Who Can Win a Single-Elimination Tournament?

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 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 = Θ(ln n/n); prior work only had such results for p ≥ Ω( ln n/n). We also establish new results for significantly more general generative models.

Robust Bounds on Choosing from Large Tournaments

“…A result of this flavor has been shown byKim et al (2017) in the context of single-elimination winners.…”

“…For example, the uniform random model (Fey 2008; Scott and Fey 2012) corresponds to taking p i, j = 1/2 for all i, j in the general random model. The Condorcet random model (Frank 1968;Łuczak et al 1996;Vassilevska Williams 2010;Kim et al 2017) corresponds to taking p i, j = p for all i < j in the general random model, for some fixed value of p. Following standard terminology, we say that an event occurs "with high probability" or "almost surely" if the probability that the event occurs converges to 1 as n, the number of alternatives, goes to infinity.…”

Refining Tournament Solutions via Margin of Victory

“…A long line of work has investigated various forms of bribery and manipulation in tournaments. This includes manipulating the tournament bracket to help a certain candidate win the tournament (Vu, Altman, and Shoham 2009;Vassilevska Williams 2010;Kim, Suksompong, and Vassilevska Williams 2017;Aziz et al 2018) and bribing players to intentionally lose matches (Russell and Walsh 2009;Kim and Vassilevska Williams 2015;Mattei et al 2015;Konicki and Vassilevska Williams 2019). In particular, Russell and Walsh (2009) considered a model where only a given subset of edges can be reversed, while other edges are assumed to be fixed.…”

Weak transitivity and agenda control for extended stepladder tournaments

“…Strengthening the notion of a king node was shown to allow for efficiently constructible seedings making kings winners in balanced knockout tournaments (cf. Kim et al 2017;Kim and Williams 2015). We refer the reader to the work of Williams (2016) for an excellent and detailed survey of the recent literature.…”