Voting, the Symmetric Group,and Representation Theory

摘要

By using only a handful of simple combinatorial objects (e.g., tabloids) and somebasic ideas from representation theory (e.g., Schur's Lemma), we are able to recastand extend some of Saari's well-known results. For example, we recover a result con-cerning the important relationship between the Borda count and pairwise voting whenvoters return full rankings of the candidates (Theorem 6). We then extend this result toa situation in which voters return partial rankings of the candidates (Theorem 9). In theprocess, we construct an infinite family of "Borda-like" voting procedures. With thehelp of our main theorem (Theorem 1), we also, for example, address the relationshipbetween positional voting and approval voting (Theorem 2).