Who Can Win a Single-Elimination Tournament?

TL;DR

This paper investigates conditions under which a player can be guaranteed to win a single-elimination tournament through seeding adjustments, generalizes previous results, and explores probabilistic models for tournament outcomes.

Contribution

It provides new sufficient conditions for guaranteed wins, relates SE winners to other tournament solutions, and extends probabilistic models to broader settings.

Findings

Guaranteed existence of a seeding for a player to win under certain conditions

All players in a Condorcet Random Model tournament are SE winners at very low noise levels

Extended results to more general tournament generation models

Abstract

A single-elimination (SE) tournament is a popular way to select a winner in both sports competitions and in elections. A natural and well-studied question is the tournament fixing problem (TFP): given the set of all pairwise match outcomes, can a tournament organizer rig an SE tournament by adjusting the initial seeding so that their favorite player wins? We prove new sufficient conditions on the pairwise match outcome information and the favorite player, under which there is guaranteed to be a seeding where the player wins the tournament. Our results greatly generalize previous results. We also investigate the relationship between the set of players that can win an SE tournament under some seeding (so called SE winners) and other traditional tournament solutions. In addition, we generalize and strengthen prior work on probabilistic models for generating tournaments. For instance, we show that \emph{every} player in an $n$ player tournament generated by the Condorcet Random Model will be an SE winner even when the noise is as small as possible, $p=\Theta(\ln n/n)$; prior work only had such results for $p\geq \Omega(\sqrt{\ln n/n})$. We also establish new results for significantly more general generative models.