Symmetrically multilateral-bargained allocations in multi-sided assignment markets

We extend the notion of symmetrically pairwise-bargained (SPB) allocations (Rochford, J Econ Theory, 34:262–281, 1984) to balanced assignment games with more than two sides. A symmetrically multilateral-bargained (SMB) allocation is a core allocation such that any agent’s payoff remains invariant after a negotiation process between all agents based on what they could receive—and use as a threat—in their preferred alternative matching to any given optimal matching. We prove that, for balanced multi-sided assignment games, the set of SMB is always nonempty and that, unlike the two-sided case, it does not coincide in general with the kernel (Davis and Maschler, Naval Res Logist Q 12:223–259, 1965). We also give an answer to an open question formulated by Rochford by introducing a kernel-based set whose intersection with the core coincides with the set of SMB.