Minimal Stable Sets in Tournaments

TL;DR

This paper introduces a systematic methodology for defining tournament solutions through minimal stable sets, creating an infinite hierarchy that includes known solutions and a new one, the minimal extending set.

Contribution

It develops a unified framework for tournament solutions using maximal qualified subsets and minimal stable sets, introducing the new minimal extending set.

Findings

Hierarchy includes top cycle, uncovered set, Banks set, and more.

Introduces the minimal extending set, conjectured to refine existing solutions.

Provides a systematic approach to extend and unify tournament solution concepts.

Abstract

We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncovered set, the Banks set, the minimal covering set, the tournament equilibrium set, the Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set.