A maximum theorem for incomplete preferences

We extend Berge's Maximum Theorem to allow for incomplete preferences. We provide a Maximum Theorem for a fixed preference that can be represented with a finite multi-utility consisting of continuous and strictly quasiconcave functions. We apply this result to study the continuity properties of the set of Walrasian equilibria in exchange economies in which agents have incomplete preferences and the set of Pareto efficient outcomes in strategic games with varying strategy spaces. We also provide a generalization that relaxes the multi-utility assumption and a more abstract theorem that allows for changing preferences. The latter result is based on a new continuity condition on the domains of comparability of a preference that clarifies why incompleteness often leads to failures of the maximum theorem.(c) 2023 Elsevier B.V. All rights reserved.