Updating variational (Bewley) preferences

This paper investigates dynamically consistent updates of weighted unanimity rules given by variational Bewley preferences (VBP), a class of incomplete preferences characterized by pairs of utility index over consequences and ambiguity index over priors. We show that these dynamically consistent updates are also VBP that preserves the same ranking over consequences and feature conditional ambiguity index obtained through the full Bayesian update of the unconditional one. Moreover, we study conditional forced-choice relations, which capture choices that a decision-maker must take after learning some event. Formally, they are monotone continuous weak orders. We show that any conditional forced-choice relation that preserves the ambiguity attitude of a given unconditional VBP has a variational representation with the same affine utility function over consequences and the ambiguity index of its corresponding dynamically consistent update. Thus, our result provides a novel foundation for full Bayesian updating of variational preferences.