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The Source of Uncertainty in Probabilistic Preferences Over Gambles

Godfrey Cadogan

posted on 15 December 2011

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Probabilistic preference models predict that a subject makes different choices with different probabilities when repeatedly faced with the same or similar situation(s). However, they do not explain why choice is probabilistic. This paper provides an explanation. First, we prove that a gamble is a statistical ensemble or sample function of a random field with canonical Luce-Gibbs measure. And we employ entropy measures of uncertainty to characterize the underlying function space. Second, we find that under the Luce-Gibbs measure, maximum entropy for unconstrained unobserved probability distributions predicts that subjects have Von Neuman Morgenstern utility. Therefore, probability weighting is inapplicable to ambiguity aversion. Third, when unobserved probability distributions are constrained by finite moments, maximum entropy predicts that the source of probabilistic preference is a behavioural quantum wave function embedded in probability weighting functionals (MaxEnt-PWF), from which probability amplitudes are computed. The [standing] waves have fixed point probability 1/e, and they subsume the Prelec class of pwfs. Fourth, for application, we show how a simple affine transformation of the MaxEnt-PWF produces a sinusoidal inverted S-shaped probability weighting functional consistent with likelihood insensitivity reported in recent source function theory of uncertainty. However, our model reveals a tautologous fixed point probability puzzle--vizly, the fixed point has maximum entropy even though it is invariant. In fact, we prove that there exist a cluster set of fixed points for MaxEnt probability weighting functionals that include Prelec's generalized pwf's which are not constrained to 1/e.