Rankings for Bipartite Tournaments via Chain Editing

Abstract: 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.