Negative Dependence in Knockout Tournaments

TL;DR

This paper investigates various forms of negative dependence in knockout tournaments, establishing new dependence properties for different tournament models with random and non-random draws, correcting previous proofs and extending theoretical understanding.

Contribution

It introduces and proves new negative dependence properties for knockout tournaments, including negative regression dependence and tail dependencies, with corrections to earlier results.

Findings

S is NA and NRTD in non-random draw tournaments with equal strength

S is not generally NRD or NLTD in non-random draw tournaments

Established negative dependence properties for random draw tournaments

Abstract

Negative dependence in tournaments has received attention in the literature. The property of negative orthant dependence (NOD) was proved for different tournament models with a special proof for each model. For general round-robin tournaments and knockout tournaments with random draws, Malinovsky and Rinott (2023) unified and simplified many existing results in the literature by proving a stronger property, negative association (NA). For a knockout tournament with a non-random draw, Malinovsky and Rinott (2023) presented an example to illustrate that S is NOD but not NA. However, their proof is not correct. In this paper, we establish the properties of negative regression dependence (NRD), negative left-tail dependence (NLTD) and negative right-tail dependence (NRTD) for a knockout tournament with a random draw and with players being of equal strength. For a knockout tournament with a non-random draw and with equal strength, we prove that S is NA and NRTD, while S is, in general, not NRD or NLTD.