Fair Seeding in Knockout Tournaments

Abstract

We investigated the existence of fair seeding in knockout tournaments. We define two fairness criteria, both adapted from the literature: envy-freeness and order preservation. We show how to achieve the first criterion in tournaments whose structure is unconstrained, and prove an impossibility result for balanced tournaments. For the second criterion we have a similar result for unconstrained tournaments, but not for the balanced case. We provide instead a heuristic algorithm which we show through experiments to be efficient and effective. This suggests that the criterion is achievable also in balanced tournaments. However, we prove that it again becomes impossible to achieve when we add a weak condition guarding against the phenomenon of tournament dropout.

Dynamic Programming Algorithms for Computing Optimal Knockout Tournaments

“…A comprehensive analysis of knockout tournaments is proposed in [3,4]. Traditionally, the method for designing a tournament involves two stages: (i) tournament structure design and (ii) seeding.…”

“…An important aspect concerns the factors that can be used to evaluate a tournament design. This problem has been also considered in previous works [3,4]. An interesting discussion of economic aspects of tournament attractiveness, such as spectator interest, is provided by [7].…”

“…The tournament is carried out in a series of rounds. If there is a number N of players equal to a power of 2, for example N = 8 = 2 3 then the tournament tree is a complete binary tree with all the players entering the tournament in the first round; however, in the general case, the number of players might not be a power of 2, for example N = 9. In this case some of the players will receive waivers thus entering the tournament directly in the the second round, while the rest of the players will enter the tournament in the first round.…”

What a fairer 24 team UEFA Euro could look like

“…He proves that the canonical knockout bracket fails to satisfy the second axiom and suggests a variant that satisfies all three seeding axioms, in which subgroups of teams are randomly shuffled. Vu and Shoham (2011) introduce two alternative criteria for fairness (envy-freeness and order preservation) and investigate several impossibility results. Karpov (2015) develops an axiomatic theory of knockout tournaments, gives axiomatic justification for various seedings methods, suggests two new seeding methods (equal gap seeding and increasing competitive intensity seeding), and provides many useful references.…”

On the complexity of bribery and manipulation in tournaments with uncertain information

“…In the deterministic case Lang et al investigate the problem of setting an agenda for a sequential majority vote (identical to scheduling match-ups in a caterpillar tournament) such that a particular outcome wins [30]. Vu et al initiated the investigation of the question of setting the schedule, subject to various sets of constraints on fairness and interesting-ness, often finding the problems computationally hard [51][52][53]. Williams found that, for certain dominant entrants in a cup tournament it is easy to find a schedule such that a particular entrant will win [56].…”