The existing literature on optimal auctions focuses on optimizing the expected revenue of the seller, and is appropriate for risk-neutral sellers. In this paper, we identify good mechanisms for risk-averse sellers. As is standard in the economics literature, we model the risk-aversion of a seller by endowing the seller with a monotone concave utility function. We then seek robust mechanisms that are approximately optimal for all sellers, no matter what their levels of risk-aversion are. We have two main results for multi-unit auctions with unit-demand bidders whose valuations are drawn i.i.d. from a regular distribution. First, we identify a posted-price mechanism called the Hedge mechanism, which gives a universal constant factor approximation; we also show for the unlimited supply case that this mechanism is in a sense the best possible. Second, we show that the VCG mechanism gives a universal constant factor approximation when the number of bidders is even only a small multiple of the number of items. Along the way we point out that Myerson's characterization of the optimal mechanisms fails to extend to utility-maximization for risk-averse sellers, and establish interesting properties of regular distributions and monotone hazard rate distributions.
Finally, we mention papers that inspire our proof techniques. @cite_11 proposes posted-price mechanisms, and it uses Myerson's mechanism to guide the selection of the prices. We use a similar idea in . Bulow and Klemperer @cite_4 shows that the VCG mechanism with @math extra bidders yields better expected revenue than the optimal mechanism so long as the bidder valuations are drawn i.i.d. from a regular distribution. @cite_8 extends the result of Bulow and Klemperer @cite_4 to matroid settings, and introduces the problem of designing markets with good revenue properties. We use ideas from these papers to bound the performance of the VCG mechanism in . The characterization of regular distributions in terms of concave revenue functions is implicit in Myerson @cite_3 , and is used explicitly in @cite_11 and @cite_10 .
@cite_4 @cite_8 @cite_3 @cite_10 @cite_11
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The VCG mechanism is the gold standard for combinatorial auctions (CAs), and it maximizes social welfare. In contrast, the revenue-maximizing (aka optimal) CA is unknown, and designing one is NP-hard. Therefore, research on optimal CAs has progressed into special settings. Notably, Levin [1997] derived the optimal CA for complements when each agent's private type is one-dimensional. (This does not fall inside the well-studied "single-parameter environment".) We introduce a new research avenue for increasing revenue where we poke holes in the allocation space--based on the bids--and then use a welfare-maximizing allocation rule within the remaining allocation set. In this paper, the first step down this avenue, we introduce a new form of "reserve pricing" into CAs. We show that Levin's optimal revenue can be 2-approximated by using "monopoly reserve prices" to curtail the allocation set, followed by welfare-maximizing allocation and Levin's payment rule. A key lemma of potential independent interest is that the expected revenue from any truthful allocation-monotonic mechanism equals the expected virtual valuation; this generalizes Myerson's lemma [1981] from the single-parameter environment. Our mechanism is close to the gold standard and thus easier to adopt than Levin's. It also requires less information about the prior over the bidders' types, and is always more efficient. Finally, we show that the optimal revenue can be 6- approximated even if the "reserve pricing" is required to be symmetric across bidders. Revenue maximization in multi-item settings is notoriously elusive. This paper studies a class of two-item auctions which we call a mixed-bundling auction with reserve prices (MBARP). It calls VCG on an enlarged set of agents by adding the seller---who has reserve valuations for each bundle of items---and a fake agent who receives nothing nor has valuations for any item or bundle, but has a valuation for pure bundling allocations, i.e., allocations where the two items are allocated to a single agent. This is a strict subclass of several known classes of auctions, including the affine maximizer auction (AMA), λ-aution, and the virtual valuations combinatorial auction (VVCA). As we show, a striking feature of MBARP is that its revenue can be represented in a simple closed form as a function of the parameters. Thus, we can solve first-order conditions on the parameters and obtain the optimal MBARP. The optimal MBARP yields significantly higher revenue than prior auctions for which the revenue-maximizing parameters could be solved for in closed form: separate Myerson auctions, pure-bundling Myerson auction, VCG, and mixed-bundling auction without reserve prices. Its revenue even exceeds that obtained via simulation within broader classes: VVCA and AMA. The focus of classic mechanism design has been on truthful direct-revelation mechanisms. In the context of combinatorial auctions the truthful direct-revelation mechanism that maximizes social welfare is the VCG mechanism. For many valuation spaces computing the allocation and payments of the VCG mechanism, however, is a computationally hard problem. We thus study the performance of the VCG mechanism when bidders are forced to choose bids from a subspace of the valuation space for which the VCG outcome can be computed efficiently. We prove improved upper bounds on the welfare loss for restrictions to additive bids and upper and lower bounds for restrictions to non-additive bids. These bounds show that the welfare loss increases in expressiveness. All our bounds apply to equilibrium concepts that can be computed in polynomial time as well as to learning outcomes. We exhibit a property of the VCG mechanism that can help explain the surprising rarity with which it is used even in settings with unit demand: a relative lack of robustness to inaccuracies in the choice of its parameters. For a standard position auction environment in which the auctioneer may not know the precise relative values of the positions, we show that under both complete and incomplete information a non-truthful mechanism supports the truthful outcome of the VCG mechanism for a wider range of these values than the VCG mechanism itself. The result for complete information concerns the generalized second-price mechanism and lends additional theoretical support to the use of this mechanism in practice. Particularly interesting from a technical perspective is the case of incomplete information, where a surprising combinatorial equivalence helps us to avoid confrontation with an unwieldy differential equation.