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 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 tricritical 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.