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下的一级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相关均衡即为唯一的严格贝叶斯纳什均衡。值得关注的是，这一结果不依赖于先验分布的假设。该证明基于无穷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，对于一级价格拍卖以及类型和投标空间离散化至21个点的情况，只要先验累积分布函数是凹的，离散化规模的增大实际上会增强贝叶斯粗相关均衡相对于贝叶斯纳什均衡的集中度。这种集中现象也在全支付拍卖中观察到，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类别的均衡确实是可学习的。