The Core of Games on Distributive Lattices

Cooperative games under precedence constraints have been introduced by Faigle and Kern [1], [4] as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a distributive game, the first one is called the precore and is a direct generalization of the classical definition. It contains the set of imputations and may be unbounded, which makes its application questionable. A second definition is proposed, imposing normalization at each stage, causing the core to be a convex bounded set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their coincidence in the convex case remain valid.