On the Complexity of Equilibrium Computation in First-Price Auctions

TL;DR

This paper investigates the computational complexity of finding equilibria in first-price auctions with continuous values and discrete bids, establishing PPAD- and FIXP-completeness results and providing an efficient algorithm for special cases.

Contribution

It proves the PPAD- and FIXP-completeness of computing approximate and exact equilibria in such auctions, and offers an efficient solution for fixed bidder and bid counts.

Findings

Computing an $oldsymbol{ ext{ extit{ε}}}$-equilibrium is PPAD-complete.

Computing an exact equilibrium is FIXP-complete.

An efficient algorithm exists for fixed numbers of bidders and bids.

Abstract

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.