Discrete Single-Parameter Optimal Auction Design
Yiannis Giannakopoulos, Johannes Hahn
TL;DR
The paper develops a unified, discrete-mechanism design framework for single-parameter auctions with finite value supports, recasting Myerson's revenue-maximization through LP duality to obtain a discrete analogue of virtual welfare maximization and ironing. It then generalizes to arbitrary convex feasibility spaces via a KKT system, introducing generalized ironed virtual values and a unified payment rule that works for both DSIC and BIC objectives, with integrality guaranteed under TU/TDI conditions. A key insight is that the DSIC and BIC frameworks align under revenue maximization, and ironing ensures monotone allocations, enabling deterministic (integral) optimal auctions even in broader environments. The framework is demonstrated on a tree-flow network with capacity constraints, yielding a combinatorial algorithm that computes the optimal allocation and payments via KKT conditions and edge-pricing. Overall, the work provides a transparent, polyhedral route to discrete optimal auctions and broadens applicability to networked, convex-constrained settings with practical implications for discrete mechanism design.
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.