Hoping for the best while preparing for the worst in the face of uncertainty

Abstract

We propose and axiomatize a new model of incomplete preferences under uncertainty, hope-and-prepare preferences: one act is preferred to another if and only if both its optimistic evaluation -- the welfare level attained in the best-case scenario-- and its pessimistic evaluation --the welfare level attained in the worst-case scenario-- rank it above the other. Both evaluations are computed over sets of probability distributions. We make the case that, compared to existing incomplete criteria under ambiguity, hope-and-prepare preferences address the trade-off between conviction and decisiveness in a new way, which is more favorable to decisiveness. We characterize a completion of an incomplete hope-and-prepare preference relation admitting an (asymmetric) $\alpha$-\textit{maxmin expected utility} representation, in which $\alpha$ is unique. Conversely, given a standard $\alpha$-MEU preference and a fixed value of $\alpha$, we recover the concordant hope-and-prepare preference that it completes.