We consider the problem of designing truthful auctions, when the bidders' valuations have a public and a private component. In particular, we consider combinatorial auctions where the valuation of an agent $i$ for a set $S$ of items can be expressed as $v_if(S)$, where $v_i$ is a private single parameter of the agent, and the function $f$ is publicly known. Our motivation behind studying this problem is two-fold: (a) Such valuation functions arise naturally in the case of ad-slots in broadcast media such as Television and Radio. For an ad shown in a set $S$ of ad-slots, $f(S)$ is, say, the number of {\em unique} viewers reached by the ad, and $v_i$ is the valuation per-unique-viewer. (b) From a theoretical point of view, this factorization of the valuation function simplifies the bidding language, and renders the combinatorial auction more amenable to better approximation factors. We present a general technique, based on maximal-in-range mechanisms, that converts any $α$-approximation non-truthful algorithm ($α\leq 1$) for this problem into $Ω(\fracα{\log{n}})$ and $Ω(α)$-approximate truthful mechanisms which run in polynomial time and quasi-polynomial time, respectively.
We study the efficiency of sequential first-price item auctions at (subgame perfect) equilibrium. This auction format has recently attracted much attention, with previous work establishing positive results for unit-demand valuations and negative results for submodular valuations. This leaves a large gap in our understanding between these valuation classes. In this work we resolve this gap on the negative side. In particular, we show that even in the very restricted case in which each bidder has either an additive valuation or a unit-demand valuation, there exist instances in which the ineffici…
We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent depends on the entire signal profile, ${\bf s}=(s_1,\ldots,s_n)$. The literature in economics shows that the interdependent model gives rise to strong impossibility results, and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasib…
In most of microeconomic theory, consumers are assumed to exhibit decreasing marginal utilities. This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valu…
We introduce draft auctions, which is a sequential auction format where at each iteration players bid for the right to buy items at a fixed price. We show that draft auctions offer an exponential improvement in social welfare at equilibrium over sequential item auctions where predetermined items are auctioned at each time step. Specifically, we show that for any subadditive valuation the social welfare at equilibrium is an $O(\log^2(m))$-approximation to the optimal social welfare, where $m$ is the number of items. We also provide tighter approximation results for several subclasses. Our welfa…
We study efficiency loss in Bayesian revenue optimal auctions. We quantify this as the worst case ratio of loss in the realized social welfare to the social welfare that can be realized by an efficient auction. Our focus is on auctions with single-parameter buyers and where buyers' valuation sets are finite. For binary valued single-parameter buyers with independent (not necessarily identically distributed) private valuations, we show that the worst case efficiency loss ratio (ELR) is no worse than it is with only one buyer; moreover, it is at most 1/2. Moving beyond the case of binary valuati…
Auction theory traditionally assumes that bidders' valuation distributions are known to the auctioneer, such as in the celebrated, revenue-optimal Myerson auction. However, this theory does not describe how the auctioneer comes to possess this information. Recently, Cole and Roughgarden [2014] showed that an approximation based on a finite sample of independent draws from each bidder's distribution is sufficient to produce a near-optimal auction. In this work, we consider the problem of learning bidders' valuation distributions from much weaker forms of observations. Specifically, we consider …
The focus of classic mechanism design has been on truthful direct-revelation mechanisms. In the context of combinatorial auctions the truthful direct-revelation mechanism that maximizes social welfare is the VCG mechanism. For many valuation spaces computing the allocation and payments of the VCG mechanism, however, is a computationally hard problem. We thus study the performance of the VCG mechanism when bidders are forced to choose bids from a subspace of the valuation space for which the VCG outcome can be computed efficiently. We prove improved upper bounds on the welfare loss for restrict…
Revealed preference techniques are used to test whether a data set is compatible with rational behaviour. They are also incorporated as constraints in mechanism design to encourage truthful behaviour in applications such as combinatorial auctions. In the auction setting, we present an efficient combinatorial algorithm to find a virtual valuation function with the optimal (additive) rationality guarantee. Moreover, we show that there exists such a valuation function that both is individually rational and is minimum (that is, it is component-wise dominated by any other individually rational, vir…
In mechanism design, it is challenging to design the optimal auction with correlated values in general settings. Although value distribution can be further exploited to improve revenue, the complex correlation structure makes it hard to acquire in practice. Data-driven auction mechanisms, powered by machine learning, enable to design auctions directly from historical auction data, without relying on specific value distributions. In this work, we design a learning-based auction, which can encode the correlation of values into the rank score of each bidder, and further adjust the ranking rule to…
We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with a known distribution of valuations. We show that a tight lower bound on the seller's expected revenue is $1/e$ times the geometric expectation of the buyer's valuation, and that this bound is uniquely achieved for the equal revenue distribution. We show also that when the valuation's expectation and geometric expectation are close, then the seller's expected revenue is close to the expected valuation.
We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e/(e-1) for first-price auctions with fractionally subadditive valuations which reveals the worst-case price dist…
We show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that provides $m^{\frac 1 2 -ε}$ approximation to the social welfare and uses value queries only must use exponentially many value queries, where $m$ is the number of items. In contrast, ignoring incentives there exist constant ratio approximation algorithms for this problem. Our approach is based on a novel \emph{direct hardness} approach and completely skips the notoriously hard characterization step. The characterization step was the main obstacle for proving impossibility resu…
The simultaneous multiple-round auction (SMRA) and the combinatorial clock auction (CCA) are the two primary mechanisms used to sell bandwidth. Under truthful bidding, the SMRA is known to output a Walrasian equilibrium that maximizes social welfare provided the bidder valuation functions satisfy the gross substitutes property. Recently, it was shown that the combinatorial clock auction (CCA) provides good welfare guarantees for general classes of valuation functions. This motivates the question of whether similar welfare guarantees hold for the SMRA in the case of general valuation functions.…
One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions \cite{D11}, and even for truthful-in-expectation mechanisms \cite{DughmiV11}. However, both of these results are based on information-theoretic arguments for valuations given…
We study the efficiency guarantees in the simple auction environment where the auctioneer has one unit of divisible good to be distributed among a number of budget constrained agents. With budget constraints, the social welfare cannot be approximated by a better factor than the number of agents by any truthful mechanism. Thus, we follow a recent work by Dobzinski and Leme (ICALP 2014) to approximate the liquid welfare, which is the welfare of the agents each capped by her/his own budget. We design a new truthful auction with an approximation ratio of $\frac{\sqrt{5}+1}{2} \approx 1.618$, impro…
We study the efficiency of sequential multiunit auctions with two-buyers and complete information. For general valuation functions, we show that the price of anarchy is exactly $1/T$ for auctions with $T$ items for sale. For concave valuation functions, we show that the price of anarchy is bounded below by $1-1/e\simeq 0.632$. This bound is asymptotically tight as the number of items sold tends to infinity.
Complements between goods - where one good takes on added value in the presence of another - have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-compleme…
We consider auctions in which the players have very limited knowledge about their own valuations. Specifically, the only information that a Knightian player $i$ has about the profile of true valuations, $θ^*$, consists of a set of distributions, from one of which $θ_i^*$ has been drawn. The VCG mechanism guarantees very high social welfare both in single- and multi-good auctions, so long as Knightian players do not select strategies that are dominated. With such Knightian players, however, we prove that the VCG mechanism guarantees very poor social welfare in unrestricted combinatorial auction…
We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations…
University of Zurich（苏黎世大学）；Google Research（谷歌研究院）
We cast the problem of combinatorial auction design in a Bayesian framework in order to incorporate prior information into the auction process and minimize the number of rounds to convergence. We first develop a generative model of agent valuations and market prices such that clearing prices become maximum a posteriori estimates given observed agent valuations. This generative model then forms the basis of an auction process which alternates between refining estimates of agent valuations and computing candidate clearing prices. We provide an implementation of the auction using assumed density …
We study signaling in Bayesian ad auctions, in which bidders' valuations depend on a random, unknown state of nature. The auction mechanism has complete knowledge of the actual state of nature, and it can send signals to bidders so as to disclose information about the state and increase revenue. For instance, a state may collectively encode some features of the user that are known to the mechanism only, since the latter has access to data sources unaccessible to the bidders. We study the problem of computing how the mechanism should send signals to bidders in order to maximize revenue. While t…
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 efficiency of simple auctions in the presence of complements. [DMSW15] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of $O(\log m)$ for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinemen…
The planner wants to give k identical, indivisible objects to the top k valuation agents at zero costs. Each agent knows her own valuation of the object and whether it is among the top k. Modify the (k+1)st-price sealed-bid auction by introducing a small participation fee and the option not to participate in it. This simple mechanism implements the desired outcome in iteratively undominated strategies. Moreover, no pair of agents can profitably deviate from the equilibrium by coordinating their strategies or bribing each other.
We present a machine learning-powered iterative combinatorial auction (MLCA). The main goal of integrating machine learning (ML) into the auction is to improve preference elicitation, which is a major challenge in large combinatorial auctions (CAs). In contrast to prior work, our auction design uses value queries instead of prices to drive the auction. The ML algorithm is used to help the auction decide which value queries to ask in every iteration. While using ML inside a CA introduces new challenges, we demonstrate how we obtain a design that is individually rational, satisfies no-deficit, h…
We study the asymptotic average-case efficiency of static and anonymous posted prices for $n$ agents and $m(n)$ multiple identical items with $m(n)=o\left(\frac{n}{\log n}\right)$. When valuations are drawn i.i.d from some fixed continuous distribution (each valuation is a vector in $\Re_+^m$ and independence is assumed only across agents) we show: (a) for any "upper mass" distribution there exist posted prices such that the expected revenue and welfare of the auction approaches the optimal expected welfare as $n$ goes to infinity; specifically, the ratio between the expected revenue of our po…
Second-price auctions with deposits are frequently used in blockchain environments. An auction takes place on-chain: bidders deposit an amount that fully covers their bid (but possibly exceeds it) in a smart contract. The deposit is used as insurance against bidders not honoring their bid if they win. The deposit, but not the bid, is publicly observed during the bidding phase of the auction. The visibility of deposits can fundamentally change the strategic structure of the auction if bidding happens sequentially: Bidding is costly since deposit are costly to make. Thus, deposits can be used as…
In a seminal paper, McAfee (1992) presented a truthful mechanism for double auctions, attaining asymptotically-optimal gain-from-trade without any prior information on the valuations of the traders. McAfee's mechanism handles single-parametric agents, allowing each seller to sell a single unit and each buyer to buy a single unit. This paper presents a double-auction mechanism that handles multi-parametric agents and allows multiple units per trader, as long as the valuation functions of all traders have decreasing marginal returns. The mechanism is prior-free, ex-post individually-rational, do…
This paper establishes the Price of Stability (PoS) for First Price Auctions, for all equilibrium concepts that have been studied in the literature: Bayes Nash Equilibrium $\subsetneq$ Bayes Correlated Equilibrium $\subsetneq$ Bayes Coarse Correlated Equilibrium} $\bullet$ Bayes Nash Equilibrium: For independent valuations, the tight PoS is $1 - 1/ e^{2} \approx 0.8647$, matching the counterpart Price of Anarchy (PoA) bound \cite{JL22}. For correlated valuations, the tight $\PoS$ is $1 - 1 / e \approx 0.6321$, matching the counterpart PoA bound \cite{ST13,S14}. This result indicates that, in t…
In markets such as digital advertising auctions, bidders want to maximize value rather than payoff. This is different to the utility functions typically assumed in auction theory and leads to different strategies and outcomes. We refer to bidders who maximize value as value bidders. While simple single-object auction formats are truthful, standard multi-object auction formats allow for manipulation. It is straightforward to show that there cannot be a truthful and revenue-maximizing deterministic auction mechanism with value bidders and general valuations. Approximation has been used as a mean…
The literature on the Price of Anarchy (PoA) of simple auctions employs a no-overbidding assumption but has completely overlooked the no-underbidding phenomenon, which is evident in empirical studies on variants of the second price auction. In this work, we provide a theoretical foundation for the no-underbidding phenomenon. We study the PoA of simultaneous 2nd price auctions (S2PA) under a new natural condition of {\em no underbidding}, meaning that agents never bid on items less than their marginal values. We establish improved (mostly tight) bounds on the PoA of S2PA under no underbidding f…
We study the design of mechanisms in combinatorial auction domains. We focus on settings where the auction is repeated, motivated by auctions for licenses or advertising space. We consider models of agent behaviour in which they either apply common learning techniques to minimize the regret of their bidding strategies, or apply short-sighted best-response strategies. We ask: when can a black-box approximation algorithm for the base auction problem be converted into a mechanism that approximately preserves the original algorithm's approximation factor on average over many iterations? We present…
Along with substantial progress made recently in designing near-optimal mechanisms for multi-item auctions, interesting structural questions have also been raised and studied. In particular, is it true that the seller can always extract more revenue from a market where the buyers value the items higher than another market? In this paper we obtain such a revenue monotonicity result in a general setting. Precisely, consider the revenue-maximizing combinatorial auction for $m$ items and $n$ buyers in the Bayesian setting, specified by a valuation function $v$ and a set $F$ of $nm$ independent ite…
The current art in optimal combinatorial auctions is limited to handling the case of single units of multiple items, with each bidder bidding on exactly one bundle (single minded bidders). This paper extends the current art by proposing an optimal auction for procuring multiple units of multiple items when the bidders are single minded. The auction minimizes the cost of procurement while satisfying Bayesian incentive compatibility and interim individual rationality. Under appropriate regularity conditions, this optimal auction also satisfies dominant strategy incentive compatibility.
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 \emph{Best-Guess} reduction, can be applied to address several central questions in optimal revenue auction theory such as the power of randomization, and Bayesian versus dominant-strategy implementations. First, when the items have independent valuation distributions, we present a deterministic mechanism called {\it Deterministic Best-Guess} that yields at least a constant fraction of the optimal revenue by any randomized …
We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either …
Modern ad auctions allow advertisers to target more specific segments of the user population. Unfortunately, this is not always in the best interest of the ad platform. In this paper, we examine the following basic question in the context of second-price ad auctions: how should an ad platform optimally reveal information about the ad opportunity to the advertisers in order to maximize revenue? We consider a model in which bidders' valuations depend on a random state of the ad opportunity. Different from previous work, we focus on a more practical, and challenging, situation where the space of …
We revisit the problem of designing the profit-maximizing single-item auction, solved by Myerson in his seminal paper for the case in which bidder valuations are independently distributed. We focus on general joint distributions, seeking the optimal deterministic incentive compatible auction. We give a geometric characterization of the optimal auction, resulting in a duality theorem and an efficient algorithm for finding the optimal deterministic auction in the two-bidder case and an NP-completeness result for three or more bidders.
Motivated by applications such as stock exchanges and spectrum auctions, there is a growing interest in mechanisms for arranging trade in two-sided markets. Existing mechanisms are either not truthful, or do not guarantee an asymptotically-optimal gain-from-trade, or rely on a prior on the traders' valuations, or operate in limited settings such as a single kind of good. We extend the random market-halving technique used in earlier works to markets with multiple kinds of goods, where traders have gross-substitute valuations. We present MIDA: a Multi Item-kind Double-Auction mechanism. It is pr…
We present a new type of monotone submodular functions: \emph{multi-peak submodular functions}. Roughly speaking, given a family of sets $\cF$, we construct a monotone submodular function $f$ with a high value $f(S)$ for every set $S \in {\cF}$ (a "peak"), and a low value on every set that does not intersect significantly any set in $\cF$. We use this construction to show that a better than $(1-\frac{1}{2e})$-approximation ($\simeq 0.816$) for welfare maximization in combinatorial auctions with submodular valuations is (1) impossible in the communication model, (2) NP-hard in the computational…
Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard for combinatorial auctions, the Generalized Vickrey Auction (GVA). Traditional analysis of these mechanisms - in particular, their truth revelation properties - assumes that the optimization problems are solved precisely. In reality, these optimization problems can usually be solved only in an approximate fas…
We study a class of iterative combinatorial auctions which can be viewed as subgradient descent methods for the problem of pricing bundles to balance supply and demand. We provide concrete convergence rates for auctions in this class, bounding the number of auction rounds needed to reach clearing prices. Our analysis allows for a variety of pricing schemes, including item, bundle, and polynomial pricing, and the respective convergence rates confirm that more expressive pricing schemes come at the cost of slower convergence. We consider two models of bidder behavior. In the first model, bidders…
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 optim…
Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders' valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and de…
