Learning Risk Preferences in Markov Decision Processes: an Application to the Fourth Down Decision in the National Football League

TL;DR

This paper models NFL coaches' fourth down decisions as risk-sensitive Markov decision processes, revealing their conservative risk preferences and how these vary by field position and over time.

Contribution

It introduces an inverse optimization framework to infer coaches' risk preferences from actual decision data in NFL games.

Findings

Coaches' decisions align with low-quantile risk preferences.

Risk tolerance increases in opponent's half.

League-wide risk preferences have become more aggressive over time.

Abstract

For decades, National Football League (NFL) coaches' observed fourth down decisions have been largely inconsistent with prescriptions based on statistical models. In this paper, we develop a framework to explain this discrepancy using an inverse optimization approach. We model the fourth down decision and the subsequent sequence of plays in a game as a Markov decision process (MDP), the dynamics of which we estimate from NFL play-by-play data from the 2014 through 2022 seasons. We assume that coaches' observed decisions are optimal but that the risk preferences governing their decisions are unknown. This yields an inverse decision problem for which the optimality criterion, or risk measure, of the MDP is the estimand. Using the quantile function to parameterize risk, we estimate which quantile-optimal policy yields the coaches' observed decisions as minimally suboptimal. In general, we find that coaches' fourth-down behavior is consistent with optimizing low quantiles of the next-state value distribution, which corresponds to conservative risk preferences. We also find that coaches exhibit higher risk tolerances when making decisions in the opponent's half of the field as opposed to their own half, and that league average fourth down risk tolerances have increased over time.