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Stable sets of weak tournaments

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. | Yugoslav Journal of Operations Research 14 (2004), Number 1, 33-40 STABLE SETS OF WEAK TOURNAMENTS Somdeb LAHIRI School of Economic and Business Sciences University of Witwatersrand at Johannesburg South Africa lahiris@sebs.wits.ac.za Received: October 2003 / Accepted: January 2004 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. Keywords: Stable sets, weak tournaments, acyclic, quasi-transitive. 1. INTRODUCTION An abiding problem in choice theory has been the one that characterizes those choice functions which are obtained as a result of some kind of optimization. Specifically, the endeavour has concentrated largely on finding a binary relation (if there be any) whose best elements coincide with observed choices. An adequate survey of this line of research till the mid eighties is available in Moulin [1985]. Miller [1977], [1980], introduces the concept of a tournament, which is an asymmetric and complete binary relation. Such binary relations arise very naturally in majority voting situations, where one candidate defeats another by a strict majority of votes. A consequence of majority voting and hence of the tournament it generates on the set of alternatives is the well known Condorcet paradox: the tournament may fail to exhibit transitivity and thus no alternative qualifies as a best alternative. This paradoxical situation called for alternative solution concepts for tournaments, which were .