In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles. We believe this approach will be useful in other Bayesian mechanism design contexts.
Much of the literature in economics considers the case where the auctioneer has one item (or multiple copies of the item). In the absence of budget constraints, Myerson @cite_24 presents the characterization of any BIC mechanism in terms of expected allocation made to a bidder: This allocation must be monotone in the revealed valuation of the bidder, and furthermore, the expected price is given by applying the VCG calculation to the expected allocation. This yields a linear-time computable optimal revenue-maximizing auction that is both BIC and DSIC. The key issue with budget constraints is that the allocations need to be thresholded in order for the prices to be below the budgets @cite_10 @cite_18 @cite_0 . However, even in this case, the optimal BIC auction follows from a polymatroid characterization that can be solved by the Ellipsoid algorithm and an all-pay condition @cite_9 @cite_0 . By all-pay , we mean that the bidder pays a fixed amount given his revealed type, regardless of the allocation made. This also yields a DSIC mechanism that is @math approximation to optimal BIC revenue @cite_6 , but the result holds only for homogeneous items.
@cite_18 @cite_9 @cite_6 @cite_24 @cite_0 @cite_10
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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. 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. In this paper, we introduce a novel approach for reducing the k-item n-bidder auction with additive valuation to k-item 1-bidder auctions. This approach, called the Best-Guess reduction, can be applied to address several central questions in optimal revenue auction theory such as the relative strength of simple versus complex mechanisms, the power of randomization, and Bayesian versus dominant-strategy implementations. First, when the items have independent valuation distributions, we present a deterministic mechanism called Deterministic Best-Guess that yields at least a constant fraction of the optimal revenue by any randomized mechanism. This also gives the first simple mechanism that achieves constant fraction optimal revenue for such multi-buyer multi-item auctions. Second, if all the nk valuation random variables are independent, the optimal revenue achievable in dominant strategy incentive compatibility (DSIC) is shown to be at least a constant fraction of that achievable in Bayesian incentive compatibility (BIC). Third, when all the nk values are identically distributed according to a common one-dimensional distribution F, the optimal revenue is shown to be expressible in the closed form Θ(k(r + ∫0mr(1 − F(x)n)dx)) where r = supx≥0 x(1 − F(x)n) and m = lk nr; this revenue is achievable by a simple mechanism called 2nd-Price Bundling. All our results apply to arbitrary distributions, regular or irregular. A central issue in applying auction theory in practice is the problem of dealing with budget-constrained agents. A desirable goal in practice is to design incentive compatible, individually rational, and Pareto optimal auctions while respecting the budget constraints. Achieving this goal is particularly challenging in the presence of nontrivial combinatorial constraints over the set of feasible allocations. Toward this goal and motivated by AdWords auctions, we present an auction for polymatroidal environments satisfying these properties. Our auction employs a novel clinching technique with a clean geometric description and only needs an oracle access to the submodular function defining the polymatroid. As a result, this auction not only simplifies and generalizes all previous results, it applies to several new applications including AdWords Auctions, bandwidth markets, and video on demand. In particular, our characterization of the AdWords auction as polymatroidal constraints might be of independent interest. This allows us to design the first mechanism for Ad Auctions taking into account simultaneously budgets, multiple keywords and multiple slots. We show that it is impossible to extend this result to generic polyhedral constraints. This also implies an impossibility result for multiunit auctions with decreasing marginal utilities in the presence of budget constraints.