Imperfections in Random Tournaments

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A tournament T on a set V of n players is an orientation of the edges of the complete graph Kn on V; Twill be called a random tournament of the directions of these edges are determined by a sequence {Yj: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.

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Combinatorics Probability and Computing