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.

翻译：在成对对决中出现的胜负模式（例如体育竞技、游戏、消费者研究、配对比较研究以及人类与动物的社会等级）通常通过概率模型进行分析，这些模型可量化竞争者的实力或预测未来比赛结果。本研究将该方法扩展至两个附加特征：一是导致爆冷获胜的随机性或运气要素，二是衡量游戏或等级体系复杂度的"竞争深度"变量。通过将所构建模型拟合至大规模数据集，我们对多种游戏、体育项目及社会情境中的深度与运气进行了估算。总体发现：社会竞争倾向于"深度"模式——即具有显著的多层级分级结构，但同时存在非零概率的爆冷获胜可能性，即使处于绝对劣势的挑战者也可能赢得支配权之争。相比之下，体育竞技与游戏的竞争往往呈"浅层"特征，且在大多数情况下，除等级体系本身浅层性所隐含的意外结果外，鲜有证据支持爆冷获胜的存在。