A Maximum Theorem for Incomplete Preferences

TL;DR

This paper extends Berge's Maximum Theorem to incomplete preferences, introducing a new continuity property for domains of comparability that ensures the preservation of maximal elements in limit problems.

Contribution

It provides a generalized maximum theorem accommodating incomplete preferences with a novel continuity condition for domains of comparability.

Findings

A simple version of the maximum theorem for convex feasible sets and fixed preferences.

Conditions under which limits of maximal elements are maximal in the limit problem.

The new continuity property is sufficient and, under certain conditions, necessary for optimality preservation.

Abstract

We extend Berge's Maximum Theorem to allow for incomplete preferences. We first provide a simple version of the Maximum Theorem for convex feasible sets and a fixed preference. Then, we show that if, in addition to the traditional continuity assumptions, a new continuity property for the domains of comparability holds, the limits of maximal elements along a sequence of decision problems are maximal elements in the limit problem. While this new continuity property for the domains of comparability is not generally necessary for optimality to be preserved by limits, we provide conditions under which it is necessary and sufficient.