Algebraic Voting Theory & Representations of $S_m \wr S_n$

TL;DR

This paper employs algebraic methods to analyze positional voting procedures in committee selection, revealing the structure of information loss and demonstrating voting paradoxes with different weighting schemes.

Contribution

It introduces an algebraic framework using module homomorphisms to decompose voter preferences and analyze voting procedures in a novel way.

Findings

Decomposition of preference spaces into simple modules

Identification of information loss in voting procedures

Existence of voting paradoxes with different weightings

Abstract

We consider the problem of selecting an $n$-member committee made up of one of $m$ candidates from each of $n$ distinct departments. Using an algebraic approach, we analyze positional voting procedures, including the Borda count, as $\mathbb{Q}S_m \wr S_n$-module homomorphisms. In particular, we decompose the spaces of voter preferences and election results into simple $\mathbb{Q}S_m \wr S_n$-submodules and apply Schur's Lemma to determine the structure of the information lost in the voting process. We conclude with a voting paradox result, showing that for sufficiently different weighting vectors, applying the associated positional voting procedures to the same set of votes can yield arbitrarily different election outcomes.