Scoring Strategies for the Underdog: A general, quantitative method for determining optimal sports strategies

TL;DR

This paper introduces a quantitative, analytical approach to determine optimal sports strategies for underdogs by balancing risk and reward, using statistical models to maximize winning probabilities.

Contribution

It develops a general method for calculating optimal offensive strategies based on variance and mean, applicable across various sports scenarios.

Findings

Optimal strategies often involve counterintuitive risk-taking.

The method accurately predicts the value of different play choices.

Strategies like stalling and specific play calls can be quantitatively justified.

Abstract

When facing a heavily-favored opponent, an underdog must be willing to assume greater-than-average risk. In statistical language, one would say that an underdog must be willing to adopt a strategy whose outcome has a larger-than-average variance. The difficult question is how much risk a team should be willing to accept. This is equivalent to asking how much the team should be willing to sacrifice from its mean score in order to increase the score's variance. In this paper a general, analytical method is developed for addressing this question quantitatively. Under the assumption that every play in a game is statistically independent, both the mean and the variance of a team's offensive output can be described using the binomial distribution. This description allows for direct calculations of the winning probability when a particular strategy is employed, and therefore allows one to calculate optimal offensive strategies. This paper develops this method for calculating optimal strategies exactly and then presents a simple heuristic for determining whether a given strategy should be adopted. A number of interesting and counterintuitive examples are then explored, including the merits of stalling for time, the run/pass/Hail Mary choice in American football, and the correct use of Hack-a-Shaq.