## Abstract

Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a nonempty subset of the alternatives, play an important role in the mathematical social sciences at large. For any given tournament solution \(S\), there is another tournament solution Open image in new window which returns the union of all inclusion-minimal sets that satisfy \(S\)-retentiveness, a natural stability criterion with respect to \(S\). Schwartz’s tournament equilibrium set (\({ TEQ }\)) is defined recursively as Open image in new window . In this article, we study under which circumstances a number of important and desirable properties are inherited from \(S\) to Open image in new window . We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that “approximate” \({ TEQ }\), which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding \({ TEQ }\), which establishes Open image in new window —a refinement of the top cycle—as an interesting new tournament solution.

## Notes

### Acknowledgments

This material is based on work supported by the Deutsche Forschungsgemeinschaft under grants BR 2312/3-3, BR 2312/6-1, BR 2312/7-1, and FI 1664/1-1. Preliminary versions of the results were presented at the Workshop on Algorithmic Aspects of Game Theory and Social Choice (Auckland, February 2010), the Dagstuhl Seminar on Computational Foundations of Social Choice (Dagstuhl, March 2010), the Doctoral School on Computational Social Choice (Estoril, April 2010), and the 9th International Joint Conference on Autonomous Agents and Multi-Agent Systems (Toronto, May 2010).

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