Hoping for the best while preparing for the worst in the face of uncertainty: a new type of incomplete preferences

We propose and axiomatize a new model of incomplete preferences under uncertainty, which we call $\mathit{\text{hope-and-prepare}}\mathit{\text{preferences}}$. An act is considered more desirable than an other act when, and only when, both an optimistic evaluation, computed as the welfare level attained in a best-case scenario, and a pessimistic one, computed as the welfare level attained in a worst-case scenario, rank the former above the latter. Our comparison criterion involves multiple priors, as best and worst cases are determined among 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 also characterize a completion of an incomplete hope-and-prepare preference relation admitting an (asymmetric) $\alpha$-$\mathit{\text{maxmin}}\mathit{\text{expected}}\mathit{\text{utility}}$ representation, in which $\alpha$ is unique.