Deepest voting on rankings

TL;DR

This paper introduces a unified framework for ranking-based voting rules using depth functions on permutations, connecting mathematical properties with voting behaviors to enhance understanding of deep voting procedures.

Contribution

It develops a comprehensive framework for deepest voting rules on permutations using depth functions and links their mathematical properties to voting rule behaviors.

Findings

Classical voting rules expressed as deepest voting procedures

Connections established between depth function properties and voting behaviors

Framework applicable to various distances on permutations

Abstract

This article aims to present a unified framework for ranking-based voting rules based on the use of depth functions on permutations, as a counterpart of deepest voting rules on evaluation introduced in Aubin et al. [2022]. It introduces the notion of depth functions, in continuous sets and in permutation sets, the later using the notion of Fr{\'e}chet means. Deepest voting procedures are then formally defined, and some classical voting rules are expressed as deepest voting procedures, using a large variety of distances on the set of permutations. Links are done between the depth functions mathematical properties and some behaviours of the voting rule, such as Neutrality, Anonymity, Universality, Condorcet winner/loser property and so on.