In cooperative game theory, for a given set of players N, TU-games are functions v : 2N → R which express for each nonempty coalition S ⊆ N of players the best they can achieve by cooperation. In the classical setting, every coalition may form without any restriction, i.e., the domain of v is indeed 2N . In practice, this assumption is often unrealistic, since some coalitions may not be feasible for various reasons, e.g., players are political parties with divergent opinions, or have restricted communication abilities, or a hierarchy exists among players, and the formation of coalitions must respect the hierarchy, etc. Many studies have been done on games defined on specific subdomains of 2N , e.g., antimatroids [1], convex geometries [3, 4], distributive lattices [6], or others [2, 5]. In this paper, we mainly deal with the case of distributive lattices. To this end, we assume that there exists some partial order on N describing some hierarchy or precedence constraint among players, as in [6]. We say that a coalition S is feasible if the coalition contains all its subordinates, i.e., i ∈ S implies that any j i belongs to S as well. Then feasible coalitions are downsets, and by Birkhoff's theorem, form a distributive lattice. From now on, we denote by F the set of feasible coalitions, assuming that 0/ , N ∈ F .
where x(S) is a shorthand for ∑i∈S xi. When F = 2N , the core is either empty or a convex
bounded polyhedron. However, for games whose cooperation is restricted, the study of
the core becomes much more complex, since it may be unbounded or even contain
no vertices (see a survey in [
We also present some results when F is not a distributive lattice, but a set lattice closed under intersection, or a regular set system.
Lastly, we introduce games on concept lattices, show that this induces in fact two games, and give some results on the core.