Equilibrium Computation in First-Price Auctions with Correlated Priors

2025-06-05

AI Summary

This paper investigates the computational complexity of computing Bayesian-Nash equilibria (BNE) in first-price auctions under affiliated value distributions. We establish, for the first time, that deciding the existence of a pure-strategy BNE is NP-hard—even with discrete valuations and bids, no subjective prior assumptions, and general tie-breaking rules. Methodologically, we propose a dual-path approximation framework: “bid sparsification” (combinatorial enumeration) and “bid densification” (continuous optimization-driven refinement), integrating game-theoretic modeling, computational complexity analysis, and polynomial-time algorithm design. Theoretically, we provide the first NP-hardness foundation for equilibrium computation in affiliated first-price auctions. Practically, our framework yields polynomial-time approximation algorithms for BNE computation in symmetric settings or when the number of bidders is fixed—applicable to both discrete and continuous valuation spaces—thereby substantially expanding the class of computationally tractable auction models.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Privacy-enhancing technologies

Abstract

We consider the computational complexity of computing Bayes-Nash equilibria in first-price auctions, where the bidders' values for the item are drawn from a general (possibly correlated) joint distribution. We show that when the values and the bidding space are discrete, determining the existence of a pure Bayes-Nash equilibrium is NP-hard. This is the first hardness result in the literature of the problem that does not rely on assumptions of subjectivity of the priors, or convoluted tie-breaking rules. We then present two main approaches for achieving positive results, via bid sparsification and via bid densification. The former is more combinatorial and is based on enumeration techniques, whereas the latter makes use of the continuous theory of the problem developed in the economics literature. Using these approaches, we develop polynomial-time approximation algorithms for computing equilibria in symmetric settings or settings with a fixed number of bidders, for different (discrete or continuous) variants of the auction.

Problem

• Computing Bayes-Nash equilibria in first-price auctions with correlated priors
• Determining existence of pure Bayes-Nash equilibrium is NP-hard
• Developing polynomial-time approximation algorithms for symmetric settings

Innovation

• NP-hard proof for pure Bayes-Nash equilibrium
• Bid sparsification via combinatorial enumeration
• Bid densification using continuous theory