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.
A strict ranking of n items may profitably be viewed as a permutation of the objects. In particular, social preference functions may be viewed as having both input and output be such rankings (or possibly ties among several such rankings). A natural combinatorial object for studying such functions is the permutahedron, because pairwise comparisons are viewed as particularly important.In this paper, we use the representation theory of the symmetry group of the permutahedron to analyze a large class of such functions. Our most important result characterizes the Borda Count and the Kemeny Rule as members of a highly symmetric one-parameter family of social preference functions.
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