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Negative dependence in knockout tournaments

Part of: Multivariate analysis Distribution theory - Probability

Published online by Cambridge University Press:  04 December 2025

Yuting Su ,
Zhenfeng Zou Open the ORCID record for Zhenfeng Zou [Opens in a new window]  and
Taizhong Hu Open the ORCID record for Taizhong Hu [Opens in a new window]
Yuting Su*
Affiliation:
University of Science and Technology of China
Zhenfeng Zou*
Affiliation:
University of Science and Technology of China
Taizhong Hu*
Affiliation:
University of Science and Technology of China
*
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China.
***Postal address: School of Public Affairs, University of Science and Technology of China, Hefei, Anhui 230026, China. Email address: zfzou@ustc.edu.cn
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China.
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Abstract

Negative dependence in tournaments has received attention in the literature. The property of negative orthant dependence (NOD) was proved for different tournament models with a special proof for each model. For general round-robin tournaments and knockout tournaments with random draws, Malinovsky and Rinott (2023) unified and simplified many existing results in the literature by proving a stronger property, negative association (NA). For a knockout tournament with a non-random draw, they presented an example to illustrate that ${\boldsymbol{S}}$ is NOD but not NA. However, their proof is not correct. In this paper, we establish the properties of negative regression dependence (NRD), negative left-tail dependence (NLTD), and negative right-tail dependence (NRTD) for a knockout tournament with a random draw and with players being of equal strength. For a knockout tournament with a non-random draw and with equal strength, we prove that ${\boldsymbol{S}}$ is NA and NRTD, while ${\boldsymbol{S}}$ is, in general, not NRD or NLTD.

Keywords

Negative regression dependencenegative left-tail dependencenegative right-tail dependencenegative associationnegative supermodular dependence

MSC classification

Primary: 62H05: Characterization and structure theory
Secondary: 60E15: Inequalities; stochastic orderings

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Adler, I., Cao, Y., Karp, R., Peköz, E. A. and Ross, S. M. (2017). Random knockout tournaments. Operat. Res. 65, 1589–1596.10.1287/opre.2017.1657CrossRefGoogle Scholar
Alam, K. and Saxena, K. M. L. (1981). Positive dependence in multivariate distributions. Commun. Statist. Theory Meth. 10, 1183–1196.Google Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD (first printed in 1975).Google Scholar
Bäuerle, N. (1997). Monotonicity results for MR/GI/1 queues. J. Appl. Prob. 34, 514–524.10.2307/3215390CrossRefGoogle Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1985). A concept of negative dependence through stochastic ordering. Statist. Prob. Lett. 3, 81–86.10.1016/0167-7152(85)90029-XCrossRefGoogle Scholar
Bruss, F. T. and Ferguson, T. S. (2018). Testing equality of players in a round-robin tournament. Math. Sci. 43, 125–136.Google Scholar
Chen, Y., Embrechts, P. and Wang, R. (2025). An unexpected stochastic dominance: Pareto distributions, dependence, and diversification. Operat. Res. 73, 1336–1344.10.1287/opre.2022.0505CrossRefGoogle Scholar
Cheung, K. C. and Lo, A. (2014). Characterizing mutual exclusivity as the strongest negative multivariate dependence structure. Insurance Math. Econom. 55, 180–190.10.1016/j.insmatheco.2014.01.001CrossRefGoogle Scholar
Christofides, T. C. and Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association. J. Multivariate Anal. 88, 138–151.10.1016/S0047-259X(03)00064-2CrossRefGoogle Scholar
Dubhashi, D. and Ranjan, D. (1998). Balls and bins: A study in negative dependence. Random Structures Algorithms 13, 99–124.10.1002/(SICI)1098-2418(199809)13:2<99::AID-RSA1>3.0.CO;2-M3.0.CO;2-M>CrossRefGoogle Scholar
Hu, T. (2000). Negatively superadditive dependence of random variables with applications. Chinese J. Appl. Prob. Statist. 16, 133–144.Google Scholar
Hu, T. and Pan, X. (1999). Preservation of multivariate dependence under multivariate claim models. Insurance Math. Econom. 25, 171–179.10.1016/S0167-6687(99)00032-3CrossRefGoogle Scholar
Hu, T. and Xie, C. (2006). Negative dependence in the balls and bins experiment with applications to order statistics. J. Multivariate Anal. 97, 1342–1354.10.1016/j.jmva.2005.09.008CrossRefGoogle Scholar
Hu, T. and Yang, J. (2004). Further developments on sufficient conditions for negative dependence of random variables. Statist. Prob. Lett. 66, 369–381.10.1016/j.spl.2003.10.023CrossRefGoogle Scholar
Joag-dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.10.1214/aos/1176346079CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities, II: Multivariate reverse rule distributions. J. Multivariate Anal. 10, 499–516.10.1016/0047-259X(80)90066-4CrossRefGoogle Scholar
Lauzier, J.-G., Lin, L. and Wang, R. (2023). Pairwise counter-monotonicity. Insurance Math. Econom. 111, 279–287.10.1016/j.insmatheco.2023.05.006CrossRefGoogle Scholar
Malinovsky, Y. and Moon, J. W. (2022). On the negative dependence inequalities and maximal score in round-robin tournaments. Statist. Prob. Lett. 185, 109432.10.1016/j.spl.2022.109432CrossRefGoogle Scholar
Malinovsky, Y. and Rinott, Y. (2023). On tournaments and negative dependence. J. Appl. Prob. 60, 945–954.10.1017/jpr.2022.104CrossRefGoogle Scholar
Moon, J. W. (2013). Topics on Tournaments. Available at https://www.gutenberg.org/ebooks/42833.Google Scholar
Puccetti, G. and Wang, R. (2015). Extremal dependence concepts. Statist. Sci. 30, 485–517.10.1214/15-STS525CrossRefGoogle Scholar
Ross, S. M. (2022). Team’s seasonal win probabilities. Prob. Eng. Inf. Sci. 36, 988–998.10.1017/S026996482100019XCrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.10.1007/978-0-387-34675-5CrossRefGoogle Scholar
Wang, B. and Wang, R. (2016). Joint mixability. Math. Operat. Res. 41, 808–826.10.1287/moor.2015.0755CrossRefGoogle Scholar