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 that where only a single sample is available. In this paper, we take a first step toward understanding the middle ground that bridges these two approaches: that of a fixed number of samples greater than one. In a variety of contexts, we ask what is possibly the most fundamental question in this direction: "are two samples really better than one sample?". We present a few surprising negative results, and complement them with our main result: showing that the worst-case, over all regular distributions, expected-revenue guarantee of the Empirical Revenue Maximization algorithm given two samples is greater than that of this algorithm given one sample. The proof is technically challenging, and provides the first result that shows that some deterministic mechanism constructed using two samples can guarantee more than one half of the optimal revenue.
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…
We study the envy free pricing problem faced by a seller who wishes to maximize revenue by setting prices for bundles of items. If there is an unlimited supply of items and agents are single minded then we show that finding the revenue maximizing envy free allocation/pricing can be solved in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. We define an allocation/pricing as \textit{multi envy free} if no agent wishes to replace her allocation with the union of the allocations of some set of other agents and her price with the sum of their prices. We…
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 study probabilistic single-item second-price auctions where the item is characterized by a set of attributes. The auctioneer knows the actual instantiation of all the attributes, but he may choose to reveal only a subset of these attributes to the bidders. Our model is an abstraction of the following Ad auction scenario. The website (auctioneer) knows the demographic information of its impressions, and this information is in terms of a list of attributes (e.g., age, gender, country of location). The website may hide certain attributes from its advertisers (bidders) in order to create thicke…
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…
The intuition that profit is optimized by maximizing marginal revenue is a guiding principle in microeconomics. In the classical auction theory for agents with linear utility and single-dimensional preferences, Bulow and Roberts (1989) show that the optimal auction of Myerson (1981) is in fact optimizing marginal revenue. In particular Myerson's 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 c…
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…
We consider the Item Pricing problem for revenue maximization in the limited supply setting, where a single seller with $n$ items caters to $m$ buyers with unknown subadditive valuation functions who arrive in a sequence. The seller sets the prices on individual items. Each buyer buys a subset of yet unsold items that maximizes her utility. Our goal is to design pricing strategies that guarantee an expected revenue that is within a small factor $α$ of the maximum possible social welfare -- an upper bound on the maximum revenue that can be generated. Most earlier work has focused on the unlimit…
Inspired by Internet ad auction applications, we study the problem of allocating a single item via an auction when bidders place very different values on the item. We formulate this as the problem of prior-free auction and focus on designing a simple mechanism that always allocates the item. Rather than designing sophisticated pricing methods like prior literature, we design better allocation methods. In particular, we propose quasi-proportional allocation methods in which the probability that an item is allocated to a bidder depends (quasi-proportionally) on the bids. We prove that correspond…
Signaling is an important topic in the study of asymmetric information in economic settings. In particular, the transparency of information available to a seller in an auction setting is a question of major interest. We introduce the study of signaling when conducting a second price auction of a probabilistic good whose actual instantiation is known to the auctioneer but not to the bidders. This framework can be used to model impressions selling in display advertising. We study the problem of computing a signaling scheme that maximizes the auctioneer's revenue in a Bayesian setting. While the …
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 …
It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue optimization can be computationally efficiently reduced to welfare optimization in all multi-dimensional Bayesian auction problems with arbitrary (possibly combinatorial) feasibility constraints and independent additive bidders with arbitrary (possibly combinatorial) demand constraints. This reduction provides a poly-time solution to the optimal mechanism design problem in all auction settings where welfare optimization can be solved efficiently, but it is fragile to approximation and cannot provide solutions to settings wh…
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 …
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…
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…
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…
Peking University（北京大学）；Microsoft Research Asia（微软亚洲研究院）
Sponsored search is an important monetization channel for search engines, in which an auction mechanism is used to select the ads shown to users and determine the prices charged from advertisers. There have been several pieces of work in the literature that investigate how to design an auction mechanism in order to optimize the revenue of the search engine. However, due to some unrealistic assumptions used, the practical values of these studies are not very clear. In this paper, we propose a novel \emph{game-theoretic machine learning} approach, which naturally combines machine learning and ga…
Sponsored search becomes an easy platform to match potential consumers' intent with merchants' advertising. Advertisers express their willingness to pay for each keyword in terms of bids to the search engine. When a user's query matches the keyword, the search engine evaluates the bids and allocates slots to the advertisers that are displayed along side the unpaid algorithmic search results. The advertiser only pays the search engine when its ad is clicked by the user and the price-per-click is determined by the bids of other competing advertisers.
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…
We provide a reduction from revenue maximization to welfare maximization in multi-dimensional Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson's result to this setting. We also show that every feasible Bayesian auction can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder's type t_i is transformed into a virtual type f_i(t_i), via a bidder-specific function…
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…
We consider prior-free auctions for revenue and welfare maximization when agents have a common budget. The abstract environments we consider are ones where there is a downward-closed and symmetric feasibility constraint on the probabilities of service of the agents. These environments include position auctions where slots with decreasing click-through rates are auctioned to advertisers. We generalize and characterize the envy-free benchmark from Hartline and Yan (2011) to settings with budgets and characterize the optimal envy-free outcomes for both welfare and revenue. We give prior-free mech…
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…
Revenue maximization (RM) is one of the most important problems on online social networks (OSNs), which attempts to find a small subset of users in OSNs that makes the expected revenue maximized. It has been researched intensively before. However, most of exsiting literatures were based on non-adaptive seeding strategy and on simple information diffusion model, such as IC/LT-model. It considered the single influenced user as a measurement unit to quantify the revenue. Until Collaborate Game model appeared, it considered activity as a basic object to compute the revenue. An activity initiated b…
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…
We provide sufficient conditions for revenue maximization in a two-good monopoly where the buyer's values for the items come from independent (but not necessarily identical) distributions over bounded intervals. Under certain distributional assumptions, we give exact, closed-form formulas for the prices and allocation rule of the optimal selling mechanism. As a side result we give the first example of an optimal mechanism in an i.i.d. setting over a support of the form $[0,b]$ which is not deterministic. Since our framework is based on duality techniques, we were also able to demonstrate how s…
Combinatorial Auctions are a central problem in Algorithmic Mechanism Design: pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare, revenue, or profit). The problem has been well-studied in the case of limited supply (one copy of each item), and in the case of digital goods (the seller can produce additional copies at no cost). Yet in the case of resources---oil, labor, computing cycles, etc.---neither of these abstractions is just right: additional supplies of these resources can be found, but at increasing difficult…
