Temperature as a measure of complexity in games and social hierarchies

Patterns of wins and losses in pairwise contests, such as occur in sports and games, social hierarchies, and other contexts, are often analyzed using probabilistic models that allow us to quantify the strength of competitors or to predict the outcome of future contests. For example, the popular Bradley-Terry model, familiar as the basis for the Elo chess ratings, uses a sigmoid "score function" for this purpose. We extend this approach in a physically motivated way: we interpret the model's sigmoid function as a Fermi-Dirac distribution and then infer the corresponding temperature by fitting the model to observational data. Examining data sets from a variety of settings we find that sports and games generically have a higher temperature than human social hierarchies, which in turn have a higher temperature than animal social hierarchies. We argue that the inferred temperature gives insight into the rigidity, depth, or complexity of the various hierarchies. We also generalize the model to include a "luck" component, which represents the inherent randomness of a contest regardless of difference in skill. We find that sports and games mostly have little evidence of "luck" in this formal sense, but animal hierarchies appear to have a pronounced luck component.