Stable Families of Coalitions and Normal Hypergraphs

Abstract

The core of a game is defined as the set of outcomes acceptable for all coalitions. This is probably the simplest and most natural concept of cooperative game theory. However, the core can be empty because there are too many coalitions. Yet, some players may not like or know each other, so they cannot form a coalition. Let K be a fixed family of coalitions. The K-core is defined as the set of outcomes acceptable for all the coalitions from K. The family K is called stable if the K-core is not empty for any normal form game (or equivalently, for any game in generalized characteristic function form). Normal hypergraphs can be characterized by several equivalent properties, e.g. they are dual to the clique hypergraphs of perfect graphs. We prove that a family K of coalitions is stable iff K as a hypergraph is normal. 1 The authors gratefully acknowledge the partial support of DIMACS, AFOSR (Grant F49620-93-1-0041) and ONR (Grants N00014-92-J-1375 and N00014-92-J-4083). 2 On leave from..