Nonempty Core-Type Solutions Over Balanced Coalitions In Tu-Games

In this paper we introduce two related core-type solutions for games with transferable utility (TU-games) the$\mathcal{B}$-core and the$\mathcal{M}$-core. The elements of the solutions are pairs$(x, \mathcal{B}), $wherex, as usual, is a vector representing a distribution of utility and$\mathcal{B}$is a balanced family of coalitions, in the case of the$\mathcal{B}$-core, and a minimal balanced one, in the case of the$\mathcal{M}$-core, describing a plausible organization of the players to achieve the vectorx. Both solutions extend the notion of classical core but, unlike it, they are always nonempty for any TU-game. For the$\mathcal{M}$-core, which also exhibits a certain kind of "minimality" property, we provide a nice axiomatic characterization in terms of the four axioms nonemptiness (NE), individual rationality (IR), superadditivity (SUPA) and a weak reduced game property (WRGP) (with appropriate modifications to adapt them to the new framework) used to characterize the classical core. However, an additional axiom, the axiom of equal opportunity is required. It roughly states that if$(x, \mathcal{B})$belongs to the$\mathcal{M}$-core then, any other admissible element of the form$(x, \mathcal{B}^{\prime})$should belong to the solution too.