In first-price and all-pay auctions under the standard symmetric independent private-values model, we show that the unique Bayesian Coarse Correlated Equilibrium with symmetric, differentiable and strictly increasing bidding strategies is the unique strict Bayesian Nash Equilibrium. Interestingly, this result does not require assumptions on the prior distribution. The proof is based on a dual bound of the infinite-dimensional linear program. Numerical experiments without restrictions on bidding strategies show that for first-price auctions and discretisations up to 21 of the type and bid space, increasing discretisation sizes actually increase the concentration of Bayesian Coarse Correlated Equilibrium over the Bayesian Nash Equilibrium, so long as the prior c.d.f. is concave. Such a concentration is also observed for all-pay auctions, independent of the prior distribution. Overall, our results imply that the equilibria of these important class of auctions are indeed learnable.

翻译：在标准对称独立私有价值模型的第一价格拍卖和全支付拍卖中，我们证明了具有对称、可微且严格递增投标策略的唯一贝叶斯粗糙相关均衡即为唯一的严格贝叶斯纳什均衡。有趣的是，该结论并不需要对先验分布做任何假设。证明基于无限维线性规划的对偶界。在不对投标策略施加限制的数值实验中表明：对于第一价格拍卖，当类型和出价空间离散化至21个节点时，只要先验分布函数为凹函数，增大离散化规模实际上会增强贝叶斯粗糙相关均衡向贝叶斯纳什均衡的收敛性。这种收敛性在全支付拍卖中同样存在，且与先验分布无关。总体而言，我们的结果表明这类重要拍卖机制的均衡确实是可学习的。