Scaling in tournaments

E. Ben-Naim; S. Redner; F. Vazquez

doi:10.1209/0295-5075/77/30005

All issues

Volume 77 / No 3 (February 2007)

EPL, 77 3 (2007) 30005

Abstract

Issue		EPL Volume 77, Number 3, February 2007


Article Number		30005
Number of page(s)		5
Section		General
DOI		https://doi.org/10.1209/0295-5075/77/30005
Published online		24 January 2007

EPL, 77 (2007) 30005

Scaling in tournaments

E. Ben-Naim¹, S. Redner² and F. Vazquez¹^,2

¹ Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
² Department of Physics, Boston University - Boston, MA 02215, USA

Received: 26 July 2006
Accepted: 8 December 2006

Abstract

We study a stochastic process that mimics single-game elimination tournaments. In our model, the outcome of each match is stochastic: the weaker player wins with upset probability $q\leqslant 1/2$ , and the stronger player wins with probability $1-q$ . The loser is eliminated. Extremal statistics of the initial distribution of player strengths governs the tournament outcome. For a uniform initial distribution of strengths, the rank of the winner, $x_{*}$ , decays algebraically with the number of players, N, as $x_{*}\sim N^{-\beta }$ . Different decay exponents are found analytically for sequential dynamics, $\beta _{{\rm seq}}=1-2q$ , and parallel dynamics, $\beta_{\rm par}=1+\frac{\ln (1-q)}{\ln 2}$ . The distribution of player strengths becomes self-similar in the long time limit with an algebraic tail. Our theory successfully describes statistics of the US college basketball national championship tournament.

PACS: 01.50.Rt – Physics tournaments and contests / 02.50.-r – Probability theory, stochastic processes, and statistics / 89.75.Da – Systems obeying scaling laws

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