Cores and Stable Blockings

TLDR: In this article, the authors investigate stable blocking relations, i.e., those outcomes which are rejected by no coalition of agents, and show that the existence of such outcomes depends only on the stability of the blocking generated by a given mechanism.

Abstract: This chapter is devoted to the issue of stable outcomes, that is those outcomes which are rejected by no coalition of agents. The existence of such outcomes depends only on the stability of the blocking generated by a given mechanism. We investigate here stable blocking relations. We begin with a few examples and give some useful instruments (Section 4.1). In Sections 4.2–4.4 we discuss three classes of stable blockings: additive blockings, almost additive blockings, and convex blockings. The main finding is that for almost additive blockings a family of coalitions which reject alternatives out of the core, can be equipped with a laminar structure (Theorem (4.4.7)). Section 4.5 reviews a series of necessary conditions to warrant the stability of a given blocking. In particular, convexity and almost-additivity turn out to be necessary for the stability of maximal blockings. In Section 4.6, we develop a veto-procedure in order to find elements in the core. The procedure yields single-element outcomes for any maximal convex blocking.