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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, 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. Competition in sports and games, by contrast, tends to be shallow, and in most cases, there is little evidence of upset wins.