Monometrics and their role in the rationalisation of ranking rules

• A new tool for the aggregation of rankings is introduced: monometrics.
• A procedure for hierarchically combining monometrics is proposed.
• The characterization of ranking rules in terms of a consensus state and a monometric is addressed.
• The search for the closest profile of rankings in a consensus state is presented as an optimization problem.

The aggregation of rankings is a long-standing problem that consists of, given a profile of rankings, obtaining the single ranking that best represents the nature of this given profile. Under the name of metric rationalisation of ranking rules, it has been proven that most ranking rules can be characterized as minimizing the distance to a consensus state for some appropriate distance function. In this paper, we propose to consider monometrics instead of distance functions. Although these concepts are closely related, monometrics better capture the nature of the problem, as the purpose of a monometric is to preserve a given betweenness relation. This is obviously only meaningful when an interesting betweenness relation is fixed, for instance, the one based on reversals in rankings proposed by Kemeny. In this way, ranking rules can be characterized in terms of a consensus state and a monometric.