Discrete Single-Parameter Optimal Auction Design

2024-06-12

Algorithmic Game Theory

AI Summary

This paper studies discrete optimal auction design in finite-support, single-parameter environments subject to convex feasibility constraints—such as tree-structured capacity networks—with the objective of jointly optimizing a convex combination of revenue and social welfare. Methodologically, it develops a generalized virtual surplus maximization framework grounded in KKT conditions; leverages total unimodularity to derive sufficient conditions for integral optimal allocations; and extends Myerson’s payment formula to arbitrary discrete single-parameter settings. Key contributions include: (i) the first rigorous proof of universal equivalence between dominant-strategy incentive compatibility (DSIC) and Bayesian incentive compatibility (BIC) in discrete single-parameter environments; (ii) a combinatorial characterization of (potentially randomized) optimal mechanisms for tree-capacity-constrained auctions; and (iii) a unified explanation of deterministic optimality results. The approach integrates strong duality theory, polyhedral analysis, and a generalized notion of virtual values.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Optimal Auction Design
• Discrete Bidding
• Capacity-constrained Tree Auctions

Innovation

• Optimal Auction Strategies
• Bayesian Incentive Compatibility
• Constrained Discrete Auction Environment