Equilibrium problems in Bayesian auction games can be described as systems of differential equations. Depending on the model assumptions, these equations might be such that we do not have a rigorous mathematical solution theory. The lack of analytical or numerical techniques with guaranteed convergence for the equilibrium problem has plagued the field and limited equilibrium analysis to rather simple auction models such as single-object auctions. Recent advances in equilibrium learning led to algorithms that find equilibrium under a wide variety of model assumptions. We analyze first- and second-price auctions where simple learning algorithms converge to an equilibrium. The equilibrium problem in auctions is equivalent to solving an infinite-dimensional variational inequality (VI). Monotonicity and the Minty condition are the central sufficient conditions for learning algorithms to converge to an equilibrium in such VIs. We show that neither monotonicity nor pseudo- or quasi-monotonicity holds for the respective VIs. The second-price auction's equilibrium is a Minty-type solution, but the first-price auction is not. However, the Bayes--Nash equilibrium is the unique solution to the VI within the class of uniformly increasing bid functions, which ensures that gradient-based algorithms attain the equilibrium in case of convergence, as also observed in numerical experiments.

翻译：贝叶斯拍卖博弈中的均衡问题可表述为微分方程组系统。根据模型假设的不同，这些方程可能缺乏严格的数学解理论。均衡问题在解析或数值技术上缺乏收敛性保证，这一困境长期困扰该领域，使得均衡分析仅限于单物品拍卖等简单模型。近期均衡学习领域的进展催生了能在多种模型假设下找到均衡的算法。本文分析了简单学习算法能收敛至均衡的首价拍卖与次价拍卖。拍卖中的均衡问题等价于求解无限维变分不等式。单调性与Minty条件是保证学习算法在此类变分不等式中收敛至均衡的核心充分条件。我们证明对应变分不等式既不满足单调性，也不满足伪单调性或拟单调性。次价拍卖的均衡属于Minty型解，而首价拍卖则不然。然而，在一致递增报价函数类中，贝叶斯-纳什均衡是该变分不等式的唯一解，这确保了基于梯度的算法在收敛时能达到均衡，数值实验也验证了这一结论。