Three-valued representative systems

Abstract

A representative system defined onn voters or propositionsi = 1,⋯,n is a functionF: {1,0, -1}ⁿ → {1,0, -1} which is monotonic (D ≥ E ⇒ F(D) ≥ F(E)), unanimous (F(1,⋯, 1) = 1), dual (F(-D) = -F(D)), and satisfies a positivity property which says that the set of all non-zero vectors in {1, 0, -1}ⁿ for whichF(D) = 0 can be partitioned into two dual subsets each of which has the property that ifD andE are in the subset thenD _i+E _i > 0 for somei. Representative systems can be defined recursively from the coordinate projectionsS _i (D) = D _i using sign functions, and in this format they are interpreted as hierarchical voting systems in which outcomes of votes in “lower” levels act as votes in “higher” levels of the system. For each positive integern, μ(n) is defined as the smallest positive integer such that all representative systems defined on {1, 0, -1}ⁿ can be characterized byμ(n) or fewer hierarchical levels. The functionμ is nondecreasing inn, unbounded above, and satisfiesμ(n) ≤ n−1 for alln. In addition,μ(n) = n−1 forn ∈ {1, 2, 3, 4}, and it is conjectured thatμ does not continue to grow linearly asn increases.