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

Luck, skill, and depth of competition in games and social hierarchies