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

TL;DR

This paper offers a constructive proof of Border's theorem for reduced-form auctions, enabling efficient design of revenue-optimal multi-item auctions with asymmetric bidders and additive valuations.

Contribution

It provides a linear-constraint characterization of feasible reduced forms and a constructive method to induce ex-post allocation rules, advancing auction design theory.

Findings

Linear number of constraints suffices for feasibility

Polynomial-time algorithms for revenue optimization

Characterization of feasible multi-item allocation rules

Abstract

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 items. Appealing to our constructive Border's theorem, we obtain polynomial-time algorithms for computing the revenue-optimal auction. Appealing to our characterization of feasible reduced forms, we characterize feasible multi-item allocation rules.