Tight Liquid Welfare Guarantees for Auctions with Budgets via LP Duality

2026-09-30

AI Summary

This study addresses the challenge of evaluating social welfare efficiency in budget-constrained auctions by proposing a unified analytical framework grounded in linear programming duality. By introducing a novel smoothness concept tailored to budgeted environments alongside classical reduction techniques, this work reduces Price of Anarchy (PoA) proofs under coarse correlated equilibria to the problem of identifying dual-feasible solutions. This paradigm enables concise and tight analyses across diverse auction formats, yielding tight welfare guarantees for multiple mechanisms. Notably, the derived bounds match the best-known upper bound for uniform-price auctions. Ultimately, this framework provides a general-purpose theoretical tool for mechanism design under budget constraints.

Abstract

We study the efficiency of auctions with budget-constrained bidders and derive tight price of anarchy (PoA) bounds for coarse correlated equilibria (CCE). To this end, we develop a general framework that reduces proving PoA guarantees to finding feasible solutions to the dual of a linear program. The LP is formulated over a small number of per-bidder equilibrium statistics, incorporates constraints implied by the auction setting and equilibrium behavior, and captures a novel smoothness notion tailored to budget-constrained environments; we provide a reduction from classical smoothness to this new notion. Our approach enables simple and tight PoA analyses across a broad range of auction formats. Using our framework, we obtain tight liquid welfare guarantees for simultaneous first-price, second-price, and all-pay auctions, simultaneous auctions with restricted uniform bidding interfaces, discriminatory and uniform price multi-unit auctions, and generalized first-price position auctions. Beyond budget-constrained settings, we derive new lower bounds for the budget-free uniform price auction and generalized second-price auction; in particular, our lower bound for the uniform price auction matches the upper bound of $3.146$ by de Keijzer et al. (2013).

Problem

• budget-constrained auctions
• price of anarchy
• liquid welfare
• coarse correlated equilibria

Innovation

• Linear Programming Duality
• Price of Anarchy
• Liquid Welfare
• Budget-Constrained Auctions
• Smoothness