A statistical theory of optimal decision-making in sports betting

The recent legalization of sports wagering in many regions of North America has renewed attention on the practice of sports betting. Although considerable effort has been previously devoted to the analysis of sportsbook odds setting and public betting trends, the principles governing optimal wagering have received less focus. Here the key decisions facing the sports bettor are cast in terms of the probability distribution of the outcome variable and the sportsbook's proposition. Knowledge of the median outcome is shown to be a sufficient condition for optimal prediction in a given match, but additional quantiles are necessary to optimally select the subset of matches to wager on (i.e., those in which one of the outcomes yields a positive expected profit). Upper and lower bounds on wagering accuracy are derived, and the conditions required for statistical estimators to attain the upper bound are provided. To relate the theory to a real-world betting market, an empirical analysis of over 5000 matches from the National Football League is conducted. It is found that the point spreads and totals proposed by sportsbooks capture 86% and 79% of the variability in the median outcome, respectively. The data suggests that, in most cases, a sportsbook bias of only a single point from the true median is sufficient to permit a positive expected profit. Collectively, these findings provide a statistical framework that may be utilized by the betting public to guide decision-making.