A Constructive Approach to Reduced-Form Auctions with Applications to Multi-Item Mechanism Design

We provide a constructive proof of Border’s theorem [Bor91, HR15a] and its generalization to reduced-form auctions with asymmetric bidders [Bor07, MV10, CKM13]. Given a reduced form, we identify a subset of Border constraints that are necessary and sufficient to determine its feasibility. Importantly, the number of these constraints is linear in the total number of bidder types. In addition, we provide a characterization result showing that every feasible reduced form can be induced by an ex-post allocation rule that is a distribution over ironings of the same total ordering of the union of all bidders’ types. We show how to leverage our results for single-item reduced forms to design auctions with heterogeneous items and asymmetric bidders with valuations that are additive over Supported by NSERC Discovery RGPIN-2015-06127, FRQNT 2017-NC-198956 and NSF Awards CCF0953960 (CAREER), CCF-1101491, and CCF-1617730. Work done in part while the author was a Research Fellow at the Simons Institute for the Theory of Computing. Supported by a Sloan Foundation Fellowship, a Microsoft Research Faculty Fellowship, and NSF Awards CCF-0953960 (CAREER), CCF-1101491, and CCF-1617730. Work done in part while the author was a Research Fellow at the Simons Institute for the Theory of Computing. Research completed in part while the author was supported by a NSF Graduate Research Fellowship, and in part while the author was a Microsoft Research Fellow at the Simons Institute for the Theory of Computing.