Incomplete Variational Preferences

We examine incomplete preference structures in a framework that allows for various relaxations of the independence axiom. We derive preference representations in terms of willingness-to-pay measures, and demonstrate how these representation can be used to determine preference incompleteness and to elicit preferences empirically. Decision makers faced with an array of choices that are poorly understood or diffi cult to compare may experience “...sensations of indecision or vacillation, which we may be reluctant to identify with indifference”(Savage 1954). In such settings, a number of authors (Aumann 1962, Kannai 1963, Fishburn 1965, Bewley 1986 and 2002, Dubra, Maccheroni, and Ok 2001, Mandler 2004, Baucells and Shapley 2008, and Galabaatar and Karni 2013) have argued that it may be inappropriate to impose the completeness axiom on rational choice behavior. Relaxing the completeness axiom while maintaining the other standard axioms, including transitivity and ‘irrelevance of independent alternatives’, yields a decision criterion that requires one alternative to dominate another for a set of probability measures before it can be regarded as preferred (Aumann 1962, Bewley 1986 and 2002). Even for complete preferences, it is widely understood that the independence axiom can be unduly restrictive. A large number of generalizations of expected utility theory have been proposed, and nearly all weaken the independence axiom (for example, Dekel 1986, Schmeidler 1989, Gilboa and Schmeidler 1989, Gul and Lantto 1990, Maccheroni, Marinacci, and Rustichini 2006). Further diffi culties arise when preferences are incomplete. Standard arguments for independence, such as Savage’s ‘sure-thing principle’require that decision makers rank, not only the choices open to them, but all possible choices. In particular, the idea of ‘compound independence’inherent in the sure-thing principle requires that decision makers can evaluate a given act by considering arbitrary compound lotteries yielding the same probability distribution over outcomes. By contrast, the appeal of the transitivity axiom is, at least for the case of individual decisions, enhanced in the case of incompleteness. Because no requirement then exists to rank alternatives that may be diffi cult to compare, there is less risk that combining two choices using transitivity will yield a ranking incompatible with actual preferences. Regardless of which axioms are maintained, very few attempts have been made to examine incomplete preference structures empirically. We are aware of only two such studies (Danan and Ziegelmeyer 2006 and Cettolin and Riedl 2016). We conjecture that one reason for this empirical paucity is the lack of empirically implementable tests possessing a solid theoretical foundation on which empirical analysis of incomplete preferences can be based.