Points Gained in Football: Using Markov Process-Based Value Functions to Assess Team Performance

To develop a novel approach for performance assessment, this paper considers the problem of computing value functions in professional American football. We provide a theoretical justification for using a dynamic programming approach to estimating value functions in sports by formulating the problem as a Markov chain for two asymmetric teams. We show that the Bellman equation has a unique solution equal to the bias of the underlying infinite horizon Markov reward process. This result provides, for the first time in the sports analytics literature, a precise interpretation of the value function as the expected number of points gained or lost over and above the steady state points gained or lost. We derive a martingale representation for the value function that provides justification, in addition to the analysis of our empirical transition probabilities, for using an infinite horizon approximation of a finite horizon game. Using more than 160,000 plays from the 2013–2016 National Football League seasons, we derive an empirical value function that forms our points gained performance assessment metric, which quantifies the value created or destroyed on any play relative to expected performance. We show how this metric provides new insight into factors that distinguish performance. For example, passing plays generate the most points gained, whereas running plays tend to generate negative value. Short passes account for the majority of the top teams’ success and the worst teams’ poor performance. Other insights include how teams differ by down, quarter, and field position. The paper concludes with a case study of the 2019 Super Bowl and suggests how the key concepts might apply outside of sports.