Adjusting prices in the multiple-partners assignment game

Starting with an initial price vector, prices are adjusted in order to eliminate the excess demand and at the same time to keep the transfers to the sellers as low as possible. In each step of the auction, to which set of sellers should those transfers be made is the key issue in the description of the algorithm. We assume additively separable utilities and introduce a novel distinction by considering multiple sellers owing multiple identical objects and multiple buyers with an exogenously defined quota, consuming more than one object but at most one unit of a seller`s good and having multi-dimensional payoffs. This distinction induces a necessarily more complicated construction of the over-demanded sets than the constructions of these sets for the other assignment games. For this approach, our mechanism yields the buyer-optimal competitive equilibrium payoff, which equals the buyer-optimal stable payoff. The symmetry of the model allows to getting the seller-optimal stable payoff and the seller-optimal competitive equilibrium payoff can then be also derived.