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 Maccheroni et al. (Econometrica 74(6):1447–1498, 2006) axiomatisation of variational preference relations and Föllmer’s and Schied’s (Finance Stoch 6(4):429–447, 2002; Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematics, vol. 27, 2nd edn. de Gruyter, Berlin, 2004) representation theorem for concave monetary utility functionals. The cases of both finite and infinite electorates are considered. 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):217–237, 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.