The only admissible way of merging arbitrary e-values

Abstract

In this paper it is proved that the only admissible way of merging arbitrary e -values is to use a weighted arithmetic average. This result completes the picture of merging methods for arbitrary e -values and generalizes the result of Vovk & Wang (2021) that the only admissible way of symmetrically merging e -values is to use the arithmetic average combined with a constant. Although the proved statement is naturally anticipated, its proof relies on a sophisticated application of optimal transport duality and a minimax theorem.