Representation Theory of the Symmetric Group in Voting Theory and Game Theory

TL;DR

This survey explores how the representation theory of the symmetric group informs voting and game theory, revealing new insights and relationships such as between the Borda count and Kemeny rule.

Contribution

It provides a comprehensive overview of the application of symmetric group representations in voting and game theory, highlighting novel connections and research questions.

Findings

Relationship between Borda count and Kemeny rule uncovered

Representation-theoretic approach to symmetric solution concepts explained

New research questions identified within the framework

Abstract

This paper is a survey of some of the ways in which the representation theory of the symmetric group has been used in voting theory and game theory. In particular, we use permutation representations that arise from the action of the symmetric group on tabloids to describe, for example, a surprising relationship between the Borda count and Kemeny rule in voting. We also explain a powerful representation-theoretic approach to working with linear symmetric solution concepts in cooperative game theory. Along the way, we discuss new research questions that arise within and because of the representation-theoretic framework we are using.