Hostname: page-component-5d59c44645-mrcq8 Total loading time: 0 Render date: 2024-03-02T23:00:47.649Z Has data issue: false hasContentIssue false

Imperfections in Random Tournaments

Published online by Cambridge University Press:  01 March 1997

A. D. BARBOUR
Affiliation:
Abteilung für Angewandte Mathematik, Universität Zürich-Irchel, Winterthurerstrasse 190, CH 8057 Zürich, Switzerland (e-mail: adb@amath.unizh.ch)
ANANT P. GODBOLE
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295, USA (e-mail: anant@mtu.edu)
JINGHUA QIAN
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, USA (e-mail: jqian@diamond.tufts.edu)

Abstract

A tournament T on a set V of n players is an orientation of the edges of the complete graph Kn on V; T will be called a random tournament if the directions of these edges are determined by a sequence {Yj[ratio ]j = 1, …, (n2)} of independent coin flips. If (y, x) is an edge in a (random) tournament, we say that y beats x. A set AV, |A| = k, is said to be beaten if there exists a player yA such that y beats x for each xA. If such a y does not exist, we say that A is unbeaten. A (random) tournament on V is said to have property Sk if each k-element subset of V is beaten. In this paper, we use the Stein–Chen method to show that the probability distribution of the number W0 of unbeaten k-subsets of V can be well-approximated by that of a Poisson random variable with the same mean; an improved condition for the existence of tournaments with property Sk is derived as a corollary. A multivariate version of this result is proved next: with Wj representing the number of k-subsets that are beaten by precisely j external vertices, j = 0, 1, …, b, it is shown that the joint distribution of (W0, W1, …, Wb) can be approximated by a multidimensional Poisson vector with independent components, provided that b is not too large.

Type
Research Article
Copyright
1997 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)