  Patterns of wins and losses in pairwise contests, such as occur in sports and
games, consumer research and paired comparison studies, and human and animal
social hierarchies, are commonly analyzed using probabilistic models that allow
one to quantify the strength of competitors or predict the outcome of future
contests. Here we generalize this approach to incorporate two additional
features: an element of randomness or luck that leads to upset wins, and a
"depth of competition" variable that measures the complexity of a game or
hierarchy. Fitting the resulting model to a large collection of data sets we
estimate depth and luck in a range of games, sports, and social situations. In
general, we find that social competition tends to be "deep," meaning it has a
pronounced hierarchy with many distinct levels, but also that there is often a
nonzero chance of an upset victory, meaning that dominance challenges can be
won even by significant underdogs. Competition in sports and games, by
contrast, tends to be shallow and in most cases there is little evidence of
upset wins, beyond those already implied by the shallowness of the hierarchy.
