Designing fair 8- and 16-team knockout tournaments

In an eight-team single-elimination tournament without reseeding, teams are seeded from best (1) to worst (8). Teams 1/8, 4/5, 2/7 and 3/6 are paired in the first round, with the 1/8 winner facing the 4/5 winner in the second round and so on. However, such tournaments are potentially unfair in the sense that inferior teams can be more likely to advance to certain stages of the tournament than better teams. For instance, if the top five teams are comparable in strength and are markedly better than the bottom three teams, then seeds 2 and 3 may be more likely to advance to the finals than team 1. We assign each team a unique power value and assume that the victory probability in a match-up is proportional to the teams' powers.We investigate properties of fair tournaments and formulate a non-linear optimization model that prescribes a fair tournament given the relative strengths of the teams involved. Although the problem is highly non-convex, we demonstrate how to consistently obtain all fair tournaments for 8- and 16-team problems.