Representations of votes facilitating monotonicity-based ranking rules: From votrix to votex

We propose a new point of view of the long-standing problem where several voters have expressed a (strict) linear order (or ranking) over a set of candidates. For a ranking a≻b≻ca≻b≻c to represent a group's opinion, it would be natural that the strength with which a≻ca≻c is supported should not be less than both the strength with which a≻ba≻b and the strength with which b≻cb≻c are supported. This intuitive property is called monotonicity and it has been recently addressed for the first time in the context of social choice. In this paper, two different representations of votes (the votrix and the votex) are considered. The former one is a formalization of the well-known reciprocal matrix of pairwise comparisons between candidates already introduced by Condorcet. The latter one is an extension of this reciprocal matrix considering hitherto unexploited information. These two representations lead to two monotonicity-based ranking rules.