The (im)possibility of collective risk measurement: Arrovian aggregation of variational preferences

Abstract

This paper studies collective decision making when individual preferences can be represented by convex risk measures. It addresses the question whether there exist non-dictatorial aggregation functions of convex risk measures satisfying the following Arrow-type rationality axioms: a weak form of universality, systematicity (a strong variant of independence), and the Pareto principle. Herein, convex risk measures are identified with variational preferences on account of the For finite electorates, we prove a variational analogue of Arrow's impossibility theorem. For infinite electorates, the possibility of rational aggregation depends on an equicontinuity condition for the variational preference profiles; we prove a variational analogue of Fishburn's possibility theorem and point to potential analogues of Campbell's impossibility theorem. The proof methodology is based on a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke (J Math Econ 24(3): 1995). Diverse applications of these results are conceivable, in particular (a) the constitutional design of panels of risk managers and (b) microfoundations for (macroeconomic) multiplier preferences.

Aggregation of Monotonic Bernoullian Archimedean Preferences: Arrovian Impossibility Results

“…Herzberg [15]). However, due to the simple and elegant axiomatisation of MBA preferences, the proof of the impossibility result in this paper is much more appealing.…”

Aggregating infinitely many probability measures

“…the set of all expected-utility preferences for some set of states of the world or the set of all multiple-prior preferences) into a single variational preference ordering (e.g. an expected-utility preference ordering on that set of states of the world), there will be no aggregation rule satisfying the analogues of Arrow's responsiveness axioms, as was shown in Herzberg [23,22]. Note that these impossibility statements can be established both for the case of profiles of a given finite length (the analogue of Arrow's [2] theorem) and for the case of profiles of any given infinite length (the analogue of Campbell's [5] theorem), using a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke [35] and systematically elaborated by Herzberg and Eckert [25,26].…”