Winning a Tournament According to Bradley-Terry Probability Model

  • Shuyang Gao The George Washington University
  • Hosam Mahmoud
Keywords: Bradley-Terry model;, Multivariate Gaussian law;, Combinatorial probability;, Quantitative analysis of tournaments;


We analyze the chances of winning a tournament under the assumption that the probabilities of winning individual matches follow Bradley-Terry model [2]. We present an exact solution and show a few examples of its use. The examples are from California volleyball tournaments, the round of sixteen in the World Cup and the Champions League, the group stage of the Association of Tennis Professionals tournament, and the volleyball SuperLega in Italy. The computational complexity of the solution grows exponentially fastwith the number of teams and we seek approximations via multivariate Gaussian laws.


Billingsley, P. (2012). Probability and Measure, Anniversary Ed., Wiley.

Bradley, R. and Terry, M. (1952). Rank analysis of incomplete block designs: I. The Method of Paired Comparisons.

Biometrika 39, 324–345.

Chetrite, R., Diel, R. and Lerasle, M. (2017). The number of potential winners in Bradley-Terry model in random environment. The Annals of Applied Probability, 27(3), 1372–1394.

Fayomi, A., Majeed, R., Algarni, A., Akhtar, S., Jamal, F. and Nasir, J. (2022). Forecasting Tennis Match Results Using the Bradley-Terry Model. International Journal of Photoenergy.

Genz, A., Bretz, F., Miwa T., Mi, X., Leisch F., Scheipl, F. and Hothorn, T. (2020). mvtnorm: Multivariate Normal and t Distributions. R package version 1.1–1.

Ross, S. (2022). Team’s seasonal win probabilities. Probability in the Engineering and Informational Sciences, 36(4), 988–998.

Whelan, J. and Klein, J. (2021). Bradly-Terry modeling with multiple games outcomes with applications to college hockey. Mathematics for Applications, 10, 157–177.

How to Cite
Gao, S., & Mahmoud , H. (2023). Winning a Tournament According to Bradley-Terry Probability Model. Statistics, Optimization & Information Computing, 11(2), 332-344.
Research Articles