How Does Sport Score Work?

In sport, score is a quantitative measure of the relative performance in competition. In most team sports the victory is determined by accruing a set number of points, or goals, more than your opponents; individual-based sports such as tennis and golf also have scoring systems based on duration, distance or height.

Modern sports produce large quantities of detailed data describing not only competition outcomes and team characteristics, but also the dynamics of scoring events within each game. Despite this growing interest in quantifying and modeling the scoring dynamics of various teams, relatively little is known about what patterns or principles cut across different sports.

Using a rich data set of scoring events from multiple seasons of college and professional (American) football, basketball, hockey and baseball, we find that the timing of scoring events across these four sports follows a simple Poisson process with a sport-specific rate. Similarly, the frequency with which a team wins an event over its opponent follows a first-order Bernoulli process, where the parameter effectively varies as a function of the size of a team’s lead.

Both of these empirical findings are supported by analytical models, in which the tempo and balance processes are described as Markov chains with first-order memory. This memory ensures that the timing of scoring events in a given moment is independent of the timing of previous events, but also prevents earlier events from constraining or driving later ones. The resulting model provides empirically valid predictions of both the rate and timing of scoring events in each of our four sports, as well as a framework for evaluating competing theories of scoring behavior.