Randomness in Competitions

ELI BEN-NAIM; N. W. HENGARTNER; SIDNEY REDNER; FEDERICO VAZQUEZ

JOURNAL OF STATISTICAL PHYSICS

SPRINGER

Lugar: New York; Año: 2013 vol. 151 p. 458 - 474

0022-4715

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 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, teamsplay 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.