We study the computational complexity of computing Bayes-Nash equilibria in first-price auctions with discrete value distributions and discrete bidding space, under general subjective beliefs. It is known that such auctions do not always have pure equilibria. In this paper we prove that the problem of deciding their existence is NP-complete, even for approximate equilibria. On the other hand, it can be shown that mixed equilibria are guaranteed to exist; however, their computational complexity has not been studied before. We establish the PPAD-completeness of computing a mixed equilibrium and we complement this by an efficient algorithm for finding symmetric approximate equilibria in the special case of iid priors. En route to these results, we develop a computational equivalence framework between continuous and discrete first-price auctions, which can be of independent interest, and which allows us to transfer existing positive and negative results from one setting to the other. Finally, we show that correlated equilibria of the auction can be computed in polynomial time.

翻译：我们研究了在一般主观信念下，具有离散价值分布和离散竞价空间的第一价格拍卖中计算贝叶斯-纳什均衡的计算复杂性。已知此类拍卖并不总是存在纯策略均衡。本文证明，即使对于近似均衡，判断其存在性的问题是NP完全的。另一方面，可以证明混合策略均衡必然存在；然而，其计算复杂性此前尚未被研究。我们建立了计算混合均衡的PPAD完全性，并在此基础上，针对独立同分布先验的特殊情况，提出了一种寻找对称近似均衡的高效算法。在推导这些结果的过程中，我们开发了一个连续与离散第一价格拍卖之间的计算等价框架，该框架具有独立的研究价值，并允许我们将已有的正面和负面结论从一个场景迁移到另一个场景。最后，我们证明该拍卖中的关联均衡可以在多项式时间内计算。