Scaling in Tournaments

TL;DR

This paper models stochastic single-elimination tournaments where weaker players can win, analyzing how the initial strength distribution influences the winner's rank and revealing different decay behaviors for sequential and parallel match dynamics.

Contribution

It introduces a probabilistic model of tournaments with stochastic outcomes and derives analytical decay exponents for winner ranks under different dynamics.

Findings

Winner's rank decays algebraically with the number of players.

Distinct decay exponents for sequential and parallel dynamics.

Model accurately describes US college basketball tournament statistics.

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<=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_* ~ N^(-beta). Different decay exponents are found analytically for sequential dynamics, beta_seq=1-2q, and parallel dynamics, beta_par=1+[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.