Recent trends in using representations in voting theory – committees and cyclic orders (Karl-Dieter Crisman, Gordon College)

One of the most important axioms in analyzing voting systems is that of “neutrality”, which stipulates that the system should treat all candidates symmetrically. Even though this doesn’t always directly apply (such as in primary systems or those with intentional incumbent protection), it is extremely important both in theory and practice.

If the voting systems in question additionally are tabulated using some sort of points, we can translate the notion of neutrality into invariance under an action of the symmetric group on a vector space. This means we can exploit representation theory to analyze them, and this has been successfully done in a number of social choice contexts from cooperative games to voting on full rankings.

In this talk, we describe recent progress in extending this technique to two interesting situations. First we consider work of Barcelo et al. on voting for committees of representatives (such as for departments in a college), where the wreath product of two symmetric groups comes into play. Then we look at work by Crisman et al. regarding voting on “cyclic orders”, or ways to seat people around a table, which implicitly has both left and right actions of the symmetric group to consider.