We prove that the only admissible way of merging e-values is to use a weighted arithmetic average. This result completes the picture of merging methods for e-values, and generalizes the result of Vovk and Wang (2021, Annals of Statistics, 49(3), 1736--1754) 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.

翻译：我们证明了合并e值的唯一可容许方法是使用加权算术平均。这一结果完善了e值合并方法的理论框架，并推广了Vovk和Wang（2021，《统计年鉴》，49(3)，1736--1754）关于对称合并e值的唯一可容许方法是使用算术平均与常数组合的结论。尽管该结论在预期之中，但其证明依赖于最优传输对偶理论和极小极大定理的复杂应用。