The Core of Cooperative TU Games with Bihierarchies

The coexistence of two hierarchies is ubiquitous in bureaucracies, economic organizations, and markets. In this paper, we study a class of transferable utility games in which agents can form a coalition only if they are in the same team within some hierarchy, and in which the collection of feasible coalitions forms a bihierarchy. We call such a game a bihierarchical game. Using a network ow approach, we provide a necessary and sufficient condition for the non-emptiness of the core of a constrained bihierarchical game. The condition identifies pivotal blocking coalitions by partitioning all feasible coalitions into insiders and outsiders and finding the boundary coalitions along the two hierarchies. We also show that, for unconstrained bihierarchical games, the core is non-empty if and only if the game satis_es a weak form of superadditivity.