We consider the problem of computing a (pure) Bayes-Nash equilibrium in the first-price auction with continuous value distributions and discrete bidding space. We prove that when bidders have independent subjective prior beliefs about the value distributions of the other bidders, computing an $\varepsilon$-equilibrium of the auction is PPAD-complete, and computing an exact equilibrium is FIXP-complete.

翻译：我们考虑了在第一次价格拍卖中计算(纯)巴耶斯-纳什平衡的问题,第一次价格拍卖有连续的价值分配和离散的投标空间。 我们证明,当投标人对其他投标人的价值分配有独立主观的先入为主的信念时,计算拍卖的美元(varepsilon$-equiquiblium)是完整的PPAD,计算准确的平衡是完整的FIXP。