Stable sets of weak tournaments

Authors

  • Somdeb Lahiri School of Economic and Business Sciences University of Witwatersrand at Johannesburg South Africa

DOI:

https://doi.org/10.2298/YJOR0401033L

Keywords:

stable sets, weak tournaments, acyclic, quasi-transitive

Abstract

In this paper we obtain conditions on weak tournaments, which guarantee that every non-empty subset of alternatives admits a stable set. We also show that there exists a unique stable set for each non-empty subset of alternatives which coincides with its set of best elements, if and only if, the weak tournament is quasi-transitive. A somewhat weaker version of this result, which is also established in this paper, is that there exists a unique stable set for each non-empty subset of alternatives (: which may or may not coincide with its set of best elements), if and only if the weak tournament is acyclic.

References

Gillies, D.B., “ Solutions to general zero-sum games”, in: A.W. Tucker and R.D. Luce (eds.), Contributions to the Theory of Games, Vol.4, Princeton Univ. Press, Princeton, 1959.

Kim, K.H., and Roush, F.W., Introduction to Mathematical Consensus Theory, Lecture Notes in Pure and Applied Mathematics, Vol. 59, Marcel Dekker, Inc., 1980.

Lucas, W.F., “Von Neumann-Morgenstern stable sets”, Chapter 17 in: R. Aumann and S.Hart (eds.), Handbook of Game Theory, Vol.1, Elsevier, Amsterdam, 1992.

Miller, N., “Graph – theoretical approaches to the theory of voting”, American Journal of Political Science, 21 (4) (1977) 769-803.

Miller, N., “ A new solution set for tournaments and majority voting: further graph theoretical approaches to the theory of voting”, American Journal of Political Science, 24 (1981) 68-96.

Moulin, H., “Choice functions over a finite set: A summary”, Social Choice Welfare, 2 (1985) 147-160.

Moulin, H., “Choosing from a tournament”, Social Choice Welfare, 3 (1986) 271-291.

Peris, J.E., and Subiza, B., “Condorcet choice correspondences for weak tournaments”, Social Choice Welfare, 16 (1999) 217-231.

Roth, A., and Sotomayor, M., “Two-sided matching”, in: Econometric Society Monograph 18, Cambridge University Press, 1990.

Downloads

Published

2004-03-01

Issue

Section

Research Articles