We show that every feasible, Bayesian, multi-item multibidder mechanism for independent, additive bidders can be implemented as a mechanism that: (a) allocates every item independently of the other items; (b) for the allocation of each item it uses a strict ordering of all bidders' types; and allocates the item using a distribution over hierarchical mechanisms that iron this ordering into a non-strict ordering, and give the item uniformly at random to the bidders whose reported types dominate all other reported types according to the non-strict ordering. Combined with cyclicmonotonicity our results provide a characterization of feasible, Bayesian Incentive Compatible mechanisms in this setting. Our characterization is enabled by a new, constructive proof of Border's theorem [Bor91], and a new generalization of this theorem to independent (but not necessarily identically distributed) bidders, improving upon the results of [Bor07, CKM11]. For a single item and independent bidders, we show that every feasible reduced form auction can be implemented as a distribution over hierarchical mechanisms that are consistent with the same strict ordering of all bidders' types, which every mechanism in the support of the distribution irons to a non-strict ordering. We also give a polynomial-time algorithm for determining feasibility of a reduced form auction, or providing a separation hyperplane from the set of feasible reduced forms. To complete the picture, we provide polynomial-time algorithms to †
In particular, the results of Cai et al. REF imply that every first-order interim rule can be efficiently implemented as a distribution over virtual value maximizers.
An algorithmic characterization of multi-dimensional mechanisms
STOC '12
Computer Science Mathematics