ICML 2026 paper proves first-price auction revenue near second-highest bid under no-swap regret dynamics

A new theoretical paper from Anders Bo Ipsen and Stratis Skoulakis, accepted to ICML 2026, tackles a fundamental question in auction theory: how much revenue can sellers expect when bidders use learning algorithms that minimize swap regret? The authors analyze first-price auctions with a discrete bid set {0, 1/k, 2/k, …, 1}. They show that in any approximate correlated equilibrium (ε-CE), the auction's revenue is at least the second-highest valuation v₂ minus two correction terms: a discretization error Θ(1/k) and an approximation error Θ(εk²). This lower bound tightens as the bid grid becomes finer and as bidders' strategies get closer to exact equilibrium.

The result establishes the first polynomial convergence rates for revenue under no-swap regret dynamics—a stronger regret notion than standard no-regret that captures adaptability to correlated strategies. Concretely, if each bidder achieves the optimal swap regret of O(√(kT)) after T rounds, then the time-averaged revenue reaches the bound after O(k⁵/ε²) rounds. For auction platforms and automated bidding systems, this means that even when participants are not explicitly cooperating, the revenue loss due to learning can be bounded and made arbitrarily small with enough rounds and finer bid increments. The work bridges gaps between equilibrium analysis and practical auction dynamics, offering provable guarantees for real-world first-price ad slots and procurement auctions.