  The task of ranking individuals or teams, based on a set of comparisons
between pairs, arises in various contexts, including sporting competitions and
the analysis of dominance hierarchies among animals and humans. Given data on
which competitors beat which others, the challenge is to rank the competitors
from best to worst. Here we study the problem of computing rankings when there
are multiple, potentially conflicting modes of comparison, such as multiple
types of dominance behaviors among animals. We assume that we do not know a
priori what information each behavior conveys about the ranking, or even
whether they convey any information at all. Nonetheless we show that it is
possible to compute a ranking in this situation and present a fast method for
doing so, based on a combination of an expectation-maximization algorithm and a
modified Bradley-Terry model. We give a selection of example applications to
both animal and human competition.
