Multiunit auctions in which almost every bid wins

We consider multiunit auctions in which there are enough units so that each bidder but one wins every unit on which he bids. We characterize the equilibrium bidding strategy for three different payment rules: the pay-your-bid auction, the uniform-price auction in which the price equals the lowest winning bid, and the uniform-price auction in which the price equals the highest losing bid. We also consider the Vickrey pricing rule. In the case we examine, the auctions are all efficient and thus are revenue equivalent. The equilibria illustrate several phenomona that cannot arise in single-unit auctions. Even though the auctions are expected-revenue equivalent, different bidders may end up paying very different amounts. Also, in contrast to single-unit auctions, changing the seller's reservation price affects revenues, even if it remains below the lowest possible value to bidders.