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：我们考虑在具有连续价值分布和离散出价空间的一级价格拍卖中计算（纯）贝叶斯-纳什均衡的问题。我们证明，当竞拍者对他人价值分布持有独立主观先验信念时，计算该拍卖的$\varepsilon$-均衡是PPAD完全的，而精确计算均衡是FIXP完全的。我们还针对固定竞拍者数量和可用出价数的特殊情况，提供了一种高效算法。