Elsevier

Operations Research Letters

Volume 44, Issue 6, November 2016, Pages 706-711
Operations Research Letters

A new knockout tournament seeding method and its axiomatic justification

https://doi.org/10.1016/j.orl.2016.09.003Get rights and content

Abstract

A new set of axioms and new method (equal gap seeding) are designed. The equal gap seeding is the unique seeding that, under the deterministic domain assumption, satisfies the delayed confrontation, fairness, increasing competitive intensity and equal rank differences axioms. The equal gap seeding is the unique seeding that, under the linear domain assumption, maximizes the probability that the strongest participant is the winner, the strongest two participants are the finalists, the strongest four participants are the quarterfinalists, etc.

Introduction

A knockout tournament (also called a single elimination tournament or elimination tournament) is a system of selecting a single winner. Participants (teams or individuals) play n rounds of matches. After each round, the winners move on to the next round, the losers are dropped from the tournament, and the number of participants is reduced by half. Only balanced knockout tournaments without byes are considered in the paper. The number of participants in each round in such a tournament always equals a power of two. Knockout tournaments are widespread in sports competitions. They are frequently used as a playoff tournament in a larger tournament with several leagues, groups and conferences and/or as a qualifying tournament  [17].

A key feature of knockout tournaments is the importance of the seeding method, or simply the seeding. The seeding is a rule that, having information regarding the initial order of participants’ strengths mainly from historical data, determines the tournament bracket. There are several seedings used in different tournaments. The most popular seeding (called standard seeding) creates pairs in the first round of the strongest participant with the weakest participant, the second strongest participant with the second weakest participant, etc. The pairs in subsequent rounds are determined in a way that preserves the first two participants from the head-to-head match before the final and that delays the confrontations between strong participants until later rounds. Strong participants are rewarded for their success through such a seeding.

This paper is limited to seeding methods with predetermined tournament brackets, i.e., without random seeding (e.g., the method proposed in  [21]) and without any reseeding methods (e.g., the method proposed in  [12]). This restriction comes from the tournament practice. The World Cup, NBA playoffs, NCAA basketball tournament, the NHL’s Stanley Cup Playoffs and most professional tennis events are examples of tournaments without reseeding. Baumann et al.  [4] noted that reseeding causes teams and spectators to have to make last-minute travel plans, which both increases costs and potentially reduces demand, and that reseeded tournaments eliminate popular gambling options related to filling out full tournament brackets, potentially reducing fan interest in the tournament.

Seeding aims to ensure high competitive intensity, especially in the latter rounds of a tournament  [27], to provide incentives for the participants to maximize their performance both during the tournament and in the time period leading up to the tournament  [4] to maximize spectator interest  [6] and to ensure fairness  [11]. Such requirements create the theoretical problem of designing an ideal seeding method.

Standard seeding has clear shortcoming. There is much evidence that, in the NCAA men’s basketball tournament, weaker participants have a higher probability of winning the tournament and better chances of advancing in some rounds than certain stronger participants  [4], [5], [13], [14]. It is an example of a violation of fairness.

Theoretical studies have attempted to find a best seeding and formalize different goals of seeding. Many studies contain impossibility results. Horen and Riezman  [11] showed that there is no “fair” seeding in the sense that a better participant has a higher probability of winning in the case of eight participants and a double monotonic domain assumption. Schwenk  [21] proposed the delayed confrontation (two participants rated among the top 2k shall never meet until the number of participants has been reduced to 2k or fewer), sincerity rewarded (a stronger participant should never be penalized by being given a schedule more difficult than that of any weaker participant) and favoritism minimized (the schedule should minimize favoritism to any particular seed) axioms. This set of axioms can be satisfied only within randomized seeding methods. Vu and Shoham  [26] investigated several impossibility results. They proved that there is no envy-free seeding under a double monotonic domain assumption, there is no order-preserving seeding under a general domain assumption (envy-free and order-preserving axioms require better conditions for stronger participants).

Numerous statistical and simulation studies  [1], [8], [9], [16], [20] did not find clear support for any seeding method with a predetermined tournament bracket. Computational social choice studies [2], [24], [25] aimed to find seedings with given characteristics. Basic problems, e.g., computing the maximum possible winning probability of a given player, are NP-hard.

Game-theoretic analysis of knockout tournaments has mainly focused on incentives of exerting effort and selection properties. The problem of designing an optimal seeding is solved mainly for four-participant tournaments  [10], [15], [19], [22].

The aim of this paper is to provide general possibility results. The equal gap seeding investigated in the paper was, to the author’s best knowledge, never discussed in academic literature. Considering different domains and desirable properties, several axiomatic justifications of equal gap seeding are proposed. Because of the simplicity and several justifications, equal gap seeding is a plausible alternative to standard seeding.

This paper is closely related to decision theory literature. Eliaz et al.  [7] axiomatized procedures of choosing the two finalists. Bajraj and Ülkü  [3] axiomatized procedures of choosing the two finalists and the winner. This paper axiomatizes procedures of choosing the winner, the two finalists, the four quarterfinalists, etc.

The structure of the paper is as follows. Section  2 describes seeding methods and probability domains. Section  3 presents axiomatics and representation theorems for the equal gap method. Section  4 presents a discrete optimization approach with justifications for the equal gap seeding.

Section snippets

Framework

Let X={1,2,,2n} be the set of alternatives (participants) and n be the number of rounds in the knockout tournament (henceforth in the text tournament). The indices of the participants represent the order of the participants’ strengths, where the first participant is the strongest.

Each tournament with n2 rounds is a set which consists of two subtournaments. Each subtournament with n2 rounds is a set which also consists of two subtournaments. Denote by Tik,n a subtournament i with k rounds,

Axiomatic justification

Participants, organizers and spectators have their own interests. Participants want to play as many matches as possible. Organizers want to support spectator interest in all matches. Spectators want to see strong participants and matches with high competitive intensity. All of them are interested in a fair tournament with a strong winner. These ideals are represented in the following set of axioms.

A1. Delayed confrontation   [21]. Two participants rated among the top 2k shall never meet until

Discrete optimization approach

The main purpose of seeding is to prevent strong participants from being eliminated at the beginning of a tournament. Many studies maximize the probability of the event “participant 1 wins a tournament”. It is a narrow approach. Such events as “the strongest 2nk participants win k matches” are also desirable for tournament design. Proposition 7 proposes the general form of the objective function. This objective function represents an idea of weak fairness (axiom A2′). The probabilities that

Acknowledgments

The author would like to thank Fuad Aleskerov, Jonas Hedlund, Dmitry Dagaev, Boris Mirkin and Sylvain Béal for valuable comments. The author acknowledges the support of the University of Heidelberg Economics Department and the DeCAn Laboratory (HSE). The article was prepared within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE) and supported within the framework of a subsidy by the Russian Academic Excellence Project ‘5-100’.

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