Partition-based Stability of Coalitional Games

Abstract: We are concerned with the stability of a coalitional game, i.e., a transferable-utility (TU) cooperative game. First, the concept of core can be weakened so that the blocking of changes is limited to only those with multilateral backings. This principle of consensual blocking, as well as the traditional core-defining principle of unilateral blocking and one straddling in between, can all be applied to partition-allocation pairs. Each such pair is made up of a partition of the grand coalition and a corresponding allocation vector whose components are individually rational and efficient for the various constituent coalitions of the given partition. For the resulting strong, medium, and weak stability concepts, the first is core-compatible in that the traditional core exactly contains those allocations that are associated through this strong stability concept with the all-consolidated partition consisting of only the grand coalition. Probably more importantly, the latter medium and weak stability concepts are universal. By this, we mean that any game, no matter how ``poor'' it is, has its fair share of stable solutions. There is also a steepest ascent method to guide the convergence process to a mediumly stable partition-allocation pair from any starting partition.