Tight Welfare Guarantees for Pure Nash Equilibria of the Uniform Price Auction

We revisit the inefficiency of the uniform price auction, one of the standard multi-unit auction formats, for allocating multiple units of a single good. In the uniform price auction, each bidder submits a sequence of non-increasing marginal bids, for each additional unit, i.e., a submodular curve. The per unit price is then set to be the highest losing bid. We focus on the pure Nash equilibria of such auctions, for bidders with submodular valuation functions. Our result is a tight upper and lower bound on the inefficiency of equilibria, showing that the Price of Anarchy is bounded by 2.1885. This resolves one of the open questions posed in previous works on multi-unit auctions. We also discuss implications of our bounds for an alternative, more practical form of the auction, employing a “ uniform bidding ” interface.