Fixing Knockout Tournaments With Seeds

TL;DR

This paper examines how seed constraints affect the ability to manipulate knockout tournaments, showing that seeds can both restrict and preserve the likelihood of a player winning, depending on the scenario.

Contribution

It introduces new structural conditions for tournament manipulation considering seeds and analyzes the probabilistic and computational aspects of manipulation with seed constraints.

Findings

Seed constraints can limit manipulation strategies.

Random match outcomes still favor all players as potential winners.

Deciding manipulation feasibility with seeds is computationally complex.

Abstract

Knockout tournaments constitute a popular format for organizing sports competitions. While prior results have shown that it is often possible to manipulate a knockout tournament by fixing the bracket, these results ignore the prevalent aspect of player seeds, which can significantly constrain the chosen bracket. We show that certain structural conditions that guarantee that a player can win a knockout tournament without seeds are no longer sufficient in light of seed constraints. On the other hand, we prove that when the pairwise match outcomes are generated randomly, all players are still likely to be knockout winners under the same probability threshold with seeds as without seeds. In addition, we investigate the complexity of deciding whether a manipulation is possible when seeds are present.