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WINNER PLAYS COMPETITION MODELS

Published online by Cambridge University Press:  18 October 2019

Yang Cao Open the ORCID record for Yang Cao [Opens in a new window]  and
Sheldon M. Ross
Yang Cao
Affiliation:
Department of Industrial and Systems Engineering, University of Southern California Los Angeles CA 90089, USA E-mail: cao573@usc.edu; smross@usc.edu
Sheldon M. Ross
Affiliation:
Department of Industrial and Systems Engineering, University of Southern California Los Angeles CA 90089, USA E-mail: cao573@usc.edu; smross@usc.edu
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Abstract

Suppose there are n players in an ongoing competition, with player i having value vi, and suppose that a game between i and j is won by i with probability vi/(vi + vj). Consider the winner plays competition where in each stage two players play a game, and the winner keeps playing in the next game. We consider two models for choosing its opponent, analyze both models as Markov chains, and determine their stationary probabilities as well as other quantities of interest.

Keywords

winner playsMarkov chainstationary probabilitiestime reversible
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 236 - 241
Copyright
Copyright © Cambridge University Press 2019

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References

1.Adler, I., Cao, Y., Karp, R., Peköz, E.A., & Ross, S.M. (2017). Random knockout tournaments. Operations Research 65(6): 1589–1596.CrossRefGoogle Scholar
2.Azizi, M.J., Cao, Y., & Ross, S.M. (2019). Some new selection procedures. Manuscript in preparation.Google Scholar
3.Bradley, R.A. & Terry, M.E. (1952). Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika 39(3/4): 324–345.Google Scholar
4.Cao, Y. & Ross, S.M. (2019) Winner plays structure in random knockout tournaments. Probability in the Engineering and Informational Sciences 33(4): 500–510.CrossRefGoogle Scholar
5.Caron, F. & Doucet, A. (2012). Efficient bayesian inference for generalized bradley–terry models. Journal of Computational and Graphical Statistics 21(1): 174–196.CrossRefGoogle Scholar
6.Cattelan, M., Varin, C., & Firth, D. (2013). Dynamic bradley–terry modelling of sports tournaments. Journal of the Royal Statistical Society: Series C (Applied Statistics) 62(1): 135–150.Google Scholar
7.Guiver, J. & Snelson, E. (2009). Bayesian inference for plackett-luce ranking models. In Proceedings of the 26th Annual International Conference on Machine Learning (ICML '09). ACM, New York, NY, USA, 377–384.CrossRefGoogle Scholar
8.Hastie, T. & Tibshirani, R. (1998). Classification by pairwise coupling. The Annals of Statistics 26(2): 451–471.CrossRefGoogle Scholar
9.Hunter, D.R. (2004). Mm algorithms for generalized bradley-terry models. The annals of statistics 32(1): 384–406.CrossRefGoogle Scholar
10.McHale, I. & Morton, A. (2011). A bradley-terry type model for forecasting tennis match results. International Journal of Forecasting 27(2): 619–630.CrossRefGoogle Scholar
11.Ross, S.M. (2019). Introduction to probability models (12th ed). Academic Press.Google Scholar
12.Ross, S.M. & Zhang, Z. (2019). Estimating the strength of sport teams. Manuscript in preparation.Google Scholar
13.Zermelo, E. (1929). Die berechnung der turnier-ergebnisse als ein maximumproblem der wahrscheinlichkeitsrechnung. Mathematische Zeitschrift 29(1): 436–460.CrossRefGoogle Scholar