Classification of Voting Rules

Chapter 3 showed that a voting rule, a on a finite set of alternatives, W, is classified by a single integer, namely the Nakamura number v(σ). If the rule is collegial, so v(σ) = ∞, then for any acyclic profile p the relation σ(p) will be acyclic. If σ is non-collegial, with v(σ) < ∞ then σ(p) will be acyclic as long as |w| < v(σ), while an acyclic profile p can always be constructed such that σ(p) is cyclic on W when |w| ≥ v(σ).