Efficiency of competitions

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., $T\sim N^3$. 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, $T_k\sim N^{{\gamma }_k}$ with ${\gamma }_k={[1-{(2∕3)}^{k+1}]}^{-1}$. 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.

Figure 1

(Color online) The average rank of the champion $⟨n_{*}⟩$ of a league with $N$ teams after $t$ games. The simulation results represent and average over ${10}^3$ independent realizations with $N={10}^2$, ${10}^3$, and ${10}^4$. A straight line of slope $-1∕2$, predicted by Eq. (7), is plotted as a reference.

Figure 2

(Color online) The average number of games $⟨T⟩$ needed for the best team to emerge as the champion of a league with $N$ teams. The simulation results, representing an average over ${10}^3$ independent realizations, are compared with the theoretical prediction (8).

Figure 3

(Color online) The rank distribution of the league winner for ordinary league format $(t=N)$. Shown is the scaled distribution $\sqrt{N}{Q}_n(t=N)$ versus the scaling variable $n∕\sqrt{N}$. The simulation data were obtained using ${10}^6$ independent Monte Carlo runs.