Attitude toward imprecise information

This paper presents an axiomatic model of decision making which incorporates objective but imprecise information. We axiomatize a decision criterion of the multiple priors (or maxmin expected utility) type. The model achieves two primary objectives. First, it explains how subjective belief varies with information. Second, it identifies an explicit attitude toward imprecision that underlies usual hedging axioms. Information is assumed to take the form of a probability-possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P. She is assumed to rank pairs of the form (P,f) where P is a probability-possibility set and f is an act mapping states into outcomes. The representation result delivers multiple-priors utility at each probability-possibility set. There is a mapping that gives for each probability-possibility set the subjective set of priors. This allows both subjective expected utility when the subjective set of priors is reduced to a singleton and the other extreme where the decision maker takes the worst case scenario in the entire probability-possibility set. We show that the relation “more averse to imprecision” is characterized by inclusion of the sets of priors, irrespective of the utility functions that capture risk attitude. We characterize, under extra axioms, a more precise functional form, in which the subjective set of priors is obtained by (i) solving for the “mean value” of the probability-possibility set and (ii) shrinking the probability-possibility set toward the mean value to a degree determined by preference.