Beyond Ellsberg's paradox. Modeling the effets of vagueness in risky decisions

Ellsberg?s famous paradox [1961] focused attention on the importance of the precision of the probabilities underlying risky choice. Following his seminal work numerous studies have demonstrated that people are generally averse to imprecisely specified (vague) probabilities. In many important risky decisions, information about both probabilities and outcomes is imprecise, and previous research has shown vagueness effects for outcomes to be similar to those for probabilities. The current work investigates the joint effects of vagueness on the two attributes. For our purposes, gambles are considered equally vague if the ranges of their possible expected values are identical, and gambles varying in precision are considered equated if their mean expected values are equal. We summarize results from three previous studies where subjects provided Certainty Equivalents (CEs) for, or made pair-wise choices between, equated options that varied in terms of their probabilities, their outcomes, and the precision with which each dimension was specified. We propose a new model for decision making with vaguely specified attributes that generalizes the Prospect Theory model. The model posits a new "vagueness resolution" operation in the editing phase. Consistent with Ellsberg?s formulation, this operation represents a range of values (probabilities or outcomes) by a weighted average of its end points. The results of this operation are used in the evaluation phase to reach decisions (e.g., choices, CEs) regarding vague prospects. The weights attached to the two focal (end) points reflect their relative salience, and can be used to classify preference patterns into vagueness neutral, vagueness avoiding and vagueness seeking behavior. The new model was fit successfully to results from the three studies reviewed. The estimated parameters are consistent with, and confirm, the patterns uncovered by the qualitative analysis. We conclude with a general discussion of the properties of the new model.