  Consider $n$ players whose "scores" are independent and identically
distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F$. We
pay special attention to the cases where (i) $F$ is geometric with parameter
$p\to0$ and (ii) $F$ is uniform on $\{1,2,...,N\}$; the latter case clearly
corresponds to the classical occupancy problem. The quantities of interest to
us are, first, the $U$-statistic $W$ which counts the number of "ties" between
pairs $i,j$; second, the univariate statistic $Y_r$, which counts the number of
strict $r$-way ties between contestants, i.e., episodes of the form
${X_i}_1={X_i}_2=...={X_i}_r$; $X_j\ne {X_i}_1;j\ne i_1,i_2,...,i_r$; and, last
but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},...,Y_B)$. We
provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$
under some general conditions. New results on the joint distribution of cell
counts in the occupancy problem are derived as a corollary.
