 
Summary: Voting paradoxes and digraphs realizations
Noga Alon
Abstract
A family of permutations F forms a realization of a directed graph T = (V, E) if for every
directed edge uv of T, u precedes v in more than half of the permutations. The quality q(F, T)
of the realization is the minimum, over all directed edges uv of T, of the ratio (F(u, v) 
F(v, u))/F, where F(x, y) is the number of permutations in F in which x precedes y. The
study of this quantity is motivated by questions about voting schemes in which each individual
has a linear ordering of all candidates, and the individual preferences are combined to decide
between any pair of possible candidates by applying the majority vote. It is shown that every
simple digraph T on n vertices, with no antiparallel edges, admits a realization F with quality
at least c/
n for some absolute positive constant c, and this is tight up to the constant factor c.
1 Introduction
All directed graphs considered here are finite, simple (that is, have no loops and no parallel edges),
and have no antiparallel edges. The densest digraphs of this type are tournaments. A tournament
on a set V of n vertices is a directed graph on V in which for every pair of distinct vertices u, v V ,
either uv or vu is a directed edge, but not both. Let T = (V, E) be a digraph, and let F be a
collection of (not necessarily distinct) permutations of V . We say that F is a realization of T if for
