 Transitive tournament

A transitive tournament is a tournament whose vertices admit an ordering $v_1,\dots ,v_n$ with every edge directed from $v_i$ to $v_j$ when $i<j$. Equivalently, it has no directed cycle, or no directed triangle. The implication from no triangle follows by shortening any shortest directed cycle of length greater than three using its chord.

Its outdegrees are $n-1,n-2,\dots ,0$, so the ordering records a consistent ranking with no cyclic preference. Every three-vertex tournament is either transitive or a directed triangle. This makes transitive tournaments the simplest model of a complete pairwise comparison that agrees with a total order.