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. We also provide an efficient algorithm for solving a special case of the problem, for a fixed number of bidders and available bids.

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,它具有连续的价值分配和分散的投标空间。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主观前置信念时,对拍卖的美元-平衡进行计算是完整的,对准确的平衡进行计算是完整的。我们还为固定数目的投标人和现有投标人提供了高效的算法,以解决问题的一个特殊案例。</s>