  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.
