Better redistribution with inefficient allocation in multi-unit auctions with unit demand

For the problem of allocating one or more items among a group of competing agents, the Vickrey-Clarke-Groves (VCG) mechanism is strategy-proof and efficient. However, the VCG mechanism is not strongly budget balanced: in general, value flows out of the system of agents in the form of VCG payments, which reduces the agents' utilities. In many settings, the objective is to maximize the sum of the agents' utilities (taking payments into account). For this purpose, several VCG redistribution mechanisms have been proposed that redistribute a large fraction of the VCG payments back to the agents, in a way that maintains strategy-proofness and the non-deficit property. Unfortunately, sometimes even the best VCG redistribution mechanism fails to redistribute a substantial fraction of the VCG payments. This results in a low total utility for the agents, even though the items are allocated efficiently. In this paper, we study strategy-proof allocation mechanisms that do not always allocate the items efficiently. It turns out that by allocating inefficiently, more payment can sometimes be redistributed, so that the net effect is an increase in the sum of the agents' utilities. Our objective is to design mechanisms that are competitive with the omnipotent perfect allocation in terms of the agents' total utility. We define linear allocation mechanisms. We propose an optimization model for simultaneously finding an allocation mechanism and a payment redistribution rule which together are optimal, given that the allocation mechanism is required to be either one of, or a mixture of, a finite set of specified linear allocation mechanisms. Finally, we propose several specific (linear) mechanisms that are based on burning items, excluding agents, and (most generally) partitioning the items and agents into groups. We show or conjecture that these mechanisms are optimal among various classes of mechanisms.