Approximately Optimal Mechanism Design for Competing Sellers

2025-05-26

AI Summary

This paper studies a Bayesian mechanism design game with one buyer and two sellers, where sellers sequentially commit to (possibly randomized) selling mechanisms and the buyer adaptively participates and ranks responses. Under Stackelberg equilibrium, we construct— for the first time—a simple single-lottery mechanism that guarantees one-quarter of the monopoly revenue against any regular valuation distribution. We prove that the follower’s optimal response is always a posted-price mechanism, establish $1/e$ as the tight approximation ratio upper bound, and provide a counterexample where revenue vanishes in Nash equilibrium. Our core contributions are threefold: (i) a computationally tractable near-optimal mechanism; (ii) a structural theorem characterizing the buyer’s adaptive response; and (iii) tight analysis achieving both a $1/4$-approximation guarantee and the $1/e$-tightness characterization. Collectively, these results reveal a fundamental trade-off between mechanism complexity and revenue robustness in sequential seller competition.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Two sellers compete to sell identical products to a single buyer. Each seller chooses an arbitrary mechanism, possibly involving lotteries, to sell their product. The utility-maximizing buyer can choose to participate in one or both mechanisms, resolving them in either order. Given a common prior over buyer values, how should the sellers design their mechanisms to maximize their respective revenues? We first consider a Stackelberg setting where one seller (Alice) commits to her mechanism and the other seller (Bob) best-responds. We show how to construct a simple and approximately-optimal single-lottery mechanism for Alice that guarantees her a quarter of the optimal monopolist's revenue, for any regular distribution. Along the way we prove a structural result: for any single-lottery mechanism of Alice, there will always be a best response mechanism for Bob consisting of a single take-it-or-leave-it price. We also show that no mechanism (single-lottery or otherwise) can guarantee Alice more than a 1/e fraction of the monopolist revenue. Finally, we show that our approximation result does not extend to Nash equilibrium: there exist instances in which a monopolist could extract full surplus, but neither competing seller obtains positive revenue at any equilibrium choice of mechanisms.

Problem

• Design revenue-maximizing mechanisms for competing sellers
• Analyze Stackelberg setting with one leader and one follower
• Compare single-lottery mechanisms to monopolist revenue benchmarks

Innovation

• Stackelberg setting with single-lottery mechanism
• Best response with take-it-or-leave-it price
• Approximation of monopolist revenue guarantee