  Negative dependence of sequences of random variables is often an interesting
characteristic of their distribution, as well as a useful tool for studying
various asymptotic results, including central limit theorems, Poisson
approximations, the rate of increase of the maximum, and more. In the study of
probability models of tournaments, negative dependence of participants'
outcomes arises naturally with application to various asymptotic results. In
particular, the property of negative orthant dependence was proved in several
articles for different tournament models, with a special proof for each model.
In this note we unify these results by proving a stronger property, negative
association, a generalization leading to a very simple proof. We also present a
natural example of a knockout tournament where the scores are negatively
orthant dependent but not negatively associated. The proof requires a new
result on a preservation property of negative orthant dependence that is of
independent interest.
