An impossibility theorem for paired comparisons

TL;DR

This paper proves an impossibility theorem showing no ranking method can satisfy both independence of irrelevant matches and self-consistency simultaneously in paired comparison problems, except in specific cases like round-robin tournaments.

Contribution

It introduces an axiomatic framework for the universal ranking problem and establishes a fundamental impossibility result under broad conditions.

Findings

No ranking method satisfies both properties simultaneously in general.

The impossibility does not occur in round-robin tournaments.

A positive result is obtained using macrovertices to restrict independence.

Abstract

In several decision-making problems, alternatives should be ranked on the basis of paired comparisons between them. We present an axiomatic approach for the universal ranking problem with arbitrary preference intensities, incomplete and multiple comparisons. In particular, two basic properties -- independence of irrelevant matches and self-consistency -- are considered. It is revealed that there exists no ranking method satisfying both requirements at the same time. The impossibility result holds under various restrictions on the set of ranking problems, however, it does not emerge in the case of round-robin tournaments. An interesting and more general possibility result is obtained by restricting the domain of independence of irrelevant matches through the concept of macrovertex.