Symmetry groups for social preference functions

TL;DR

This paper explores the possible symmetry groups of social preference functions, providing a complete characterization for neutrality groups and conditions for anonymity groups, with implications for Boolean function representability.

Contribution

It introduces the concepts of anonymity, neutrality, and symmetry groups for social preference functions and characterizes which permutation groups can serve as these groups.

Findings

Complete description of neutrality groups.

Sufficient conditions for anonymity groups.

Connections to Boolean function representability.

Abstract

We introduce the anonymity group, the neutrality group and the symmetry group of a social preference function. Inspired by a problem posed by Kelly in 1991 and remained unsolved, we investigate the problem of recognizing which permutation groups may arise as anonymity, neutrality and symmetry group of a social preference function. A complete description is found for the neutrality groups and a sufficient condition, which largely encompasses the problem, is found for the anonymity groups. Using the concept of orbit extension of a group $U$, we formulate manageable necessary conditions for being $U$ an anonymity or a symmetry group. Our research deeply interacts with problems of representability by Boolean functions shedding light on them.