Hu Fu

Simultaneous auctions are (almost) efficient

Simultaneous item auctions are simple and practical procedures for allocating items to bidders with potentially complex preferences. In a simultaneous auction, every bidder submits independent bids on all items simultaneously. The allocation and prices are then resolved for each item separately, based solely on the bids submitted on that item. We study the efficiency of Bayes-Nash equilibrium (BNE) outcomes of simultaneous first- and second-price auctions when bidders have complement-free (a.k.a. subadditive) valuations. While it is known that the social welfare of every pure Nash equilibrium (NE) constitutes a constant fraction of the optimal social welfare, a pure NE rarely exists, and moreover, the full information assumption is often unrealistic. Therefore, quantifying the welfare loss in Bayes-Nash equilibria is of particular interest. Previous work established a logarithmic bound on the ratio between the social welfare of a BNE and the expected optimal social welfare in both first-price auctions (Hassidim et al., 2011) and second-price auctions (Bhawalkar and Roughgarden, 2011), leaving a large gap between a constant and a logarithmic ratio. We introduce a new proof technique and use it to resolve both of these gaps in a unified way. Specifically, we show that the expected social welfare of any BNE is at least 1/2 of the optimal social welfare in the case of first-price auctions, and at least 1/4 in the case of second-price auctions.

The Simple Economics of Approximately Optimal Auctions

The intuition that profit is optimized by maximizing marginal revenue is a guiding principle in microeconomics. In the classical auction theory for agents with quasi-linear utility and single-dimensional preferences, BR89 show that the optimal auction of M81 is in fact optimizing marginal revenue. In particular Myersons virtual values are exactly the derivative of an appropriate revenue curve. This paper considers mechanism design in environments where the agents have multi-dimensional and non-linear preferences. Understanding good auctions for these environments is considered to be the main challenge in Bayesian optimal mechanism design. In these environments maximizing marginal revenue may not be optimal, and furthermore, there is sometimes no direct way to implement the marginal revenue maximization mechanism. Our contributions are three fold: we characterize the settings for which marginal revenue maximization is optimal (by identifying an important condition that we call revenue linearity), we give simple procedures for implementing marginal revenue maximization in general, and we show that marginal revenue maximization is approximately optimal. Our approximation factor smoothly degrades in a term that quantifies how far the environment is from an ideal one (i.e., where marginal revenue maximization is optimal). Because the marginal revenue mechanism is optimal for well-studied single-dimensional agents, our generalization immediately extends many approximation results for single-dimensional agents to more general preferences. Finally, one of the biggest open questions in Bayesian algorithmic mechanism design is in developing methodologies that are not brute-force in size of the agent type space (usually exponential in the dimension for multi-dimensional agents). Our methods identify a sub problem that, e.g., for unit-demand agents with values drawn from product distributions, enables approximation mechanisms that are polynomial in the dimension.

Prior-independent auctions for risk-averse agents

We study simple and approximately optimal auctions for agents with a particular form of risk-averse preferences. We show that, for symmetric agents, the optimal revenue (given a prior distribution over the agent preferences) can be approximated by the first-price auction (which is prior independent), and, for asymmetric agents, the optimal revenue can be approximated by an auction with simple form. These results are based on two technical methods. The first is for upper-bounding the revenue from a risk-averse agent. The second gives a payment identity for mechanisms with pay-your-bid semantics.

Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions

Bayesian optimal auctions via multi- to single-agent reduction

f_1

Optimal auctions with correlated bidders are easy

,

Sketching valuation functions

f_2

Inapproximability for VCG-based combinatorial auctions

and

Conditional equilibrium outcomes via ascending price processes with applications to combinatorial auctions with item bidding

f_3: \mathbb{F}_2^k \to \{0, 1\}

On the complexity of computing an equilibrium in combinatorial auctions

are said to be triangle free if there is no