Closure Properties of Independence Concepts for Continuous Utilities

This paper examines nine independence concepts for ordinal and expected utilities: utility and preference independence, weak separability (a uniqueness of nonstrict conditional preferences), as well as generalizations that allow for complete indifference or reversal of preferences. Some of these conditions are closed under set-theoretic operations, which simplifies the verification of such assumptions in a practical decision analysis. Preference independence is closed under union without assumptions of strict essentiality, which are however necessary for closure under differences of preference independence and its mentioned generalizations. The well-known additive representation of a utility function is used, in a corrected form with a self-contained proof, to show closure under symmetric difference. This generalizes a classical result of Gorman (1968), and supplements its proof. Generalized utility independence has all, whereas weak separability has no closure properties. An independent set of any kind is utility or preference independent if this holds for a subset. Counterexamples are given throughout to show that the results are as strong as possible. The approach consistently uses conditional functions instead of preference relations, based on simple topological notions like connectivity and continuity.