The Limits of Auctions under Ex-Ante Collusion

We study revenue-maximizing auction design when bidders can collude ex ante—i.e., before each bidder learns his value or decides whether to participate in the auction.  We show that bidders' ex-post private information about values does not preclude them from colluding efficiently, and as a result, the seller cannot guarantee a higher expected revenue than from posting an optimal price and rationing the good.  However, a posted-price mechanism leaves the seller vulnerable to inefficient collusion.  We construct mechanisms that guarantee the non-collusive expected revenue of an optimal posted price against a broad range of forms of ex-ante collusion—including collusion by rings comprised of subsets of bidders, rings that can commit to punish bidders who refuse to join, and rings of which some bidders may be unaware.  We show that achieving such a guarantee requires protecting bidders' ex-ante expected utilities against all such forms of collusion—a property that we term strong collusion-robustness.  With symmetric bidders, our strongly collusion-robust mechanism simply combines a posted price with a first-price knockout auction conducted by the seller on behalf of the bidders.