Imprecise Information and Second-Order Beliefs

A decision problem under uncertainty is often given with a piece of objective but imprecise information about the states of the world such as in the Ellsberg urn. By incorporating such information into the smooth ambiguity model of Seo (2009), we characterize a class of smooth ambiguity representations whose second-order beliefs are consistent with the objective information. As a corollary, we provide an axiomatization for the second-order expected utility, which has been studied by Nau (2001), Neilson (2009), Grant, Polak, and Strzalecki (2009), Strzalecki (2011), and Ghirardato and Pennesi (2019). In our model, attitude toward uncertainty can be disentangled from a perception about uncertainty and connected with attitude toward reduction of compound lotteries.