Efficient Equilibrium Computation in Symmetric First-Price Auctions

2026-03-25

AI Summary

This work investigates the computational complexity of computing Bayesian–Nash equilibria in symmetric single-item first-price auctions under independent and identically distributed (i.i.d.) bidder valuations. The paper presents a polynomial-time algorithm in the white-box model for continuous bid spaces and a query-efficient algorithm in the black-box model for finite bid sets. By integrating numerical optimization with game-theoretic analysis, the study achieves, for the first time, efficient equilibrium computation in both settings. These results fully characterize the computational complexity of equilibrium finding in this canonical auction format and resolve a long-standing open problem in auction theory.

Technology Category

Game Theory and Economic Paradigms: Mechanism DesignMultiagent Systems: Mechanism DesignSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchangesGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

We study the complexity of computing Bayes-Nash equilibria in single-item first-price auctions. We present the first efficient algorithms for the problem, when the bidders' values for the item are independently drawn from the same continuous distribution, under both established variants of continuous and finite bidding sets. More precisely, we design polynomial-time algorithms for the white-box model, where the distribution is provided directly as part of the input, and query-efficient algorithms for the black-box model, where the distribution is accessed via oracle calls. Our results settle the computational complexity of the problem for bidders with i.i.d. values.

Problem

• Bayes-Nash equilibrium
• first-price auction
• computational complexity
• symmetric bidders
• i.i.d. valuations

Innovation

• Bayes-Nash equilibrium
• first-price auction
• polynomial-time algorithm
• query-efficient algorithm
• symmetric auctions