Voting, the Symmetric Group, and Representation Theory

Abstract

Abstract. We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied QSn-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.

Linear rank tests of uniformity: understanding inconsistent outcomes and the construction of new tests

“…A rejection of H 0 could then be explained using the computed summary statistic M(P) because, by construction, M(P) captures all of the information necessary to compute P S 2 . For convenience, we will refer to ker(M) ⊥ as the effective space of M (Daugherty, Eustis, Minton, and Orrison 2009).…”

Decomposition behavior in aggregated data sets

“…Likewise, each sub-vector space is orthogonal to the others, so this is a true decomposition, and one can check they are simple Σ 3 -modules (see, for instance, [8]), so this is an irreducible decomposition with respect to switching the names of the candidates. We easily obtain the decomposition of a profile vector by (left-)multiplying it by the inverse of the matrix whose rows are the vectors above (excepting B C and R C , as otherwise they are not linearly independent).…”

“…Nonetheless, we call any voting profile with only Kernel and Basic non-vanishing components pure Basic (and likewise for pure Reversal, pure Condorcet). Indeed, in the theory of voting, it turns out that pure Basic profiles may be viewed as providing the least amount of paradox (see for instance [8,22,23]), a particularly nice subspace. Hence it is reasonable to search for profiles with as large a (relative) Basic portion as possible, or even pure Basic, in the related context of nonparametric data sets.…”

“…Recall that this is in the order: A B C, A C B, C A B, C B A, B C A, B A C. 8 That is, where A comes from the higher-stacked matrix of ranks, and hence always is ranked above the B and C ranks.…”

“…Thus, it is reasonable to look at the underlying profiles instead of the data sets per se, to see for which of them paradoxes might be minimized, since they cannot be avoided. In the context of voting theory, Saari (for instance, see [19,20,21,22,23]) has used symmetry-respecting decompositions of the profile vector space to determine what components of a voting profile may be viewed as 'causing' various types of paradoxes (see also Orrison et al in [8] for a representation-theoretic view of this).…”