  The core of a cooperative game on a set of players $N$ is one of the most
popular concept of solution. When cooperation is restricted (feasible
coalitions form a subcollection $\cF$ of $2^N$), the core may become unbounded,
which makes it usage questionable in practice. Our proposal is to make the core
bounded by turning some of the inequalities defining the core into equalities
(additional efficiency constraints). We address the following mathematical
problem: can we find a minimal set of inequalities in the core such that, if
turned into equalities, the core becomes bounded? The new core obtained is
called the restricted core. We completely solve the question when $\cF$ is a
distributive lattice, introducing also the notion of restricted Weber set. We
show that the case of regular set systems amounts more or less to the case of
distributive lattices. We also study the case of weakly union-closed systems
and give some results for the general case.
