Voting paradoxes and digraphs realizations A family of permutations F forms a realization of a directed graph T = (V, E) if for every 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 anti-parallel 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 anti-parallel 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 Collections: Mathematics