Probabilistic Transitivity in Sports

25 Pages Posted: 3 Sep 2014 Last revised: 14 Nov 2018

See all articles by Johannes Tiwisina

Johannes Tiwisina

Bielefeld University - Center for Mathematical Economics

Philipp Külpmann

University of Vienna - Vienna Center for Experimental Economics

Date Written: August 1, 2018


We seek to find the statistical model that most accurately describes empirically observed results in sports. The idea of a transitive relation concerning the team strengths is implemented by imposing a set of constraints on the outcome probabilities. We theoretically investigate the resulting optimization problem and draw comparisons to similar problems from the existing literature including the linear ordering problem and the isotonic regression problem. Our optimization problem turns out to be very complicated to solve. We propose a branch and bound algorithm for an exact solution and for larger sets of teams a heuristic method for quickly finding a "good" solution. Finally we apply the described methods to panel data from soccer, American football and tennis and also use our framework to compare the performance of empirically applied ranking schemes.

Keywords: stochastic transitivity, trinomial, geometric optimization, ranking, branch and bound, linear ordering problem, ELO, tabu search, football, soccer, tennis, bundesliga, NFL, ATP

JEL Classification: L83, C61, C63, C81

Suggested Citation

Tiwisina, Johannes and Külpmann, Philipp, Probabilistic Transitivity in Sports (August 1, 2018). Institute of Mathematical Economics Working Paper No. 520, Available at SSRN: or

Johannes Tiwisina (Contact Author)

Bielefeld University - Center for Mathematical Economics ( email )

Postfach 10 01 31
Bielefeld, D-33501

Philipp Külpmann

University of Vienna - Vienna Center for Experimental Economics ( email )

Oskar-Morgenstern-Platz 1
Vienna, Vienna 1090

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