Ensuring the boundedness of the core of games with restricted cooperation

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.

💡 Research Summary

The paper addresses a fundamental drawback of the core in cooperative games when cooperation is restricted to a sub‑collection F of coalitions: the core, defined by the inequalities x(S) ≥ v(S) for all S ∈ F together with the efficiency condition x(N)=v(N), may become unbounded. An unbounded core is undesirable because it no longer represents a realistic set of payoff vectors. The authors propose to bound the core by converting a selected subset of the defining inequalities into equalities, thereby adding supplementary efficiency constraints while preserving coalitional rationality.

The central problem is to identify a minimal set of coalitions whose corresponding inequalities, when turned into equalities, guarantee that the resulting “restricted core” is bounded for any game v. This problem is examined first in the context of distributive lattices, which are set systems closed under union and intersection and can be represented as the family of down‑sets of a partially ordered set of n elements. In this setting the recession cone C(0) of the core is completely characterized: its extreme rays are vectors of the form (1_j, −1_i) where j covers i in the underlying poset. A ray is eliminated precisely when there exists a coalition F ∈ F containing j but not i; imposing the equality x(F)=0 then forces the ray to disappear.

The authors prove that the number of equalities required to eliminate all rays equals the height h(N) of the poset (the length of a longest chain). They present an explicit algorithm (Algo 1) that iteratively removes minimal elements, adds the equality x(↓M₀)=0 for the down‑set generated by those minima, and repeats on the reduced set. The algorithm produces exactly h(N) equalities, each of minimal size, and is shown to be optimal: fewer equalities cannot suffice to bound the core.

Beyond the boundedness issue, the paper defines a “restricted Weber set” by intersecting the classical Weber set with the hyperplanes introduced by the normal collection. It is proved that the restricted core is always contained in this restricted Weber set, preserving the classic inclusion property of the unrestricted case.

The analysis is then extended to two broader families of set systems. First, regular set systems—those in which every maximal chain from ∅ to N has length n—behave similarly to distributive lattices: the same algorithm applies and the minimal number of equalities remains n. Second, weakly union‑closed systems, where the union of any two disjoint feasible coalitions is feasible, are more delicate. The authors identify additional structural conditions under which the extreme rays can still be expressed in a form amenable to the lattice‑based treatment, allowing the same bounding technique to be employed.

For the completely general case, the paper provides a characterization of the extreme rays of C(0) and shows how, when these rays have a simple “single‑difference” structure, a single equality suffices to eliminate each. While a universal polynomial‑time method for arbitrary set systems is not claimed, the authors delineate sufficient conditions that guarantee tractability.

In summary, the work offers a rigorous, constructive solution to the unbounded core problem in games with restricted cooperation. By identifying the minimal normal collection needed to bound the core, supplying an optimal algorithm for distributive lattices and regular systems, and extending the approach to weakly union‑closed and more general structures, the paper bridges a gap between theoretical cooperative game concepts and practical applications where only limited coalitions are feasible. The introduction of the restricted Weber set further integrates the new bounded core into the established framework of solution concepts, ensuring that classical inclusion results continue to hold under the added efficiency constraints.