Volume 77, Number 3, February 2007
Article Number 30005
Number of page(s) 5
Section General
Published online 24 January 2007
EPL, 77 (2007) 30005
DOI: 10.1209/0295-5075/77/30005

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 in final form 8 December 2006; published February 2007
published online 24 January 2007

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.

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

© Europhysics Letters Association 2007