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On the distribution of winners’ scores in a round-robin tournament

Part of: Graph theory Nonparametric inference

Published online by Cambridge University Press:  06 August 2021

Yaakov Malinovsky Open the ORCID record for Yaakov Malinovsky [Opens in a new window]
Yaakov Malinovsky*
Affiliation:
Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA. E-mail: yaakovm@umbc.edu
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Abstract

In a classical chess round-robin tournament, each of $n$ players wins, draws, or loses a game against each of the other $n-1$ players. A win rewards a player with 1 points, a draw with 1/2 point, and a loss with 0 points. We are interested in the distribution of the scores associated with ranks of $n$ players after ${{n \choose 2}}$ games, that is, the distribution of the maximal score, second maximum, and so on. The exact distribution for a general $n$ seems impossible to obtain; we obtain a limit distribution.

Keywords

ExtremesNegative correlation

MSC classification

Primary: 62G32: Statistics of extreme values; tail inference
Secondary: 05C20: Directed graphs (digraphs), tournaments
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 1098 - 1102
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

Berman, S.M. (1964). Limiting distribution of the maximum of a diffusion process. Annals of Mathematical Statistics 35: 319–329.CrossRefGoogle Scholar
Courant, R. & Robbins, H. (1996). What is mathematics? An elementary approach to ideas and methods, 2nd ed., revised by Ian Stewart. New York: Oxford University Press.Google Scholar
Cramér, H. (1946). Mathematical methods of statistics. Princeton: Princeton University Press.Google Scholar
Galambos, J. (1987). The asymptotic theory of extreme order statistics. 2nd ed. Malabar, FL: Krieger.Google Scholar
Grimmett, G.R. & Stirzaker, D.R. (2020). Probability and random processes, 4th ed. Oxford: Oxford University Press.Google Scholar
Gumbel, E.J. (1958). Statistics of extemes. New York: Columbia University Press.CrossRefGoogle Scholar
Huber, P.J. (1963). A remark on a paper of Trawinski and David entitled: Selection of the best treatment in a paired comparison experiment. Annals of Mathematical Statistics 34: 92–94.CrossRefGoogle Scholar
Leadbetter, M.R. (2017). Extremes under dependence–historical development and parallels with central limit theory. In Extreme events in finance, Wiley Handb. Finance Eng. Econom. Edited by Francois Longin. Hoboken: Wiley, pp. 11–23.Google Scholar
Leadbetter, M.R., Lindgren, G., & Rootzén, H. (1983). Extremes and related properties of random sequences and processes. Springer Series in Statistics. New York-Berlin: Springer-Verlag.CrossRefGoogle Scholar
Lehmann, E.L. (1998). Elements of large-sample theory. New York: Springer.Google Scholar
Malinovsky, Y. & Moon, J.W. (2021). On the negative dependence inequalities and maximal score in round-robin tournament. https://arxiv.org/abs/2104.01450.Google Scholar
Moon, J.W. (2013). Topics on tournaments. [Publicaly available on website of Project Gutenberg https://www.gutenberg.org/ebooks/42833].Google Scholar
Ross, S.M. (2021). Team's seasonal win probabilities. Probability in the Engineering and Informational Sciences. doi:10.1017/S026996482100019X. In press.CrossRefGoogle Scholar