Objective rationality and uncertainty averse preferences

As in Gilboa et al. (2010), we consider a decision maker characterized by two binary relations: greater than or similar to* and greater than or similar to<^>. The first binary relation is a Bewley preference. It models the rankings for which the decision maker is sure. The second binary relation is an uncertainty averse preference, as defined by Cerreia-Vioglio et al. (2011c). It models the rankings that the decision maker expresses if he has to make a choice. We assume that greater than or similar to<^> is a completion of greater than or similar to*. We identify axioms under which the set of probabilities and the utility index representing greater than or similar to* are the same as those representing greater than or similar to<^>. In this way, we show that Bewley preferences and uncertainty averse preferences, two different approaches to modelling decision making under Knightian uncertainty, are complementary. As a by-product, we extend the main result of Gilboa et al. (2010), who restrict their attention to maxmin expected utility completions.