We study the complexity of finding an approximate (pure) Bayesian Nash equilibrium in a first-price auction with common priors when the tie-breaking rule is part of the input. We show that the problem is PPAD-complete even when the tie-breaking rule is trilateral (i.e., it specifies item allocations when no more than three bidders are in tie, and adopts the uniform tie-breaking rule otherwise). This is the first hardness result for equilibrium computation in first-price auctions with common priors. On the positive side, we give a PTAS for the problem under the uniform tie-breaking rule.

翻译：我们研究在具有共同先验的第一价格拍卖中，当平局规则作为输入的一部分时，寻找近似（纯）贝叶斯纳什均衡的复杂性。我们证明，即使平局规则是三边的（即，当不超过三个竞拍者平局时指定物品分配，否则采用统一平局规则），该问题仍是PPAD完全的。这是关于共同先验第一价格拍卖中均衡计算的首个困难性结果。在积极方面，我们给出了统一平局规则下该问题的多项式时间近似方案（PTAS）。