The only admissible way of merging arbitrary <i>e</i>-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 &amp; 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.