Heights of Representative Systems: A Proof of Fishburn's Conjecture

Representative systems with n-voters are hierarchical choice functions from {— 1,0,1}n to {— 1,0,1} constructed as iterations of weighted majority voting. The height of a representative system is the minimal number of iterations necessary for this construction. In the paper we give an upper bound for μ(n) the maximal height of any n-voter representative system, and show that $\frac{{\mu (n)}} {n}$ goes to zero as n goes to infinity, n thus proving a conjecture made by Fishburn. Technically, the results are obtained by transferring the problem to the context of proper simple games, which have a similar hierarchical structure, and using known results on heights of simple games.