Randomness in Competitions

TL;DR

This paper models the impact of randomness in sports competitions, comparing different tournament formats and proposing a new schedule that balances fairness and efficiency.

Contribution

It introduces a simple probabilistic model to analyze fairness and efficiency in sports competitions and proposes a new elimination schedule that improves these aspects.

Findings

Single-elimination tournaments are efficient but unfair.

Leagues are fair but require many games.

A new gradual elimination schedule balances fairness and efficiency.

Abstract

We study the effects of randomness on competitions based on an elementary random process in which there is a finite probability that a weaker team upsets a stronger team. We apply this model to sports leagues and sports tournaments, and compare the theoretical results with empirical data. Our model shows that single-elimination tournaments are efficient but unfair: the number of games is proportional to the number of teams N, but the probability that the weakest team wins decays only algebraically with N. In contrast, leagues, where every team plays every other team, are fair but inefficient: the top $\sqrt{N}$ of teams remain in contention for the championship, while the probability that the weakest team becomes champion is exponentially small. We also propose a gradual elimination schedule that consists of a preliminary round and a championship round. Initially, teams play a small number of preliminary games, and subsequently, a few teams qualify for the championship round. This algorithm is fair and efficient: the best team wins with a high probability and the number of games scales as $N^{9/5}$, whereas traditional leagues require N^3 games to fairly determine a champion.