Efficiency of competitions

Abstract

League competition is investigated using random processes and scaling techniques. In our model, a weak team can upset a strong team with a fixed probability. Teams play an equal number of head-to-head matches and the team with the largest number of wins is declared to be the champion. The total number of games needed for the best team to win the championship with high certainty T grows as the cube of the number of teams N , i.e., T approximately N(3). This number can be substantially reduced using preliminary rounds where teams play a small number of games and subsequently, only the top teams advance to the next round. When there are k rounds, the total number of games needed for the best team to emerge as champion, T(k), scales as follows, T(k) approximately N(gamma(k)) with gamma(k) = [1-(2/3)(k+1)](-1). For example, gamma(k)=95,2719,8165 for k=1,2,3 . These results suggest an algorithm for how to infer the best team using a schedule that is linear in N. We conclude that league format is an ineffective method of determining the best team, and that sequential elimination from the bottom up is fair and efficient.