A Ladder Tournament

TL;DR

This paper formalizes the ranking rule of ladder tournaments, revealing their mathematical properties and relationships between rank and pivotal performance, highlighting their incomplete and non-transitive nature.

Contribution

It provides a formal analysis of ladder tournament rankings, establishing conditions for transitivity and exploring the link between rank and pivotality.

Findings

Ladder tournament rankings are generally incomplete and non-transitive.

If the ranking is complete, it is transitive with a union of transitive tournaments.

An individual's pivotability weakly increases with their rank.

Abstract

Ladder tournaments are widely used to rank individuals in real-world organizations and games. Their mathematical properties however are still poorly understood. We formalize the ranking rule generated by a ladder tournament, and we show that it is neither complete nor transitive in general. If it is complete, then it is transitive and its asymmetric component is a finite union of transitive tournaments. We also study the relationship between an individual's rank and his performance as measured by the frequency at which he is pivotal. We show an individual's pivotability is a weakly increasing function of his rank.