Comparing Multidimensional Ordinal Distributions under Partial Orderings

This paper studies the comparison of multidimensional ordinal distributions when outcome profiles are only partially ordered. Such settings arise naturally in inequality analysis whenever well-being is described by several ordinal attributes and some profile comparisons cannot be resolved without introducing additional value judgments about the relative importance of dimensions. To address this problem, the paper introduces a rank-membership representation based on the admissible linear extensions of the underlying partial order. Each profile is thereby associated with a probability distribution over ranks, which in turn yields completion-averaged rank frequencies and multidimensional Hammond coordinates. The paper shows that these coordinates coincide with the uniform average of the corresponding one-dimensional Hammond coordinates across admissible completions. We define a coordinate-wise multidimensional Hammond dominance criterion and establish its relation to unanimous completion-wise Hammond dominance. We also introduce a graded refinement based on the proportion of coordinates at which one distribution weakly dominates another and derives its basic transitivity structure. The proposed framework provides a way to extend rank-sensitive inequality comparisons to multidimensional ordinal settings with incomplete comparability, while making explicit the informational trade-off implied by averaging across admissible completions.