On the Uniqueness of Bayesian Coarse Correlated Equilibria in Standard First-Price and All-Pay Auctions

We study the Bayesian coarse correlated equilibrium (BCCE) of continuous and discretised first-price and all-pay auctions under the standard symmetric independent private-values model. Our study is motivated by the question of how the canonical Bayes-Nash equilibrium (BNE) of the auction relates to the outcomes learned by buyers utilising no-regret algorithms. Numerical experiments show that in two buyer first-price auctions the Wasserstein-2 distance of buyers' marginal bid distributions decline as O(1/n) in the discretisation size in instances where the prior distribution is concave, whereas all-pay auctions exhibit similar behaviour without prior dependence. To explain this convergence to a near-equilibrium, we study uniqueness of the BCCE of the continuous auction. Our uniqueness results translate to provable convergence of deterministic self-play to a near equilibrium outcome in these auctions. In the all-pay auction, we show that independent of the prior distribution there is a unique BCCE with symmetric, differentiable, and increasing bidding strategies, which is equivalent to the unique strict BNE. In the first-price auction, we need stronger conditions. Either the prior is strictly concave or the learning algorithm has to be restricted to strictly increasing strategies. Without such strong assumptions, no-regret algorithms can end up in low-price pooling strategies. This is important because it proves that in repeated first-price auctions such as in display ad actions, algorithmic collusion cannot be ruled out without further assumptions even if all bidders rely on no-regret algorithms.

• A first version of this paper appeared on ArXiV on January 2 nd , 2024, and a second version on June 20th. The current version incorporates reviewer feedback following acceptance to SODA'25.

1 That is, before considering deviations to a bid, they should know their own valuation for the item.