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.