  Competitive tournaments appear in sports, politics, population ecology, and
animal behavior. All of these fields have developed methods for rating
competitors and ranking them accordingly. A tournament is intransitive if it is
not consistent with any ranking. Intransitive tournaments contain
rock-paper-scissor type cycles. The discrete Helmholtz-Hodge decomposition
(HHD) is well adapted to describing intransitive tournaments. It separates a
tournament into perfectly transitive and perfectly cyclic components, where the
perfectly transitive component is associated with a set of ratings. The size of
the cyclic component can be used as a measure of intransitivity. Here we show
that the HHD arises naturally from two classes of tournaments with simple
statistical interpretations. We then discuss six different sets of assumptions
that define equivalent decompositions. This analysis motivates the choice to
use the HHD among other existing methods. Success in competition is typically
mediated by the traits of the competitors. A trait-performance model assumes
that the probability that one competitor beats another can be expressed as a
function of their traits. We show that, if the traits of each competitor are
drawn independently and identically from a trait distribution then the expected
degree of intransitivity in the network can be computed explicitly. Using this
result we show that increasing the number of pairs of competitors who could
compete promotes cyclic competition, and that increasing the correlation in the
performance of $A$ against $B$ with the performance of $A$ against $C$ promotes
transitive competition. The expected size of cyclic competition can thus be
understood by analyzing this correlation. An illustrative example is provided.
