Necessity of concavity for TU-fuzzy-core non-emptiness

Determine whether concavity of the utility functions is necessary for the non-emptiness of the TU-fuzzy core under the local TU-fuzzy-coalition stability and exchange-economy assumptions of Theorem 3.2.

Background

Theorem 3.2 establishes that a pure exchange economy satisfying Assumptions (A1) and (A2), local TU-fuzzy-coalition stability, and concavity of every utility function has a nonempty TU-fuzzy core. Corollary 3.3 shows that when there are two agents, the concavity assumption can be removed. The paper then presents a three-agent example with quasi-concave but non-concave utility functions that nevertheless has a nonempty TU-fuzzy core, demonstrating that concavity is sufficient but may not be necessary. The unresolved issue is whether concavity is required in general for the theorem's non-emptiness conclusion.