Auction Mechanisms for Social-Welfare-Maximizing Allocation Under Pairwise-Additive Negative Value Externalities

We study the auctioneer’s problem of social welfare maximization under a class of valuation functions, called the Pairwise-Additive Negative Value Externalities (PANE) functions, that captures some commonly-known forms of negative value externalities in the allocations of heterogeneous and indivisible items to bidders. We formulate the social welfare maximization problem as a binary quadratic program with linear constraints. We show that the binary relaxation of this program is a nonconvex optimization problem that (i) has integral solutions, (ii) possesses strong duality, and (iii) can be solved in polynomial time. With results (i) and (ii), we show the existence of anonymous and simple item prices that support a social-welfare-maximizing allocation of items. With result (iii), we show that a VCG mechanism (a direct mechanism) can be implemented in practice, since the VCG allocations and payments (that result in a social-welfare-maximizing allocation) can be computed in polynomial time. However, for practical settings where complete information revelation is not easy to justify, we develop an iterative auction (an indirect mechanism), based on the subgradient algorithm, that terminates in a social-welfare-maximizing allocation. This iterative auction is easy to implement in practice since it fixes prices at the level of the individual items, and the prices are not dependent on the bidders’ identities.