Metrizable preferences over preferences

A hyper-preference is a weak order over all linear orders defined over a finite set A of alternatives. An extension rule associates with each linear order p over A a hyper-preference. The well-known Kemeny extension rule ranks all linear orders over A according to their Kemeny distance to p. More generally, an extension rule is metrizable iff it extends p to a hyper-preference consistent with a distance criterion. We characterize the class of metrizable extension rules by means of two properties, namely self-consistency and acyclicity across orders. Moreover, we provide a characterization of neutral and metrizable extension rules, based on a simpler formulation of acyclicity across orders. Furthermore, we establish the logical incompatibility between neutrality, metrizability and strictness. However, we show that these three conditions are pairwise logically compatible.