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Competitive intensity and quality maximizing seedings in knock-out tournaments


Before a knock-out tournament starts, the participants are assigned to positions in the tournament bracket through a process known as seeding. There are many ways to seed a tournament. In this paper, we solve a discrete optimization problem of finding a seeding that maximizes spectator interest in a tournament when spectators are interested in matches with high competitive intensity (i.e., matches that involve teams comparable in strength) and high quality (i.e., matches that involve strong teams). We find a solution to the problem under two assumptions: the objective function is linear in quality and competitive intensity and a stronger team beats a weaker one with sufficiently high probability. Depending on parameters, only two special classes of seedings can be optimal. While one of the classes includes a seeding that is often used in practice, the seedings in the other class are very different. When we relax the assumption of linearity, we find that these classes of seedings are in fact optimal in a sizable number of cases. In contrast to existing literature on optimal seedings, our results are valid for an arbitrarily large number of participants in a tournament.

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Fig. 1
Fig. 2


  1. The share of close seedings in the set of all seedings is equal to \(\frac{2^{2^n-1}}{(2^n)!}\) while the share of distant seedings is the same number multiplied by \(\prod _{k=1}^{n-1}(2^k)!\).


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Correspondence to Dmitry Dagaev.

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Dmitry Dagaev is supported by The National Research University—Higher School of Economics Academic Fund Program in 2014/2015 (research grant No 14-01-0007).

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Dagaev, D., Suzdaltsev, A. Competitive intensity and quality maximizing seedings in knock-out tournaments. J Comb Optim 35, 170–188 (2018).

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  • Knock-out tournament
  • Seeding
  • Combinatorial optimization
  • Operations research in sports