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For decades, National Football League (NFL) coaches' observed fourth downdecisions have been largely inconsistent with prescriptions based onstatistical models. In this paper, we develop a framework to explain thisdiscrepancy using a novel inverse optimization approach. We model the fourthdown decision and the subsequent sequence of plays in a game as a Markovdecision process (MDP), the dynamics of which we estimate from NFL play-by-playdata from the 2014 through 2022 seasons. We assume that coaches' observeddecisions are optimal but that the risk preferences governing their decisionsare unknown. This yields a novel inverse decision problem for which theoptimality criterion, or risk measure, of the MDP is the estimand. Using thequantile function to parameterize risk, we estimate which quantile-optimalpolicy yields the coaches' observed decisions as minimally suboptimal. Ingeneral, we find that coaches' fourth-down behavior is consistent withoptimizing low quantiles of the next-state value distribution, whichcorresponds to conservative risk preferences. We also find that coaches exhibithigher risk tolerances when making decisions in the opponent's half of thefield than in their own, and that league average fourth down risk toleranceshave increased over the seasons in our data.

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