Cooperative TU-games: Dominance, stable sets, and the core revisited

Stable sets are introduced by von Neumann and Morgenstern (1944) as “the solution” of a cooperative game. Later on, Gillies (1953) defines the core of the game. Both notions can be established in terms of dominance. It is well known that the core may be an empty set, whereas stable sets may fail to exist, or may produce different proposals. We provide a new dominance relation so that the stable set obtained when applying this notion (the $\delta$-stable set) always exists, it is unique, and it coincides with the core of the cooperative game, whenever the core is not empty. We apply this concept to some particular classes of $TU$-games having typically an empty core: voting (majority) games, minimum cost spanning trees games with revenue, controlled capacitated networks, or $m$-sequencing games.