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.
The Bayesian setting is widely studied in the economics literature @cite_21 @cite_26 @cite_10 @cite_27 @cite_18 @cite_22 @cite_0 @cite_20 @cite_5 . In this setting, the optimal (either BIC or DSIC) mechanism can always be computed by encoding the incentive compatibility constraints in an integer program and maximizing expected revenue. However, the number of variables (and constraints) in this IP is exponential in the number of bidders, there being variables for the allocations and prices for each scenario of revealed types. Therefore, the key difficulty in the Bayesian mechanism design case is computational : Can the optimal (or approximately optimal) auction be efficiently computed and implemented?
@cite_18 @cite_26 @cite_22 @cite_21 @cite_0 @cite_27 @cite_5 @cite_10 @cite_20
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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. We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity. We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity. We study the classic mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [25]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multi-dimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time. This work can be viewed as an extension and improvement of the single-agent algorithmic pricing work of [9] to the setting of multiple agents where the designer has combinatorial feasibility constraints on which agents can simultaneously obtain each service.