---
title: "When Does Schwartz Conjecture Hold?"
authors:
  - "Matthias Mnich"
  - "Yash Raj Shrestha"
  - "Yongjie Yang"
doi: ""
url: "https://ijcai.org/proceedings/2015/91"
pdf: "https://ijcai.org/proceedings/2015/0091.pdf"
venue: "Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence"
year: 2015
pages: "603-608"
track: "Main Track"
keywords:
---

# When Does Schwartz Conjecture Hold?

Matthias Mnich, Yash Raj Shrestha, Yongjie Yang

*Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence*, pages 603-608.

## Abstract

In 1990, Thomas Schwartz proposed the conjecture that every nonempty tournament has a unique minimal TEQ-retentive set (TEQ stands for tournament equilibrium set). A weak variant of Schwartz's Conjecture was recently proposed by Felix Brandt. However, both conjectures were disproved very recently by two counterexamples. In this paper, we prove sufficient conditions for infinite classes of tournaments that satisfy Schwartz's Conjecture and Brandt's Conjecture. Moreover, we prove that TEQ can be calculated in polynomial time in several infinite classes of tournaments. Furthermore, our results reveal some structures that are forbidden in every counterexample to Schwartz's Conjecture.