Discrete Single-Parameter Optimal Auction Design

TL;DR

This paper characterizes optimal auction mechanisms in discrete single-parameter settings using LP duality, extending classical results to more general convex feasibility constraints and demonstrating their application to flow-based auctions.

Contribution

It introduces a unified LP duality framework for discrete single-parameter auctions, generalizes Myerson's results, and provides new conditions for auction optimality in complex feasibility spaces.

Findings

Re-derivation of classical auction results using LP duality.

Extension of optimal auction characterization to convex feasibility constraints.

Application to flow-based auction settings with combinatorial descriptions.

Abstract

We study the classic single-item auction setting of Myerson, but under the assumption that the buyers' values for the item are distributed over finite supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a convex combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson's payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules. Finally, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction.