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.

翻译：本文研究标准对称独立私有价值模型下连续与离散化第一价格及全支付拍卖的贝叶斯粗相关均衡。本研究的动机在于探究拍卖的经典贝叶斯-纳什均衡与买方采用无悔算法习得结果之间的关联。数值实验表明：在双买方第一价格拍卖中，当先验分布为凹函数时，买方边际出价分布的Wasserstein-$2$距离随离散化粒度呈$O(1/n)$衰减；而全支付拍卖在无先验依赖条件下亦呈现类似规律。为解释这种趋近均衡的现象，我们研究了连续拍卖贝叶斯粗相关均衡的唯一性。所得唯一性结论可转化为确定性自我博弈在这些拍卖中可证明收敛至近均衡结果的理论保证。在全支付拍卖中，我们证明：独立于先验分布存在唯一的对称、可微且递增出价策略贝叶斯粗相关均衡，该均衡等价于唯一的严格贝叶斯-纳什均衡。对于第一价格拍卖，则需要更强条件：或要求先验分布严格凹，或需将学习算法限制于严格递增策略。若无此类强假设，无悔算法可能收敛至低价聚集策略。这一发现具有重要意义，因其证明在诸如展示广告拍卖等重复第一价格拍卖中，若缺乏额外假设，即使所有投标方均采用无悔算法，算法合谋的可能性仍无法被排除。