We study auctions for a commodity in unlimited supply, e.g., a digital good. In particular we consider three desirable properties for auctions: 1. Competitive: the auction achieves a constant fraction of the optimal revenue even on worst case inputs. 2. Truthful: any bidder's best strategy is to bid the maximum value they are willing to pay. 3. Envy-free: after the auction is run, no bidder would be happier with the outcome of another bidder (for digital good auctions, this means that there is a single sale price and goods are allocated to all bidders willing to pay this price). Our main result is to show that no constant-competitive auction that is truthful and always gives outcomes are envy-free. We consider two relaxations of these requirements, allowing the auction to be untruthful with vanishingly small probability, and allowing the auction to give non-envy-free outcomes with vanishingly small probability. Under both of these relaxations we get competitive auctions.
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