Objective rationality and uncertainty averse preferences

As in Gilboa, Maccheroni, Marinacci, and Schmeidler (2010), we consider a decision maker characterized by two binary relations: 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, Maccheroni, Marinacci, and Montrucchio (2011). It models the rankings that the decision maker expresses if he has to make a choice. We assume that the first binary relation is a completion of the second one. We identify axioms under which the set of probabilities and the utility index representing the first are the same as those representing the second one. In this way, we show that Bewley preferences and uncertainty averse preferences, two different approaches in modelling decision making under Knightian uncertainty, are complementary. As a by-product, we extend the main result of Gilboa, Maccheroni, Marinacci, and Schmeidler (2011), who restrict their attention to maxmin expected utility completions.