  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.
