Voting on Cyclic Orders, Group Theory, and Ballots
Karl-Dieter Crisman, Abraham Holleran, Micah Martin, Josephine Noonan
TL;DR
This work addresses voting on cyclic orders by embedding the problem in the representation theory of the symmetric group $S_n$. It introduces ROLO and TRAD ballot families and develops neutral, points-based rules via a scoring matrix, enabling a systematic decomposition of outcome spaces into irreducible $S_n$-modules. For $n=4$, it provides a complete characterization: $\, ext{Q}CO_4 \, ext{decomposes as } S^{(4)}oxplus S^{(2,2)}oxplus S^{(2,1,1)}$, so any neutral rule is determined by three scalars and can be analyzed via invariant subspaces; ROLO serves as a concrete regular-representation example. The paper also outlines the $n=5$ case, giving a decomposition of $\, ext{Q}CO_5$ into irreducibles $S^{(5)} oxplus S^{(3,2)} oxplus S^{(3,1,1)^{oxplus 2}} oxplus S^{(2,2,1)} oxplus S^{(1,1,1,1,1)}$, illustrating how the same framework scales to larger $n$ and providing a foundation for constructing and evaluating new ballots on cyclic orders.
Abstract
A cyclic order may be thought of informally as a way to seat people around a table, perhaps for a game of chance or for dinner. Given a set of agents such as $\{A,B,C\}$, we can formalize this by defining a cyclic order as a permutation or linear order on this finite set, under the equivalence relation where $A\succ B\succ C$ is identified with both $B\succ C\succ A$ and $C\succ A\succ B$. As with other collections of sets with some structure, we might want to aggregate preferences of a (possibly different) set of voters on the set of possible ways to choose a cyclic order. However, given the combinatorial explosion of the number of full rankings of cyclic orders, one may not wish to use the usual voting machinery. This raises the question of what sort of ballots may be appropriate; a single cyclic order, a set of them, or some other ballot type? Further, there is a natural action of the group of permutations on the set of agents. A reasonable requirement for a choice procedure would be to respect this symmetry (the equivalent of neutrality in normal voting theory). In this paper we will exploit the representation theory of the symmetric group to analyze several natural types of ballots for voting on cyclic orders, and points-based procedures using such ballots. We provide a full characterization of such procedures for two quite different ballot types for $n=4$, along with the most important observations for $n=5$.