Which way up? Consistency, anti-consistency and inconsistency of social welfare and inequality partial orderings for ordinal data

In the context of distributional comparisons, we introduce consistency and anti-consistency as two properties for partial orderings de ned on ordinal variables. A consistent social welfare or inequality partial ordering regards distribution p more desirable than q if and only if it ranks the corresponding reverse-ordered distribution Rp more desirable than Rq. An anti-consistent partial ordering regards p more desirable than q if and only if the reverse- ordered distribution Rp is less desirable than Rq. First, for a broad class of social welfare and inequality partial orders, which we call linear, we characterise those relations which are robust to any given type of permutation or reversal of the categories. Deploying these results as a speci c consistency test for some prominent examples in the literature, we demonstrate the consistency of the Hammond inequality order and establish the anti-consistency of rst-order dominance, and the inconsistency of two forms of the Hammond welfare partial ordering. Then deploying consistency tests based on dominance implementation criteria, we show that, among relations not falling in the linear class, the median-preserving spreads and the bipolarisation partial orderings are both consistent, whereas the status Lorenz ordering is inconsistent.