Discriminatory price auctions with resale and optimal quantity caps

We present a model of a discriminatory price auction in which a large bidder competes against many small bidders, followed by a post‐auction resale stage in which the large bidder is endogenously determined to be a buyer or a seller. We extend results on first‐price auctions with resale to this setting and use these results to give a tractable characterization of equilibrium behavior. We use this characterization to study the policy of capping the amount that may be won by large bidders in the auction, a policy that has received little attention in the auction literature. Our analysis shows that the trade‐offs involved when adjusting these quantity caps can be understood in terms familiar to students of asymmetric first‐price single‐unit auctions. Furthermore, whether one seeks to maximize welfare or revenue can have contradictory implications for the choice of cap.