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Imperfections in Random Tournaments

Published online by Cambridge University Press:  01 March 1997

A. D. BARBOUR ,
ANANT P. GODBOLE  and
JINGHUA QIAN
A. D. BARBOUR
Affiliation:
Abteilung für Angewandte Mathematik, Universität Zürich-Irchel, Winterthurerstrasse 190, CH 8057 Zürich, Switzerland (e-mail: [email protected])
ANANT P. GODBOLE
Affiliation:
Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295, USA (e-mail: [email protected])
JINGHUA QIAN
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, USA (e-mail: [email protected])
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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 A ⊂ V, |A| = k, is said to be beaten if there exists a player y ∉ A such that y beats x for each x ∈ A. 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
Information
Combinatorics, Probability and Computing , Volume 6 , Issue 1 , March 1997 , pp. 1 - 15
Copyright
1997 Cambridge University Press

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