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Journal ArticleDOI

New Concepts in Seeding Knockout Tournaments

Frank K. Hwang1
01 Apr 1982-American Mathematical Monthly (Informa UK Limited)-Vol. 89, Iss: 4, pp 235-239
About: This article is published in American Mathematical Monthly.The article was published on 1982-04-01. It has received 74 citations till now. The article focuses on the topics: Seeding.
Citations
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Journal ArticleDOI
TL;DR: In this paper, an elimination tournament with heterogenous contestants whose ability is common-knowledge is modeled as an all-pay auction, and equilibrium efforts are in mixed strategies yielding complex dynamics: endogenous win probabilities in each match depend on other matches' outcome through the identity of the expected opponent in the next round.
Abstract: We study an elimination tournament with heterogenous contestants whose ability is common-knowledge. Each pair-wise match is modeled as an all-pay auction. Equilibrium efforts are in mixed strategies, yielding complex dynamics: endogenous win probabilities in each match depend on other matches’ outcome through the identity of the expected opponent in the next round. The designer seeds competitors according to their ranks. For tournaments with four players we find optimal seedings for three different criteria: (1) maximization of total tournament effort; (2) maximization of the probability of a final among the two top ranked teams; (3) maximization of the win probability for the top player. We also find the seedings ensuring that higher ranked players have a higher winning probability. We compare our predictions with data from NCAA basketball tournaments.

139 citations

Journal ArticleDOI
TL;DR: This study compares different draws, or pairings, of teams in single-elimination tournaments under a set of relatively nonrestricting assumptions about the participating teams' pairwise probabilities of winning.
Abstract: Tournaments are used to select a single winner from a group of participants in a sporting event or a paired-comparison experiment. This study compares different draws, or pairings, of teams in single-elimination tournaments under a set of relatively nonrestricting assumptions about the participating teams' pairwise probabilities of winning. We analyze and compare draws for four-team tournaments using various criteria, then attempt to generalize the results to eight-team tournaments. For example, only one-four team draw maximizes the probability that the best team wins for all pairwise probabilities, whereas eight draws are possibly optimal for eight-team tournaments.

77 citations

Proceedings Article
10 May 2009
TL;DR: This paper investigates the computational complexity of tournament schedule control, i.e., designing a tournament that maximizes the winning probability a target player.
Abstract: Knockout tournaments constitute a common format of sporting events, and also model a specific type of election scheme (namely, sequential pairwise elimination election). In such tournaments the designer controls the shape of the tournament (a binary tree) and the seeding of the players (their assignment to the tree leaves). In this paper we investigate the computational complexity of tournament schedule control, i.e., designing a tournament that maximizes the winning probability a target player. We start with a generic probabilistic model consisting of a matrix of pairwise winning probabilities, and then investigate the problem under two types of constraint: constraints on the probability matrix, and constraints on the allowable tournament structure. While the complexity of the general problem is as yet unknown, these various constraints -- all naturally occurring in practice -- serve to push to the problem to one side or the other: easy (polynomial) or hard (NP-complete).

61 citations


Cites background from "New Concepts in Seeding Knockout To..."

  • ...One of the most common models in the literature is the monotonic model (see for example [6, 11, 7, 14])....

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Journal ArticleDOI
TL;DR: In this paper, the effect of seeding policy for various tournament structures, while taking account of competitive balance, is investigated using Monte Carlo simulation to calculate the values of this metric, showing that, in general, seeding favours stronger competitors.
Abstract: Tournament outcome uncertainty depends on: the design of the tournament; and the relative strengths of the competitors – the competitive balance. A tournament design comprises the arrangement of the individual matches, which we call the tournament structure, the seeding policy and the progression rules. In this paper, we investigate the effect of seeding policy for various tournament structures, while taking account of competitive balance. Our methodology uses tournament outcome uncertainty to consider the effect of seeding policy and other design changes. The tournament outcome uncertainty is measured using the tournament outcome characteristic which is the probability Pq,R that a team in the top 100q pre-tournament rank percentile progresses forward from round R, for all q and R. We use Monte Carlo simulation to calculate the values of this metric. We find that, in general, seeding favours stronger competitors, but that the degree of favouritism varies with the type of seeding. Reseeding after each round favours the strong to the greatest extent. The ideas in the paper are illustrated using the soccer World Cup Finals tournament.

50 citations


Cites background from "New Concepts in Seeding Knockout To..."

  • ...Various studies have been carried out in the past to explore seeding, but this has been done only in the context of a KO tournament, see for example the work of Hwang (1982), Appleton © 2010 The Authors....

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  • ...However, in the second round, the highest surviving seed plays with the lowest surviving seed; the second-highest surviving seed plays the second-lowest surviving seed, etc. Hwang (1982) proposed this seeding policy to ensure monotonicity in seeding in each round....

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Journal ArticleDOI
TL;DR: Using a common probability model for expressing relative player strengths, this work develops an adaptive approach to pairing players each round in which the probability that the best player advances to the next round is maximized.

47 citations


Cites background or methods from "New Concepts in Seeding Knockout To..."

  • ...the following procedure, which was introduced by Hwang (1982) adapted here to the current...

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  • ...One strategy to account for the N not being a power of 2 is to carry out the following procedure, which was introduced by Hwang (1982) adapted here to the current problem....

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  • ...The variant by Hwang (1982) involves reseeding players after each round....

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  • ...The variant by Hwang (1982) involves reseeding players after each round....

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  • ...Two variants that have attempted to improve on the standard format are by Schwenk (2000) and Hwang (1982)....

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References
More filters
Journal ArticleDOI
TL;DR: Various forms of line drawing representation are described, different schemes of quantization are compared, and the manner in which a line drawing can be extracted from a tracing or a photographic image is reviewed.
Abstract: This paper describes various forms of line drawing representation, compares different schemes of quantization, and reviews the manner in which a line drawing can be extracted from a tracing or a photographic image. The subjective aspects of a line drawing are examined. Different encoding schemes are compared, with emphasis on the so-called chain code which is convenient for highly irregular line drawings. The properties of chain-coded line drawings are derived, and algorithms are developed for analyzing line drawings to determine various geometric features. Procedures are described for rotating, expanding, and smoothing line structures, and for establishing the degree of similarity between two contours by a correlation technique. Three applications are described in detail: automatic assembly of jigsaw puzzles, map matching, and optimum two-dimensional template layout

1,485 citations

Book
01 Jan 1977
TL;DR: In this article, the philosophy of selecting and ordering populations has been studied in the context of normal distribution models, and the main focus of this paper is on the following: 1. Selecting the one best population for Normal Distributions with Common Known Variance (CKV) 2.
Abstract: 1. The Philosophy of Selecting and Ordering Populations 2. Selecting the One Best Population for Normal Distributions with Common Known Variance 3. Selecting the One Best Population for Other Normal Distribution Models 4. Selecting the One Best Population Bionomial (or Bernoulli) Distributions 5. Selecting the One Normal Population with the Smallest Variance 6. Selecting the One Best Category for the Multinomial Distribution 7. Nonparametric Selection Procedures 8. Selection Procedures for a Design with Paired Comparisons 9. Selecting the Normal Population with the Best Regression Value 10. Selecting Normal Populations Better than a Control 11. Selecting the t Best Out of k Populations 12. Complete Ordering of k Populations 13. Subset Selection (or Elimination) Procedures 14. Selecting the Best Gamma Population 15. Selection Procedures for Multivariate Normal Distributions Appendix A. Tables for Normal Means Selection Problems Appendix B. Figures for Normal Means Selection Problems Appendix C. Table of the Cumulative Standard Normal Distribution F(z) Appendix D. Table of Critical Values for the Chi-Square Distribution Appendix E. Tables for Binomial Selection Problems Appendix F. Figures for Binomial Selection Problems Appendix G. Tables for Normal Variances Selection Problems Appendix H. Tables for Multinomial Selection Problems Appendix I. Curtailment Tables for the Multinomial Selection Problem Appendix J. Tables of the Incomplete Beta Function Appendix K. Tables for Nonparametric Selection Problems Appendix L. Tables for Paired-Comparison Selection Problems Appendix M. Tables for Selecting from k Normal Populations Those Better Than a Control Appendix N. Tables for Selecting the t Best Normal Populations Appendix O. Table of Critical Values of Fisher's F Distribution Appendix P. Tables for Complete Ordering Problems Appendix Q. Tables for Subset Selection Problems Appendix R. Tables for Gamma Distribution Problems Appendix S. Tables for Multivariate Selection Problems Appendix T. Excerpt of Table of Random Numbers Appendix U. Table of Squares and Square Roots Bibliography References for Applications Index for Data and Examples Name Index Subject Index.

357 citations

Journal ArticleDOI
TL;DR: It is shown that a digital arc S is the digitization of a straight line segment if and only if it has the "chord property:" the line segment joining any two points of S lies everywhere within distance 1 of S.
Abstract: It is shown that a digital arc S is the digitization of a straight line segment if and only if it has the "chord property:" the line segment joining any two points of S lies everywhere within distance 1 of S. This result is used to derive several regularity properties of digitizations of straight line segments.

335 citations

Journal ArticleDOI
TL;DR: A theory for describing and measuring the concavities of cellular complexes (digitized silhouettes) is developed that involves the use of the minimum-perimeter polygon and its convex hull.
Abstract: A theory for describing and measuring the concavities of cellular complexes (digitized silhouettes) is developed. This theory involves the use of the minimum-perimeter polygon and its convex hull.

212 citations

Journal ArticleDOI
TL;DR: It is shown that, by properly marking the virtual as well as the real vertices of an MPP, the MPP can serve as a precise representation of any regular complex, and that this representation is often an economical one.
Abstract: The minimum-perimeter polygon of a silhouette has been shown to be a means for recognizing convex silhouettes and for smoothing the effects of digitization in silhouettes. We describe a new method of computing the minimum-perimeter polygon (MPP) of any digitized silhouette satisfying certain constraints of connectedness and smoothness, and establish the underlying theory. Such a digitized silhouette is called a ``regular complex,'' in accordance with the usage in piecewise linear topology. The method makes use of the concept of a stretched string constrained to lie in the cellular boundary of the digitized silhouette. We show that, by properly marking the virtual as well as the real vertices of an MPP, the MPP can serve as a precise representation of any regular complex, and that this representation is often an economical one.

142 citations