Objective rationality and uncertainty averse preferences

As in Gilboa, Maccheroni, Marinacci, and Schmeidler \cite{GMMS}, we consider a decision maker characterized by two binary relations: $\succsim^{\ast}$ and $\succsim^{{\small \wedge}}$. 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 \cite{CMMM}. It models the rankings that the decision maker expresses if he has to make a choice. We assume that $\succsim^{{\small \wedge}}$ is a completion of $\succsim^{\ast}% $.\ We identify axioms under which the set of probabilities and the utility index representing $\succsim^{\ast}$ are the same as those representing $\succsim^{{\small \wedge}}$. 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 \cite{GMMS}, who restrict their attention to maxmin expected utility completions.