Optimal Auctions for Correlated Buyers with Sampling

Cr\'emer and McLean [1985] showed that, when buyers' valuations are drawn from a correlated distribution, an auction with full knowledge on the distribution can extract the full social surplus. We study whether this phenomenon persists when the auctioneer has only incomplete knowledge of the distribution, represented by a finite family of candidate distributions, and has sample access to the real distribution. We show that the naive approach which uses samples to distinguish candidate distributions may fail, whereas an extended version of the Cr\'emer-McLean auction simultaneously extracts full social surplus under each candidate distribution. With an algebraic argument, we give a tight bound on the number of samples needed by this auction, which is the difference between the number of candidate distributions and the dimension of the linear space they span

Improved Revenue Bounds for Posted-Price and Second-Price Mechanisms

We study revenue maximization through sequential posted-price (SPP) mechanisms in single-dimensional settings with $n$ buyers and independent but not necessarily identical value distributions. We construct the SPP mechanisms by considering the best of two simple pricing rules: one that imitates the revenue optimal mchanism, namely the Myersonian mechanism, via the taxation principle and the other that posts a uniform price. Our pricing rules are rather generalizable and yield the first improvement over long-established approximation factors in several settings. We design factor-revealing mathematical programs that crisply capture the approximation factor of our SPP mechanism. In the single-unit setting, our SPP mechanism yields a better approximation factor than the state of the art prior to our work (Azar, Chiplunkar & Kaplan, 2018). In the multi-unit setting, our SPP mechanism yields the first improved approximation factor over the state of the art after over nine years (Yan, 2011 and Chakraborty et al., 2010). Our results on SPP mechanisms immediately imply improved performance guarantees for the equivalent free-order prophet inequality problem. In the position auction setting, our SPP mechanism yields the first higher-than $1-1/e$ approximation factor. In eager second-price (ESP) auctions, our two simple pricing rules lead to the first improved approximation factor that is strictly greater than what is obtained by the SPP mechanism in the single-unit setting.Comment: Accepted to Operations Researc

Mechanisms for Risk Averse Agents, Without Loss

Auctions in which agents' payoffs are random variables have received increased attention in recent years. In particular, recent work in algorithmic mechanism design has produced mechanisms employing internal randomization, partly in response to limitations on deterministic mechanisms imposed by computational complexity. For many of these mechanisms, which are often referred to as truthful-in-expectation, incentive compatibility is contingent on the assumption that agents are risk-neutral. These mechanisms have been criticized on the grounds that this assumption is too strong, because "real" agents are typically risk averse, and moreover their precise attitude towards risk is typically unknown a-priori. In response, researchers in algorithmic mechanism design have sought the design of universally-truthful mechanisms --- mechanisms for which incentive-compatibility makes no assumptions regarding agents' attitudes towards risk. We show that any truthful-in-expectation mechanism can be generically transformed into a mechanism that is incentive compatible even when agents are risk averse, without modifying the mechanism's allocation rule. The transformed mechanism does not require reporting of agents' risk profiles. Equivalently, our result can be stated as follows: Every (randomized) allocation rule that is implementable in dominant strategies when players are risk neutral is also implementable when players are endowed with an arbitrary and unknown concave utility function for money.Comment: Presented at the workshop on risk aversion in algorithmic game theory and mechanism design, held in conjunction with EC 201

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distribution D. We show that if D is a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who are k-demand, additive up to a matroid constraint, or additive up to constraints of any downwards-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers [LY13, BILW14]. In the second part of the paper, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon that sometimes strictly increasing buyers' values for every set can strictly decrease the revenue of the optimal mechanism [HR12]. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof of approximate revenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms.Comment: Updated title and body to version included in TEAC. Adapted Theorem 5.2 to accommodate \eta-BIC auctions (versus exactly BIC

The Sample Complexity of Auctions with Side Information

Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter is a reasonable approach when the bidders can be classified into a few categories, but there are many instances where the classification of bidders is a continuum. For example, the classification of the bidders may be based on their annual income, their propensity to buy an item based on past behavior, or in the case of ad auctions, the click through rate of their ads. We introduce an alternate model that captures this aspect, where bidders are a priori identical, but can be distinguished based (only) on some side information the auctioneer obtains at the time of the auction. We extend the sample complexity approach of Dhangwatnotai et al. and Cole and Roughgarden to this model and obtain almost matching upper and lower bounds. As an aside, we obtain a revenue monotonicity lemma which may be of independent interest. We also show how to use Empirical Risk Minimization techniques to improve the sample complexity bound of Cole and Roughgarden for the non-identical but independent value distribution case.Comment: A version of this paper appeared in STOC 201