  Competing urns refers to the random experiment where m balls are dropped,
randomly and independently, into urns 1,...,n. Formally, we have a random map
$\sigma$ from {1,...,m} to {1,...,n} with the $\sigma(i)$'s i.i.d. With $x_j$
the indicator of the event that at least $t_j$ balls land in urn j (for some
threshold $t_j$), we prove conditional negative association for the random
variables $x_1,...,x_n$. We mostly deal with the more general situation in
which the $\sigma(i)$'s need not be identically distributed, proving results
which imply conditional negative association in the i.i.d. case. Some of the
results--particularly Lemma 8 on graph orientations--are thought to be of
independent interest. We also give a counterexample to a negative correlation
conjecture of D. Welsh, a strong version of a (still open) conjecture of G.
Farr.
