The solution of generalized stable sets and its refinement

Abstract

In this paper, we review some results of the solution of generalized stable sets introduced by Van Deemen (1991) as a variant of stable sets for abstract decision problems. This solution will be investigated and reviewed for the more general case of irreflexive but not necessarily asymmetric or complete dominance relations. It is proven that the fundamental properties of this solution are preserved also for this kind of dominance relations. Two main shortcomings of the solution of generalized stable sets are firstly that it may contain Pareto-suboptimal alternatives when the dominance relation is derived from pairwise majority comparison, and secondly that it may fail to discriminate among the alternatives under consideration when the dominance relation is a Hamilton cycle, i.e. a cycle that includes all alternatives. A refinement of generalized stable sets is proposed in order to address these two shortcomings. This refinement is called the solution of undisturbed generalized stable sets. It will be shown that undisturbed generalized stable sets are Pareto optimal and have discriminating power in the case of Hamilton cycles. In addition, and perhaps more important, we prove that this refinement is always a subset of the solutions of the uncovered set and of the unsurpassed set. This implies that undisturbed generalized stable set also may be seen as a refinement of both the uncovered set and the unsurpassed set.