A Ladder Tournament

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.