Heights of representative systems: A proof of Fishburn's conjecture (Q1057766)

This paper is concerned with the height of representative systems. A representative system, which was first introduced by Murakami (1966), is a map \(F: \{-1,0,1\}^ n\{-1,0,1\}\) that is constructed by a hierarchy of weighted majority voting. The height of F, \(\mu\) (F), is then defined as the minimal number of hierarchies necessary in this construction. Moreover, \(\mu\) (n) is defined as the maximal height of any n-voter representative system. In this conection, there is a famous conjecture made by Fishburn (1975, 1979) that \(\mu\) (n)/n\(\to 0\) as \(n\to \infty\). By transforming the problem to a similar one concerning proper simple games, the author attempts to give a complete yet rather lengthy proof to this conjecture.