The Annals of Applied Probability

The number of potential winners in Bradley–Terry model in random environment

Raphael Chetrite, Roland Diel, and Matthieu Lerasle

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Abstract

We consider a Bradley–Terry model in random environment where each player faces each other once. More precisely, the strengths of the players are assumed to be random and we study the influence of their distributions on the asymptotic number of potential winners. First, we prove that under moment and convexity conditions, the asymptotic probability that the best player wins is 1. The convexity condition is natural when the distribution of strengths is unbounded and, in the bounded case, when this convexity condition fails the number of potential winners grows at a rate depending on the tail of the distribution. We also study the minimal strength required for an additional player to win in this last case.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1372-1394.

Dates
Received: October 2015
Revised: July 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451225

Digital Object Identifier
doi:10.1214/16-AAP1231

Mathematical Reviews number (MathSciNet)
MR3678473

Zentralblatt MATH identifier
1372.60138

Subjects
Primary: 60K37: Processes in random environments 60G70: Extreme value theory; extremal processes 60K40: Other physical applications of random processes

Keywords
Bradley–Terry model random environment paired comparisons extreme values

Citation

Chetrite, Raphael; Diel, Roland; Lerasle, Matthieu. The number of potential winners in Bradley–Terry model in random environment. Ann. Appl. Probab. 27 (2017), no. 3, 1372--1394. doi:10.1214/16-AAP1231. https://projecteuclid.org/euclid.aoap/1500451225


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References

  • Ben-Naim, E. and Hengartner, N. W. (2007). Efficiency of competitions. Phys. Rev. E 76 026106.
  • Bradley, R. A. (1976). Science, statistics, and paired comparisons. Biometrics 32 213–232.
  • Bradley, R. A. and Terry, M. E. (1952). Rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika 39 324–345.
  • Cattelan, M. (2012). Models for paired comparison data: A review with emphasis on dependent data. Statist. Sci. 27 412–433.
  • David, H. A. (1988). The Method of Paired Comparisons, 2nd ed. Griffin’s Statistical Monographs & Courses 41. Charles Griffin & Co., London.
  • Davidson, R. (1970). On extending the Bradley–Terry model to accomodate ties in paired comparison experiments. J. Amer. Statist. Assoc. 65 317–328.
  • Davidson, R. R. and Beaver, R. J. (1977). On extending the Bradley–Terry model to incorporate within-pair order effects. Biometrics 33 693–702.
  • de Gennes, P. G. (1979). Scaling Concept in Polymer Physics. Cornell Univ. Press, Ithaca, NY.
  • Dembo, A. and Zeitouni, O. (2002). Large deviations and applications. In Handbook of Stochastic Analysis and Applications. Statist. Textbooks Monogr. 163 361–416. Dekker, New York.
  • Drewitz, A. and Ramírez, A. F. (2014). Selected topics in random walks in random environment. In Topics in Percolative and Disordered Systems. Springer Proc. Math. Stat. 69 23–83. Springer, New York.
  • Ford, L. R. Jr. (1957). Solution of a ranking problem from binary comparisons. Amer. Math. Monthly 64 28–33.
  • Glickman, M. E. and Jensen, S. T. (2005). Adaptive paired comparison design. J. Statist. Plann. Inference 127 279–293.
  • Hastie, T. and Tibshirani, R. (1998). Classification by pairwise coupling. Ann. Statist. 26 451–471.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • Hunter, D. R. (2004). MM algorithms for generalized Bradley–Terry models. Ann. Statist. 32 384–406.
  • McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989). London Mathematical Society Lecture Note Series 141 148–188. Cambridge Univ. Press, Cambridge.
  • Pisier, G. (1983). Some applications of the metric entropy condition to harmonic analysis. In Banach Spaces, Harmonic Analysis, and Probability Theory (Storrs, Conn., 1980/1981). Lecture Notes in Math. 995 123–154. Springer, Berlin.
  • Rao, P. V. and Kupper, L. L. (1967). Ties in paired-comparison experiments: A generalization of the Bradley–Terry model. J. Amer. Statist. Assoc. 62 194–204.
  • Simons, G. and Yao, Y.-C. (1999). Asymptotics when the number of parameters tends to infinity in the Bradley–Terry model for paired comparisons. Ann. Statist. 27 1041–1060.
  • Sire, C. and Redner, S. (2009). Understanding baseball team standings and streaks. Eur. Phys. J. B 67 473–481.
  • Yan, T., Yang, Y. and Xu, J. (2012). Sparse paired comparisons in the Bradley–Terry model. Statist. Sinica 22 1305–1318.
  • Zeitouni, O. (2012). Random walks in random environment. In Computational Complexity. Vols. 16 2564–2577. Springer, New York.
  • Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung. Math. Z. 29 436–460.