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Applications of combinatorial auctions (CA) as market mechanisms are prevalent in practice, yet their Bayesian Nash equilibria (BNE) remain poorly understood. Analytical solutions are known only for a few cases where the problem can be reformulated as a tractable partial differential equation (PDE). In the general case, finding BNE is known to be computationally hard. Previous work on numerical computation of BNE in auctions has relied either on solving such PDEs explicitly, calculating pointwise best-responses in strategy space, or iteratively solving restricted subgames. In this study, we present a generic yet scalable alternative multi-agent equilibrium learning method that represents strategies as neural networks and applies policy iteration based on gradient dynamics in self-play. Most auctions are ex-post nondifferentiable, so gradients may be unavailable or misleading, and we rely on suitable pseudogradient estimates instead. Although it is well-known that gradient dynamics cannot guarantee convergence to NE in general, we observe fast and robust convergence to approximate BNE in a wide variety of auctions and present a sufficient condition for convergence.

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Display ad auctions have become the predominant means to allocate user impressions on a website to advertisers. These auctions are conducted in milliseconds online, whenever a user visits a website. The impressions are typically priced via a simple second-price rule. For single-item auctions, this Vickrey payment rule is known to be incentive-compatible. However, it is unclear whether bidders should still bid truthful in an online auction where impressions (or items) arrive dynamically over time and their valuations are not separable, as is the case with campaign targets or budgets. The allocation process might not maximize welfare and the payments can differ substantially from those paid in an offline auction with a Vickrey-Clarke-Groves (VCG) payment rule or also competitive equilibrium prices. We study the properties of the offline problem and model it as a mathematical program. In numerical experiments, we find that the welfare achieved in the online auction process with truthful bidders is high compared to the theoretical worst-case efficiency, but that the bidders pay significantly more on average compared to what they would need to pay in a corresponding offline auction in thin markets with up to four bidders. However, incentives for bid shading in these second-price auctions decrease quickly with additional competition and bidders risk losing.