  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 ~ 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 ~N^(\gamma_k) with
gamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 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.
