The Shape of Incomplete Preferences

by Robert F. Nau (Fuqua School of Business Working Paper)

Abstract: The emergence of robustness as an important consideration in Bayesian statistical models has led to a renewed interest in normative models of incomplete preference orders represented by indeterminate (set-valued) probabilities and utilities. This paper presents a simple axiomatization of incomplete preferences and characterizes the shape of their representing sets of probabilities and utilities. Deletion of the completeness assumption from the axiom system of Anscombe and Aumann yields preferences represented by a convex set of state-dependent expected utilities, of which at least one must be a probability/utility pair. A strengthening of the state-independence axiom is needed to obtain representation purely in terms of probability/utility pairs.

Key words: Axioms of decision theory, Bayesian robustness, state-dependent utility, coherence, partial order, incompleteness, ambiguity, imprecision, subjective probability