When a combination of convexity and continuity forces monotonicity of preferences

We consider arbitrary subsets L of random variables defined on an arbitrary non-additive probability space (Omega, F, v). A topology tau on L satisfies Condition BU if every open set in this topology which contains X is an element of L as a member also contains as a subset some (c, epsilon)-ball around X, defined as B-c,B- epsilon (X) = {Y is an element of L|v (|X - Y| >= c) < epsilon}. Condition BU is satisfied by any topology of convergence in non-additive measure v [21,18] but also by all coarser topologies. Next we consider preference relations that are continuous with respect to any topology tau satisfying Condition BU. For non-atomic v we prove that any convex and tau-continuous preference relation over the random variables in a given local cone L must satisfy monotonicity in the local cone order. This monotonicity result comes with surprisingly strong decision theoretic implications: (i) The only tau-continuous and convex preference relation defined over all random variables is the indifference relation; (ii) Any tau-continuous and convex preference relation defined over all positive random variables must satisfy payoff-monotonicity; (iii) Any convex and payoff-monotone preference relation defined over all loss random variables must violate tau-continuity. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study examines arbitrary subsets of random variables defined on non-additive probability spaces and preference relations that are continuous with respect to specific topologies. For non-atomic v, it is proven that convex preference relations over random variables in a given local cone must satisfy monotonicity, leading to strong implications in decision theory.