Models and Rating Systems for Head-to-Head Competition

One of the most important tasks in sports analytics is the development of binary response models for head-to-head game outcomes to estimate team and player strength. We discuss commonly used probability models for game outcomes, including the Bradley–Terry and Thurstone–Mosteller models, as well as extensions to ties as a third outcome and to the inclusion of a home-field advantage. We consider dynamic extensions to these models to account for the evolution of competitor strengths over time. Full likelihood-based analyses of these time-varying models can be simplified into rating systems, such as the Elo and Glicko rating systems. We present other modern rating systems, including popular methods for online gaming, and novel systems that have been implemented for online chess and Go. The discussion of the analytic methods are accompanied by examples of where these approaches have been implemented for various gaming organizations, as well as a detailed application to National Basketball Association game outcomes.