An axiomatic approach to cultivar ranking in multi-environment trials

Multi-environment trials are central to cultivar evaluation because they reveal how candidate cultivars perform across locations, years, management conditions, and stress environments. The resulting yield matrix is a rich source of data on genotype-by-environment interaction, and a wide literature on estimation, decomposition, visualisation, and prediction of yield potential and stability has flourished. However the ultimate question of which cultivar to recommend on the basis of such a matrix is often left implicit. The question is far from trivial, and in this paper we formulate cultivar recommendation as an axiomatic ranking problem. This framework is rich enough to encompass the existing literature on stability indices, as well as any other deterministic ranking procedure. We show that many commonly used stability-based procedures can violate minimal criteria of efficiency or consistency. The result of such violations is that a cultivar with uniformly high yield could be ranked below a cultivar with uniformly low yield, or the relative ranks of two cultivars could depend on whether or not a third cultivar is present in the matrix. Our results prove that under a small number of such criteria the space of admissible rules collapses to the family of power means and their limiting cases. If we further wish to allow multiplication normalisation of yield, we are left with the geometric mean as the unique solution.