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Randomness in Competitions

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Abstract

We study the effects of randomness on competitions based on an elementary random process in which there is a finite probability that a weaker team upsets a stronger team. We apply this model to sports leagues and sports tournaments, and compare the theoretical results with empirical data. Our model shows that single-elimination tournaments are efficient but unfair: the number of games is proportional to the number of teams N, but the probability that the weakest team wins decays only algebraically with N. In contrast, leagues, where every team plays every other team, are fair but inefficient: the top \(\sqrt{N}\) of teams remain in contention for the championship, while the probability that the weakest team becomes champion is exponentially small. We also propose a gradual elimination schedule that consists of a preliminary round and a championship round. Initially, teams play a small number of preliminary games, and subsequently, a few teams qualify for the championship round. This algorithm is fair and efficient: the best team wins with a high probability and the number of games scales as N 9/5, whereas traditional leagues require N 3 games to fairly determine a champion.

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Notes

  1. Most of the results reported in this mini-review appear in [13–17]. For clarity, we present a number of additional plots including Figs. 1, 3, 5, 8. The scaling arguments in Sect. 4 are equivalent to those presented in [17].

  2. For deterministic competitions, q=0, the scaling function is exponential ψ(z)=e −z.

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Acknowledgements

We thank Micha Ben-Naim for help with data collection. We acknowledge support from DOE (DE-AC52-06NA25396) and NSF (DMR0227670, DMR0535503, & DMR-0906504).

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Authors and Affiliations

  1. Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    E. Ben-Naim

  2. Computing and Computer Science Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    N. W. Hengartner

  3. Department of Physics, Boston University, Boston, MA, 02215, USA

    S. Redner

  4. Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, 01187, Dresden, Germany

    F. Vazquez

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  1. E. Ben-Naim
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  2. N. W. Hengartner
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  3. S. Redner
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  4. F. Vazquez
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Correspondence to E. Ben-Naim.

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Ben-Naim, E., Hengartner, N.W., Redner, S. et al. Randomness in Competitions. J Stat Phys 151, 458–474 (2013). https://doi.org/10.1007/s10955-012-0648-x

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