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.
There is some work that deals with risk in the context of auctions. Eso @cite_15 identifies an optimal mechanism for a risk-averse seller, which always provides the same revenue bid vector by modifying Myerson's optimal mechanism; unfortunately, this mechanism does not satisfy ex-post (or even ex-interim) individual rationality, and charges bidders even when they lose. Maskin and Riley @cite_7 identifies the optimal Bayesian-incentive compatible mechanism for a risk-neutral seller when the are risk-averse. In our model, we identify mechanisms that are ex-post incentive compatible. So the buyers optimize their utility bidding truthfully for every realization of the valuations, and thus have no uncertainty or risk to deal with. @cite_0 studies risk-aversion in single-item auctions. Specifically, they show for both the first and second price mechanisms that the optimal reserve price reduces as the level of risk-aversion of the seller increases. In contrast, we identify the optimal truthful mechanism for a risk-averse seller in a single-item auction in (it happens to be a second price mechanism with a reserve), study auctions of two or more items and identify mechanisms that are simultaneously approximate for all risk-averse sellers.
@cite_0 @cite_15 @cite_7
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In this paper, we consider the problem of designing incentive compatible auctions for multiple (homogeneous) units of a good, when bidders have private valuations and private budget constraints. When only the valuations are private and the budgets are public, [8] show that the adaptive clinching auction is the unique incentive-compatible auction achieving Pareto-optimality. They further show that this auction is not truthful with private budgets, so that there is no deterministic Pareto-optimal auction with private budgets. Our main contribution is to show the following Budget Monotonicity property of this auction: When there is only one infinitely divisible good, a bidder cannot improve her utility by reporting a budget smaller than the truth. This implies that the adaptive clinching auction is incentive compatible when over-reporting the budget is not possible (for instance, when funds must be shown upfront). We can also make reporting larger budgets suboptimal with a small randomized modification to the auction. In either case, this makes the modified auction Pareto-optimal with private budgets. We also show that the Budget Monotonicity property does not hold for auctioning indivisible units of the good, showing a sharp contrast between the divisible and indivisible cases. The Budget Monotonicity property also implies other improved results in this context. For revenue maximization, the same auction improves the best-known competitive ratio due to Abrams [1] by a factor of 4, and asymptotically approaches the performance of the optimal single-price auction. Finally, we consider the problem of revenue maximization (or social welfare) in a Bayesian setting. We allow the bidders have public size constraints (on the amount of good they are willing to buy) in addition to private budget constraints. We show a simple poly-time computable 5.83-approximation to the optimal Bayesian incentive compatible mechanism, that is implementable in dominant strategies. Our technique again crucially needs the ability to prevent bidders from over-reporting budgets via randomization. We show the approximation result via designing a rounding scheme for an LP relaxation of the problem, which may be of independent interest. We present a simple auction mechanism which extends the second-price auction with reserve and is truthful in expectation. This mechanism is particularly effective in private value environments where the distribution of valuations are irregular. Bidders can "buy-it-now", or alternatively "take-a-chance" where the top d bidders are equally likely to win. The randomized take-a-chance allocation incentivizes high valuation bidders to buy-it-now. We show that for a large class of valuations, this mechanism achieves similar allocations and revenues as Myerson's optimal mechanism, and outperforms the second-price auction with reserve. In addition, we present an evaluation of bid data from Microsoft's AdECN platform. We find the valuations are irregular, and counterfactual experiments suggest our BIN-TAC mechanism would improve revenue by 11 relative to an optimal second-price mechanism with reserve. How should one sell an item to a buyer whose value for the item will only be realized next week? E.g. consider selling a flight to some executive who may or may not have a meeting with a client next week. Suppose that both the seller and the buyer only know a distribution, F, from which the buyer's value, v, for the item will be drawn. One way the seller could go about this sale is to make a take-it-or-leave-it offer today. The offer reads "pay the expected value today to get the item next week". A risk-neutral buyer would find this offer attractive, hence the seller would extract the full surplus. The unsettling feature of the afore-described mechanism is that, for some realizations of v, the bidder ends up with negative utility. In particular, while our mechanism is interim Individually Rational (IR), it is not ex-post IR. How could we fix this? One way is to wait until next week when the value is realized and make a take-it-or-leave-it offer of the item at an optimal monopoly price. The new mechanism is clearly ex-post IR, but its revenue could be much smaller than that of the previous one. Still, this trivial mechanism extracts the best possible revenue among all ex-post IR mechanisms, as a simple argument can establish. However, this optimality argument fails when several items are to be sold over consecutive periods. In this paper, we provide a characterization of the revenue-optimal, ex-post IR, dynamic mechanism selling k items over k periods to a bidder whose values are independent. In particular, we optimize the seller's revenue subject to the following strong individual rationality condition: at each period, the stage utility of the agent, defined to be the surplus from that period's allocation minus the agent's payment, must be non-negative. In particular, the non-negativity of the stage utilities implies that, at the end of each period, the agent's realized utility from participating in the mechanism so far is non-negative. We provide extensions to multiple bidders and an infinite horizon with discount factors. 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.