Auctions are modeled as Bayesian games with continuous type and action spaces. Determining equilibria in auction games is computationally hard in general and no exact solution theory is known. We introduce an algorithmic framework in which we discretize type and action space and then learn distributional strategies via online optimization algorithms. One advantage of distributional strategies is that we do not have to make any assumptions on the shape of the bid function. Besides, the expected utility of agents is linear in the strategies. It follows that if our optimization algorithms converge to a pure strategy, then they converge to an approximate equilibrium of the discretized game with high precision. Importantly, we show that the equilibrium of the discretized game approximates an equilibrium in the continuous game. In a wide variety of auction games, we provide empirical evidence that the approach approximates the analytical (pure) Bayes Nash equilibrium closely. This speed and precision is remarkable, because in many finite games learning dynamics do not converge or are even chaotic. In standard models where agents are symmetric, we find equilibrium in seconds. While we focus on dual averaging, we show that the overall approach converges independent of the regularizer and alternative online convex optimization methods achieve similar results, even though the discretized game neither satisfies monotonicity nor variational stability globally. The method allows for interdependent valuations and different types of utility functions and provides a foundation for broadly applicable equilibrium solvers that can push the boundaries of equilibrium analysis in auction markets and beyond.

翻译：拍卖被建模为具有连续类型和行动空间的贝叶斯博弈。确定拍卖博弈中的均衡通常在计算上具有挑战性，且目前尚无精确解理论。我们提出了一种算法框架，通过离散化类型和行动空间，利用在线优化算法学习分布策略。分布策略的一个优势是无需对投标函数形式作出任何假设。此外，智能体的期望效用关于策略是线性的。因此，如果我们的优化算法收敛到纯策略，则它们将以高精度收敛到离散化博弈的近似均衡。重要的是，我们证明了离散化博弈的均衡近似连续博弈中的均衡。在多种拍卖博弈中，我们提供了经验证据，表明该方法能够紧密逼近解析（纯）贝叶斯纳什均衡。这种速度和精度值得关注，因为在许多有限博弈中，学习动态要么不收敛，甚至呈现混沌状态。在智能体对称的标准模型中，我们可在数秒内找到均衡。尽管我们重点关注对偶平均方法，但研究表明整体方法的收敛性独立于正则化器，且替代的在线凸优化方法也能达到类似结果，尽管离散化博弈既不完全满足全局单调性也不满足变分稳定性。该方法允许相互依赖的估值和不同类型的效用函数，为广泛适用的均衡求解器奠定了基础，可推动拍卖市场及其他领域的均衡分析边界。