  Ranking the participants of a tournament has applications in voting, paired
comparisons analysis, sports and other domains. In this paper we introduce
bipartite tournaments, which model situations in which two different kinds of
entity compete indirectly via matches against players of the opposite kind;
examples include education (students/exam questions) and solo sports
(golfers/courses). In particular, we look to find rankings via chain graphs,
which correspond to bipartite tournaments in which the sets of adversaries
defeated by the players on one side are nested with respect to set inclusion.
Tournaments of this form have a natural and appealing ranking associated with
them. We apply chain editing -- finding the minimum number of edge changes
required to form a chain graph -- as a new mechanism for tournament ranking.
The properties of these rankings are investigated in a probabilistic setting,
where they arise as maximum likelihood estimators, and through the axiomatic
method of social choice theory. Despite some nice properties, two problems
remain: an important anonymity axiom is violated, and chain editing is NP-hard.
We address both issues by relaxing the minimisation constraint in chain
editing, and characterise the resulting ranking methods via a greedy
approximation algorithm.
