The statistical properties of pairwise majority voting over S alternatives is
analyzed in an infinite random population. We first compute the probability
that the majority is transitive (i.e. that if it prefers A to B to C, then it
prefers A to C) and then study the case of an interacting population. This is
described by a constrained multi-component random field Ising model whose
ferromagnetic phase describes the emergence of a strong transitive majority. We
derive the phase diagram, which is characterized by a tri-critical point and
show that, contrary to intuition, it may be more likely for an interacting
population to reach consensus on a number S of alternatives when S increases.
This effect is due to the constraint imposed by transitivity on voting
behavior. Indeed if agents are allowed to express non transitive votes, the
agents' interaction may decrease considerably the probability of a transitive
majority.