A computational analysis of the tournament equilibrium set
- 89 Downloads
A recurring theme in the mathematical social sciences is how to select the “most desirable” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions proposed so far. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive solution concept refining both the Banks set and Dutta’s minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard, and thus does not admit a polynomial-time algorithm unless P equals NP. Furthermore, we propose a heuristic that significantly outperforms the naive algorithm for computing TEQ.
KeywordsDominance Relation Maximal Element Solution Concept Condorcet Winner Naive Algorithm
Unable to display preview. Download preview PDF.
- Bouyssou D, Marchant T, Pirlot M, Tsoukiàs A, Vincke P (2006) Evaluation and decision models: stepping stones for the analyst. Springer, BerlinGoogle Scholar
- Brandt F, Harrenstein P (2009) Characterization of dominance relations in finite coalitional games. Theory Decis (to appear)Google Scholar
- Conitzer V (2006) Computing Slater rankings using similarities among candidates. In: Proceedings of the 21st national conference on artificial intelligence (AAAI). AAAI Press, pp 613–619Google Scholar
- Condorcet Marquis de Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Imprimerie Royale, 1785. Facsimile published in 1972. Chelsea, New YorkGoogle Scholar
- Gillies DB (1959) Solutions to general non-zero-sum games. In: Tucker AW, Luce RD (eds) Contributions to the Theory of Games IV, vol 40. Annals of Mathematics Studies. Princeton University Press, New Jersy, pp 47–85Google Scholar
- Laffond G, Laslier J-F, Le Breton M (1993) More on the tournament equilibrium set. Math Sci Hum 31(123): 37–44Google Scholar
- Laslier J-F (1997) Tournament solutions and majority voting. Springer, BerlinGoogle Scholar
- Papadimitriou CH (1994) Computational complexity. Addison-Wesley, ReadingGoogle Scholar