambles

Abstract

Probabilistic preference models predict that a subject makes different choices with different probabilities in repeatedly experiments with the same stimuli. This paper explains why. First, we prove that a gamble is a statistical ensemble or sample function of a random field with canonical Gibbs-Luce measure. And we employ Good-Jaynes maximum entropy principle to characterize the underlying probability function space. Second, we find that maximum entropy for unconstrained unobserved probability distributions predicts that subjects have Von Neuman Morgenstern utility (EUT). Therefore, probability weighting is inapplicable to ambiguity aversion. Third, when unobserved probability distributions are constrained by partially revealed information, i.e., finite moments, maximum entropy predicts that for the same stimuli the source of differential choice in probabilistic preference is a behavioural quantum wave function embedded in harmonic probability weighting functionals (HPWF), from which probability amplitudes (popularized by quantum decision [field] theory) are computed. This explains the “quantal effect” or why pwfs are not well behaved near their end points. Fourth, for application, we show how a simple affine transformation produces a harmonic inverted S-shaped probability weighting functional consistent with the principle of bounded subadditivity, and likelihood insensitivity reported in recent source function theory of uncertainty. To wit, probabilistic preferences are different for the same stimuli because decision weights depend on broken cycles for HPWF. However, our model also reveals a fixed point probability puzzle. Vizly, for -w(p)log(w(p))=p, we get p=1/e when w(p)=p. So the fixed point has maximum entropy even though it is invariant. We prove that the puzzle may be explained by the existence of a cluster set of fixed points for probability weighting functionals, that include Prelec’s generalized pwf’s, which are not constrained to 1/e. The cluster set represents an invariant linear subspace of probability weighting schemes that support EUT while the complementary subspace supports nonexpected utility theories.