One-seller assignment markets with multi-unit demands: core and competitive equilibrium

We consider an assignment market with one seller who owns several indivisible heterogeneous goods and many buyers each willing to buy up to a given capacity. Our aim is to study the relationship between the core of the game and the set of competitive equilibria. The core is non-empty and it has a lattice structure which contains the allocation in which every buyer gets his marginal contribution to the grand coalition. The set of competitive equilibrium price vectors also has a lattice structure and we determine the minimum and maximum competitive equilibrium prices. Necessary and sufficient conditions under which the buyers-optimal and the seller-optimal core allocations come from a competitive equilibrium are provided. In addition, we characterize in terms of the valuation matrix the coincidence between the core and the set of competitive equilibrium payoff vectors. As a consequence, we obtain that this coincidence always holds if the capacities of all buyers are large enough.