We consider (approximate) revenue maximization in auctions where the distribution on input valuations is given via "black box" access to samples from the distribution. We observe that the number of samples required -- the sample complexity -- is tightly related to the representation complexity of an approximately revenue-maximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities.
We consider the problem of maximizing revenue when selling 2 items to a single buyer with known valuation distributions. Hart and Nisan showed that selling each item separately using the optimal Myerson's price, gains at least half of the revenue attainable by optimal auction for two items. We show that in case the items have different revenues when sold separately the bound can be tightened.
We study the revenue maximization problem of a publisher selling consumer software. We assume that the publisher sells either traditional perpetual licenses, subscription licenses, or both. For our analysis, we employ a game-theoretic model, which enables us to derive the users' equilibrium strategies and the publisher's optimal pricing strategy. Via extensive numerical evaluations, we then demonstrate the sizable impact different pricing strategies have on the publisher's revenue, and we provide comparative statics for the most important settings parameters. Although in practice, many publish…
We present a polynomial-time algorithm that, given samples from the unknown valuation distribution of each bidder, learns an auction that approximately maximizes the auctioneer's revenue in a variety of single-parameter auction environments including matroid environments, position environments, and the public project environment. The valuation distributions may be arbitrary bounded distributions (in particular, they may be irregular, and may differ for the various bidders), thus resolving a problem left open by previous papers. The analysis uses basic tools, is performed in its entirety in val…
We study the revenue maximization problem with an imprecisely estimated distribution of a single buyer or several independent and identically distributed buyers given that this estimation is not far away from the true distribution. We use the earth mover's distance to capture the estimation error between those two distributions in terms of both values and their probabilities, i.e., the error in value space given a quantile, and the error in quantile space given a value. We give explicit characterization of the optimal mechanisms for the single buyer setting. For the multi-buyer case, we provid…
This paper settles the sample complexity of single-parameter revenue maximization by showing matching upper and lower bounds, up to a poly-logarithmic factor, for all families of value distributions that have been considered in the literature. The upper bounds are unified under a novel framework, which builds on the strong revenue monotonicity by Devanur, Huang, and Psomas (STOC 2016), and an information theoretic argument. This is fundamentally different from the previous approaches that rely on either constructing an $ε$-net of the mechanism space, explicitly or implicitly via statistical le…
We consider a revenue-maximizing single seller with $m$ items for sale to a single buyer whose value $v(\cdot)$ for the items is drawn from a known distribution $D$ of support $k$. A series of works by Cai et al. establishes that when each $v(\cdot)$ in the support of $D$ is additive or unit-demand (or $c$-demand), the revenue-optimal auction can be found in $\operatorname{poly}(m,k)$ time. We show that going barely beyond this, even to matroid-based valuations (a proper subset of Gross Substitutes), results in strong hardness of approximation. Specifically, even on instances with $m$ items an…
We study revenue maximization in settings where agents' values are interdependent: each agent receives a signal drawn from a correlated distribution and agents' values are functions of all of the signals. We introduce a variant of the generalized VCG auction with reserve prices and random admission, and show that this auction gives a constant approximation to the optimal expected revenue in matroid environments. Our results do not require any assumptions on the signal distributions, however, they require the value functions to satisfy a standard single-crossing property and a concavity-type co…
The literature on "mechanism design from samples," which has flourished in recent years at the interface of economics and computer science, offers a bridge between the classic computer-science approach of worst-case analysis (corresponding to "no samples") and the classic economic approach of average-case analysis for a given Bayesian prior (conceptually corresponding to the number of samples tending to infinity). Nonetheless, the two directions studied so far are two extreme and almost diametrically opposed directions: that of asymptotic results where the number of samples grows large, and th…
In this paper, we consider revenue maximization problem for a two server system in the presence of heterogeneous customers. We assume that the customers differ in their cost for unit delay and this is modeled as a continuous random variable with a distribution $F.$ We also assume that each server charges an admission price to each customer that decide to join its queue. We first consider the monopoly problem where both the servers belong to a single operator. The heterogeneity of the customer makes the analysis of the problem difficult. The difficulty lies in the inability to characterize the …
We consider markets consisting of a set of indivisible items, and buyers that have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a specific number $d_i$ of items; a bundle of size less than $d_i$ has no value, while a bundle of size greater than $d_i$ is worth no more than the most valued $d_i$ items (valuations being additive). We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess,…
We study the problem of selling $n$ items to a single buyer with an additive valuation function. We consider the valuation of the items to be correlated, i.e., desirabilities of the buyer for the items are not drawn independently. Ideally, the goal is to design a mechanism to maximize the revenue. However, it has been shown that a revenue optimal mechanism might be very complicated and as a result inapplicable to real-world auctions. Therefore, our focus is on designing a simple mechanism that achieves a constant fraction of the optimal revenue. Babaioff et al. propose a simple mechanism that …
We consider the black-box reduction from multi-dimensional revenue maximization to virtual welfare maximization. Cai et al. show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible ($\varepsilon$-BIC). We provide two new polynomial time transformations that convert any $\varepsilon$-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss. Our first transformation applies to any mechanism design setting with downward-closed outcome space and only requires sample access …
Maximizing the revenue from selling _more than one_ good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple "one-dimensional" mechanisms, such as selling the goods separately, _guarantee_ at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least $50\%$ when they are independent. For the case of $k>2$ independent goods, we show that selling them separately guarantees at least a $c/\log^2 k$ fraction of the optimal revenue; and, for inde…
In the design and analysis of revenue-maximizing auctions, auction performance is typically measured with respect to a prior distribution over inputs. The most obvious source for such a distribution is past data. The goal is to understand how much data is necessary and sufficient to guarantee near-optimal expected revenue. Our basic model is a single-item auction in which bidders' valuations are drawn independently from unknown and non-identical distributions. The seller is given $m$ samples from each of these distributions "for free" and chooses an auction to run on a fresh sample. How large …
We study an online revenue maximization problem where the consumers arrive i.i.d from some unknown distribution and purchase a bundle of products from the sellers. The classical approach generally assumes complete knowledge of the consumer utility functions, while recent works have been devoted to unknown linear utility functions. This paper focuses on the online posted-price model with unknown consumer distribution and unknown consumer utilities, given they are concave. Hence, the two questions to ask are i) when is the seller's online maximization problem concave, and ii) how to find the opt…
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is wort…
We consider the well known, and notoriously difficult, problem of a single revenue-maximizing seller selling two or more heterogeneous goods to a single buyer whose private values for the goods are drawn from a (possibly correlated) known distribution, and whose valuation is additive over the goods. We show that when there are two (or more) goods, _simple mechanisms_ -- such as selling the goods separately or as a bundle -- _may yield only a negligible fraction of the optimal revenue_. This resolves the open problem of Briest, Chawla, Kleinberg, and Weinberg (JET 2015) who prove the result for…
Mobile data demand is increasing tremendously in wireless social networks, and thus an efficient pricing scheme for social-enabled services is urgently needed. Though static pricing is dominant in the actual data market, price intuitively ought to be dynamically changed to yield greater revenue. The critical question is how to design the optimal dynamic pricing scheme, with prospects for maximizing the expected long-term revenue. In this paper, we study the sequential dynamic pricing scheme of a monopoly mobile network operator in the social data market. In the market, the operator, i.e., the …
Microsoft Research（微软研究院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Princeton University（普林斯顿大学）
We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of $n$ additive bidders whose values for $m$ heterogeneous items are drawn independently. For any such instance and any $\varepsilon>0$, we show that it is possible to learn an $\varepsilon$-Bayesian Incentive Compatible auction whose expected revenue is within $\varepsilon$ of the optimal $\varepsilon$-BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally, and…
We study the problem of multi-dimensional revenue maximization when selling $m$ items to a buyer that has additive valuations for them, drawn from a (possibly correlated) prior distribution. Unlike traditional Bayesian auction design, we assume that the seller has a very restricted knowledge of this prior: they only know the mean $μ_j$ and an upper bound $σ_j$ on the standard deviation of each item's marginal distribution. Our goal is to design mechanisms that achieve good revenue against an ideal optimal auction that has full knowledge of the distribution in advance. Informally, our main cont…
The common way to optimize auction and pricing systems is to set aside a small fraction of the traffic to run experiments. This leads to the question: how can we learn the most with the smallest amount of data? For truthful auctions, this is the \emph{sample complexity} problem. For posted price auctions, we no longer have access to samples. Instead, the algorithm is allowed to choose a price $p_t$; then for a fresh sample $v_t \sim \mathcal{D}$ we learn the sign $s_t = sign(p_t - v_t) \in \{-1,+1\}$. How many pricing queries are needed to estimate a given parameter of the underlying distribut…
We consider the problem of a revenue-maximizing seller with m items for sale to n additive bidders with hard budget constraints, assuming that the seller has some prior distribution over bidder values and budgets. The prior may be correlated across items and budgets of the same bidder, but is assumed independent across bidders. We target mechanisms that are Bayesian Incentive Compatible, but that are ex-post Individually Rational and ex-post budget respecting. Virtually no such mechanisms are known that satisfy all these conditions and guarantee any revenue approximation, even with just a sing…
Peking University（北京大学）；Technion - Israel Institute of Technology（以色列理工学院）；Hong Kong University of Science and Technology（香港科技大学）；Alibaba Group（阿里巴巴集团）；Shanghai Jiao Tong University（上海交通大学）
The Empirical Revenue Maximization (ERM) is one of the most important price learning algorithms in auction design: as the literature shows it can learn approximately optimal reserve prices for revenue-maximizing auctioneers in both repeated auctions and uniform-price auctions. However, in these applications the agents who provide inputs to ERM have incentives to manipulate the inputs to lower the outputted price. We generalize the definition of an incentive-awareness measure proposed by Lavi et al (2019), to quantify the reduction of ERM's outputted price due to a change of $m\ge 1$ out of $N$…
Today's queueing network systems are more rapidly evolving and more complex than those of even a few years ago. The goal of this paper is to study customers' behavior in an unobservable Markovian M/M/1 queue where consumers have to choose between two strategic decisions about information acquisition before joining or not the queue. According to their decision, customers decide to give up the service (balk the system) or to join the queue. We study the Nash equilibrium strategies: we compute the equilibrium and we prove its existence and uniqueness. Based on this result, we consider the problem…
Recommender Systems (RS) play a vital role in applications such as e-commerce and on-demand content streaming. Research on RS has mainly focused on the customer perspective, i.e., accurate prediction of user preferences and maximization of user utilities. As a result, most existing techniques are not explicitly built for revenue maximization, the primary business goal of enterprises. In this work, we explore and exploit a novel connection between RS and the profitability of a business. As recommendations can be seen as an information channel between a business and its customers, it is interest…
Tel Aviv University（特拉维夫大学）；Ben-Gurion University（本-古里安大学）；Google Research（谷歌研究院）
We consider a seller faced with buyers which have the ability to delay their decision, which we call patience. Each buyer's type is composed of value and patience, and it is sampled i.i.d. from a distribution. The seller, using posted prices, would like to maximize her revenue from selling to the buyer. In this paper, we formalize this setting and characterize the resulting Stackelberg equilibrium, where the seller first commits to her strategy, and then the buyers best respond. Following this, we show how to compute both the optimal pure and mixed strategies. We then consider a learning setti…
Many online companies sell advertisement space in second-price auctions with reserve. In this paper, we develop a probabilistic method to learn a profitable strategy to set the reserve price. We use historical auction data with features to fit a predictor of the best reserve price. This problem is delicate - the structure of the auction is such that a reserve price set too high is much worse than a reserve price set too low. To address this we develop objective variables, a new framework for combining probabilistic modeling with optimal decision-making. Objective variables are "hallucinated ob…
我们认为优先级自由拍卖收入和福利最大化时，代理人有一个共同的预算。我们考虑的抽象环境是有一个向下封闭的和对称的可行性约束的代理服务的概率。这些环境包括位置拍卖，其中具有降低的点击率的位置被拍卖给广告商。我们将Hartline和Yan（2011）的无嫉妒基准推广和表征到有预算的环境中，并表征福利和收入的最佳无嫉妒结果。我们给出了近似这些基准的无先验机制。在我们的机制的一个组成部分是一个成交拍卖的位置拍卖环境。该拍卖是Dobzinski et al.（2008）的多单位成交拍卖的推广，也是Goel et al.（2012）的多面体成交拍卖的特例。对于福利最大化，我们表明，这种成交拍卖是一个很好的近似的嫉妒无最优福利的位置拍卖环境。对于利润最大化，我们推广了Fiat等人（2002）的随机抽样利润提取拍卖，用于数字商品，以给出对称下闭环境中无嫉妒最优收入的10.0近似值。即使没有预算，利润最大化问题也很有趣，我们的机制是一个7.5近似，改进了Ha和Hartline（2012）的30.4界限。
We consider the revenue maximization problem with sharp multi-demand, in which $m$ indivisible items have to be sold to $n$ potential buyers. Each buyer $i$ is interested in getting exactly $d_i$ items, and each item $j$ gives a benefit $v_{ij}$ to buyer $i$. We distinguish between unrelated and related valuations. In the former case, the benefit $v_{ij}$ is completely arbitrary, while, in the latter, each item $j$ has a quality $q_j$, each buyer $i$ has a value $v_i$ and the benefit $v_{ij}$ is defined as the product $v_i q_j$. The problem asks to determine a price for each item and an alloca…
This paper considers Bayesian revenue maximization in the $k$-unit setting, where a monopolist seller has $k$ copies of an indivisible item and faces $n$ unit-demand buyers (whose value distributions can be non-identical). Four basic mechanisms among others have been widely employed in practice and widely studied in the literature: {\sf Myerson Auction}, {\sf Sequential Posted-Pricing}, {\sf $(k + 1)$-th Price Auction with Anonymous Reserve}, and {\sf Anonymous Pricing}. Regarding a pair of mechanisms, we investigate the largest possible ratio between the two revenues (a.k.a.\ the revenue gap)…
We consider the revenue maximization problem of a monopolist via a non-Myersonian approach that could generalize to multiple items and multiple buyers. Although such an approach does not lead to any closed-form solution of the problem, it does provide some insights into this problem from different angles. In particular, we consider both Bayesian (Bayesian Incentive Compatible + Bayesian Individually Rational) and Dominant-Strategy (Dominant-Strategy Incentive Compatible + ex-post Individually Rational) implementations, where all the buyers have additive valuations and quasi-linear utilities an…
We consider the problem of maximizing the expected revenue from selling $k$ homogeneous goods to $n$ unit-demand buyers who arrive sequentially with independent and identically distributed valuations. In this setting the optimal posted prices are dynamic in the sense that they depend on the remaining numbers of goods and buyers. We investigate how much revenue is lost when a single static price is used instead for all buyers and goods, and prove upper bounds on the ratio between the maximum revenue from dynamic prices and that from static prices. These bounds are tight for all values of $k$ an…
Designing revenue optimal auctions for selling an item to $n$ symmetric bidders is a fundamental problem in mechanism design. Myerson (1981) shows that the second price auction with an appropriate reserve price is optimal when bidders' values are drawn i.i.d. from a known regular distribution. A cornerstone in the prior-independent revenue maximization literature is a result by Bulow and Klemperer (1996) showing that the second price auction without a reserve achieves $(n-1)/n$ of the optimal revenue in the worst case. We construct a randomized mechanism that strictly outperforms the second pr…
