Scaling in tournamentsE. 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 , 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 . Different decay exponents are found analytically for sequential dynamics, , and parallel dynamics, . 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