Game Theory in American Football
Source: https://www.researchgate.net/publication/46554851_Optimizing_Football_Game_Play_Calling
In this paper published in the Journal of Quantitative Analysis of Sports, Jordan et al discuss the usage of game theory, Nash equilibrium in particular, in optimizing game calling in American Football. Historically, professional football coaches have used a mix of experience, instinct, personal bias, and the observable situation of the game to call plays. In American football, as well as any sport at the professional level, optimizing behavior is constantly occurring. If a certain player is subbed out or comes onto the field, if a certain player had a bad week of practice, or shows specific tendencies, coaches will optimize their play calling based on this new information. Game theory is increasingly becoming a frequently used tool in capturing the risk to reward ratio of certain strategies, as well as quantifying the difference between perceived optimal strategy and true optimal strategy.
Perceived optimal strategy is what the optimal decision appears to be, based on the limited information available. True optimal strategy is achievable when perfect information is available. In terms of football, if a defense appears to line up in a 4-3 man formation, the perceived optimal strategy for the offense would be to throw a short pass. However, the defense may suddenly drop an extra man in coverage pre-snap, switching to 4-4 zone formation. This dramatically reduces the payoff for the offense, as the true optimal strategy would then be a long pass. The payoffs for each strategy are shown in the table below.
Since football is a zero-sum two-player game, a gain of 1 yard for the offense is equivalent to a loss of 1 yard for the defense. The table above shows an example based on average expected yardage for the offense if it runs a certain play-type against a certain defensive front. Even in this simplification, there is no pure-strategy Nash equilibrium. Thus, a mixed-strategy model is used to determine what is the perceived optimal strategy for the offense, as well as the defense. NFL teams are increasingly using statistical modeling based on game theory to determine what type of play call to use to win games. In this multi-billion dollar business, any ethical advantage is worth taking. However, models are not perfect; they can help calculate what strategy is statistically optimal given the correct information, but information is often lacking. Due to the sheer number of factors at play in a single down of football, game-theory models should not be relied on as an end all be all. Nonetheless, teams at all levels, in all sports will continue to increase their application of game-theory to win games.