On Tournaments and Negative Dependence

TL;DR

This paper unifies and strengthens the understanding of negative dependence in tournament models, proving negative association and introducing a new preservation property, with implications for asymptotic probability results.

Contribution

It provides a unified proof of negative association in tournament models and introduces a new preservation property of negative orthant dependence.

Findings

Negative association holds in various tournament models.

A new preservation property of negative orthant dependence is established.

An example shows negative orthant dependence does not imply negative association.

Abstract

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.