Auction theory is of central importance in the study of markets. Unfortunately, we do not know equilibrium bidding strategies for most auction games. For realistic markets with multiple items and value interdependencies, the Bayes Nash equilibria (BNEs) often turn out to be intractable systems of partial differential equations. Previous numerical techniques have relied either on calculating pointwise best responses in strategy space or iteratively solving restricted subgames. We present a learning method that represents strategies as neural networks and applies policy iteration on the basis of gradient dynamics in self-play to provably learn local equilibria. Our empirical results show that these approximated BNEs coincide with the global equilibria whenever available. The method follows the simultaneous gradient of the game and uses a smoothing technique to circumvent discontinuities in the ex post utility functions of auction games. Discontinuities arise at the bid value where an infinite small change would make the difference between winning and not winning. Convergence to local BNEs can be explained by the fact that bidders in most auction models are symmetric, which leads to potential games for which gradient dynamics converge.

拍卖理论在市场研究中至关重要。不幸的是，我们不知道大多数拍卖游戏的均衡竞价策略。对于具有多个项目和价值相互依赖的现实市场，贝叶斯纳什均衡 (BNE) 通常是难以处理的偏微分方程系统。以前的数值技术要么依赖于计算策略空间中的逐点最佳响应，要么依赖于迭代求解受限子博弈。我们提出了一种学习方法，将策略表示为神经网络，并在自我博弈中基于梯度动态应用策略迭代来可证明地学习局部均衡。我们的实证结果表明，只要可用，这些近似的 BNE 与全球均衡一致。该方法遵循博弈的同时梯度，并使用平滑技术来规避拍卖博弈的事后效用函数的不连续性。出价出现不连续性，无限小的变化会在中奖和未中奖之间产生差异。与本地 BNE 的收敛可以通过以下事实来解释：大多数拍卖模型中的投标人是对称的，这导致梯度动态收敛的潜在博弈。