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.