Aggregation of Opinions and Risk Measures

Abstract

A ranking over a set of alternatives is an aggregation of experts' opinions (AEO) if it depends on the experts' assessments only. We study both those rankings that result from pooling Bayesian experts and those that result from pooling possibly non-Bayesian experts. In the non-Bayesian case, we allow for the simultaneous presence of experts that may display very different attitudes toward uncertainty. We show that a unique axiom along with a mild regularity condition fully characterize those AEO rankings which are "generalized averages" of experts' opinions, in the sense that the average is obtained by using a capacity rather than a probability measure. We call these rankings non-linear pools. We consider a number of special cases such as linear pools (Stone (1961)), concave/convex pools (Cres, Gilboa, and Vieille (2011)), quantiles, and pools of equally reliable experts. We then apply our results to the theory of risk measures. Our application can be viewed as a generalization of the robust approach (Glasserman and Xu (2013)) to risk measurement in that it allows both for a more general notion of "model" and more general aggregation rules. We show that a wide class of risk measures can be regarded as non-linear pools. Not only does this class include all coherent risk measures (Artzner et al. (1999)), but also measures like the Value-at-Risk, which fail subadditivity. We also briefly discuss the possibility of extending our findings to include the convex risk measures of Follmer and Schied (2002) as well as their non-subadditive extensions.

Keywords: Aggregation of opinions, Non-linear opinion pools, Non-Bayesian experts, Choquet integral, OWA aggregation, Risk measures, Value-at-risk, Robust approach to risk measurement.

JEL Classification: D7, D8, G2.