A large fraction of online advertisement is sold via repeated second price auctions. In these auctions, the reserve price is the main tool for the auctioneer to boost revenues. In this work, we investigate the following question: Can changing the reserve prices based on the previous bids improve the revenue of the auction, taking into account the long-term incentives and strategic behavior of the bidders? We show that if the distribution of the valuations is known and satisfies the standard regularity assumptions, then the optimal mechanism has a constant reserve. However, when there is uncertainty in the distribution of the valuations, previous bids can be used to learn the distribution of the valuations and to update the reserve price. We present a simple, approximately incentive-compatible, and asymptotically optimal dynamic reserve mechanism that can significantly improve the revenue over the best static reserve.
The paper is from July 2014 (our submission to WINE 2014), posted later here on the arxiv to complement the 1-page abstract in the WINE 2014 proceedings.
This letter considers the design of an auction mechanism to sell the object of a seller when the buyers quantize their private value estimates regarding the object prior to communicating them to the seller. The designed auction mechanism maximizes the utility of the seller (i.e., the auction is optimal), prevents buyers from communicating falsified quantized bids (i.e., the auction is incentive-compatible), and ensures that buyers will participate in the auction (i.e., the auction is individually-rational). The letter also investigates the design of the optimal quantization thresholds using wh…
A seller chooses a reserve price in a second-price auction to maximize worst-case expected revenue when she knows only the mean of value distribution and an upper bound on either values themselves or variance. Values are private and iid. Using an indirect technique, we prove that it is always optimal to set the reserve price to the seller's own valuation. However, the maxmin reserve price may not be unique. If the number of bidders is sufficiently high, all prices below the seller's valuation, including zero, are also optimal. A second-price auction with the reserve equal to seller's value (or…
We derive optimal strategies for a bidding agent that participates in multiple, simultaneous second-price auctions with perfect substitutes. We prove that, if everyone else bids locally in a single auction, the global bidder should always place non-zero bids in all available auctions, provided there are no budget constraints. With a budget, however, the optimal strategy is to bid locally if this budget is equal or less than the valuation. Furthermore, for a wide range of valuation distributions, we prove that the problem of finding the optimal bids reduces to two dimensions if all auctions are…
Stanford Graduate School of Business, Stanford University（斯坦福大学斯坦福商学院）
Motivated by online advertising auctions, we study auction design in repeated auctions played by simple Artificial Intelligence algorithms (Q-learning). We find that first-price auctions with no additional feedback lead to tacit-collusive outcomes (bids lower than values), while second-price auctions do not. We show that the difference is driven by the incentive in first-price auctions to outbid opponents by just one bid increment. This facilitates re-coordination on low bids after a phase of experimentation. We also show that providing information about lowest bid to win, as introduced by Goo…
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 quasi-proportional auctions, the allocation is shared among bidders in proportion to their weighted bids. The auctioneer selects a bid weight function, and bidders know the weight function when they bid. In this note, we analyze how weight functions that are exponential in the bid affect bidder behavior. We show that exponential weight functions have a pure-strategy Nash equilibrium, we characterize bids at an equilibrium, and we compare it to an equilibrium for power weight functions.
What fraction of the single item $n$ buyers setting's expected optimal revenue MyeRev can the second price auction with reserves achieve? In the special case where the buyers' valuation distributions are all drawn i.i.d. and the distributions satisfy the regularity condition, the second price auction with an anonymous reserve (ASP) is the optimal auction itself. As the setting gets more complex, there are established upper bounds on the fraction of MyeRev that ASP can achieve. On the contrary, no such upper bounds are known for the fraction of MyeRev achievable by the second price auction with…
In classic auction theory, reserve prices are known to be effective for improving revenue for the auctioneer against quasi-linear utility maximizing bidders. The introduction of reserve prices, however, usually do not help improve total welfare of the auctioneer and the bidders. In this paper, we focus on value maximizing bidders with return on spend constraints -- a paradigm that has drawn considerable attention recently as more advertisers adopt auto-bidding algorithms in advertising platforms -- and show that the introduction of reserve prices has a novel impact on the market. Namely, by ch…
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…
We study a class of manipulations in combinatorial auctions where bidders fundamentally misrepresent what goods they are interested in. Prior work has largely assumed that bidders only submit bids on their bundles of interest, which we call simple bidding: strategizing over the bid amounts, but not the bundle identities. However, we show that there exists an entire class of auction instances for which simple bids are never optimal in Bayes-Nash equilibrium, always being strictly dominated by complex bids (where bidders bid on goods they are not interested in). We show this result for the two m…
We present a general framework for designing approximately revenue-optimal mechanisms for multi-item additive auctions, which applies to both truthful and non-truthful auctions. Given a (not necessarily truthful) single-item auction format $A$ satisfying certain technical conditions, we run simultaneous item auctions augmented with a personalized entry fee for each bidder that must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types, and in particular are independent of realized bids. We bound the revenue of the resulting two-part …
We initiate the study of the social welfare loss caused by corrupt auctioneers, both in single-item and multi-unit auctions. In our model, the auctioneer may collude with the winning bidders by letting them lower their bids in exchange for a (possibly bidder-dependent) fraction $γ$ of the surplus. We consider different corruption schemes. In the most basic one, all winning bidders lower their bid to the highest losing bid. We show that this setting is equivalent to a $γ$-hybrid auction in which the payments are a convex combination of first-price and the second-price payments. More generally, …
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.
Setting an effective reserve price for strategic bidders in repeated auctions is a central question in online advertising. In this paper, we investigate how to set an anonymous reserve price in repeated auctions based on historical bids in a way that balances revenue and incentives to misreport. We propose two simple and computationally efficient methods to set reserve prices based on the notion of a clearing price and make them robust to bidder misreports. The first approach adds random noise to the reserve price, drawing on techniques from differential privacy. The second method applies a sm…
This paper proves the tight sample complexity of {\sf Second-Price Auction with Anonymous Reserve}, up to a logarithmic factor, for each of all the value distribution families studied in the literature: $[0,\, 1]$-bounded, $[1,\, H]$-bounded, regular, and monotone hazard rate (MHR). Remarkably, the setting-specific tight sample complexity $\mathsf{poly}(\varepsilon^{-1})$ depends on the precision $\varepsilon \in (0, 1)$, but not on the number of bidders $n \geq 1$. Further, in the two bounded-support settings, our learning algorithm allows {\em correlated} value distributions. In contrast, th…
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…
Simultaneous item auctions are simple procedures for allocating items to bidders with potentially complex preferences over different item sets. In a simultaneous auction, every bidder submits 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. Such procedures occur in practice (e.g. eBay) but are not truthful. We study the efficiency of Bayesian Nash equilibrium (BNE) outcomes of simultaneous first- and second-price auctions when bidders have complement-free (a.k.a. subadditive) valuations. We …
This paper develops the theory of mechanism redesign by which an auctioneer can reoptimize an auction based on bid data collected from previous iterations of the auction on bidders from the same market. We give a direct method for estimation of the revenue of a counterfactual auction from the bids in the current auction. The estimator is a simple weighted order statistic of the bids and has the optimal error rate. Two applications of our estimator are A/B testing (a.k.a., randomized controlled trials) and instrumented optimization (i.e., revenue optimization subject to being able to do accurat…
The display advertising industry has recently transitioned from second- to first-price auctions as its primary mechanism for ad allocation and pricing. In light of this, publishers need to re-evaluate and optimize their auction parameters, notably reserve prices. In this paper, we propose a gradient-based algorithm to adaptively update and optimize reserve prices based on estimates of bidders' responsiveness to experimental shocks in reserves. Our key innovation is to draw on the inherent structure of the revenue objective in order to reduce the variance of gradient estimates and improve conve…
We study the question of setting and testing reserve prices in single item auctions when the bidders are not identical. At a high level, there are two generalizations of the standard second price auction: in the lazy version we first determine the winner, and then apply reserve prices; in the eager version we first discard the bidders not meeting their reserves, and then determine the winner among the rest. We show that the two versions have dramatically different properties: lazy reserves are easy to optimize, and A/B test in production, whereas eager reserves always lead to higher welfare, b…
Over the past few years, more and more Internet advertisers have started using automated bidding for optimizing their advertising campaigns. Such advertisers have an optimization goal (e.g. to maximize conversions), and some constraints (e.g. a budget or an upper bound on average cost per conversion), and the automated bidding system optimizes their auction bids on their behalf. Often, these advertisers participate on multiple advertising channels and try to optimize across these channels. A central question that remains unexplored is how automated bidding affects optimal auction design in the…
We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data-set that contains the submitted bids of $n$ buyers in a set of auctions and the problem is to return personalized reserve prices $\textbf r$ that maximize the revenue earned on these auctions by running eager second price auctions with reserve $\textbf r$. For this problem, which is known to be APX-hard, we present a novel LP formulation and a rounding procedure which achieves a $(1+2(\sqrt{2}-1)e^{\sqrt{2}-…
A common practice in many auctions is to offer bidders an opportunity to improve their bids, known as a Best and Final Offer (BAFO) stage. This final bid can depend on new information provided about either the asset or the competitors. This paper examines the effects of new information regarding competitors, seeking to determine what information the auctioneer should provide assuming the set of allowable bids is discrete. The rational strategy profile that maximizes the revenue of the auctioneer is the one where each bidder makes the highest possible bid that is lower than his valuation of the…
We study the problem of finding personalized reserve prices for unit-demand buyers in multi-unit eager VCG auctions with correlated buyers. The input to this problem is a dataset of submitted bids of $n$ buyers in a set of auctions. The goal is to find a vector of reserve prices, one for each buyer, that maximizes the total revenue across all auctions. Roughgarden and Wang (2016) showed that this problem is APX-hard but admits a greedy $\frac{1}{2}$-approximation algorithm. Later, Derakhshan, Golrezai, and Paes Leme (2019) gave an LP-based algorithm achieving a $0.68$-approximation for the (im…
We study revenue optimization learning algorithms for repeated second-price auctions with reserve where a seller interacts with multiple strategic bidders each of which holds a fixed private valuation for a good and seeks to maximize his expected future cumulative discounted surplus. We propose a novel algorithm that has strategic regret upper bound of $O(\log\log T)$ for worst-case valuations. This pricing is based on our novel transformation that upgrades an algorithm designed for the setup with a single buyer to the multi-buyer case. We provide theoretical guarantees on the ability of a tra…
A classical trading experiment consists of a set of unit demand buyers and unit supply sellers with identical items. Each agent's value or opportunity cost for the item is their private information and preferences are quasi-linear. Trade between agents employs a double oral auction (DOA) in which both buyers and sellers call out bids or offers which an auctioneer recognizes. Transactions resulting from accepted bids and offers are recorded. This continues until there are no more acceptable bids or offers. Remarkably, the experiment consistently terminates in a Walrasian price. The main result …
Buyers (e.g., advertisers) often have limited financial and processing resources, and so their participation in auctions is throttled. Changes to auctions may affect bids or throttling and any change may affect what winners pay. This paper shows that if an A/B experiment affects only bids, then the observed treatment effect is unbiased when all the bidders in an auction are randomly assigned to A or B but it can be severely biased otherwise, even in the absence of throttling. Experiments that affect throttling algorithms can also be badly biased, but the bias can be substantially reduced if th…
In auction theory, cryptography has been used to achieve anonymity of the participants, security and privacy of the bids, secure computation and to simulate mediator (auctioneer). Auction theory focuses on revenue and Cryptography focuses on security and privacy. Involving Cryptography at base level, to enhance revenue gives entirely new perspective and insight to Auction theory, thereby achieving the core goals of auction theory. In this report, we try to investigate an interesting field of study in Auction Theory using Cryptographic primitives.
We study the efficiency of simple combinatorial auctions for the allocation of a set of items to a set of agents, with private subadditive valuation functions and budget constraints. The class we consider includes all auctions that allocate each item independently to the agent that submits the highest bid for it, and requests a payment that depends on the bids of all agents only for this item. Two well-known examples of this class are the simultaneous first and second price auctions. We focus on the pure equilibria of the induced strategic games, and using the liquid welfare as our efficiency …
With the increasing use of auctions in online advertising, there has been a large effort to study seller revenue maximization, following Myerson's seminal work, both theoretically and practically. We take the point of view of the buyer in classical auctions and ask the question of whether she has an incentive to shade her bid even in auctions that are reputed to be truthful, when aware of the revenue optimization mechanism. We show that in auctions such as the Myerson auction or a VCG with reserve price set as the monopoly price, the buyer who is aware of this information has indeed an incenti…
Good economic mechanisms depend on the preferences of participants in the mechanism. For example, the revenue-optimal auction for selling an item is parameterized by a reserve price, and the appropriate reserve price depends on how much the bidders are willing to pay. A mechanism designer can potentially learn about the participants' preferences by observing historical data from the mechanism; the designer could then update the mechanism in response to learned preferences to improve its performance. The challenge of such an approach is that the data corresponds to the actions of the participan…
This paper studies an environment of simultaneous, separate, first-price auctions for complementary goods. Agents observe private values of each good before making bids, and the complementarity between goods is explicitly incorporated in their utility. For simplicity, a model is presented with two first-price auctions and two bidders. We show that a monotone pure-strategy Bayesian Nash Equilibrium exists in the environment.
Since economic mechanisms are often applied to very different instances of the same problem, it is desirable to identify mechanisms that work well in a wide range of circumstances. We pursue this goal for a position auction setting and specifically seek mechanisms that guarantee good outcomes under both complete and incomplete information. A variant of the generalized first-price mechanism with multi-dimensional bids turns out to be the only standard mechanism able to achieve this goal, even when types are one-dimensional. The fact that expressiveness beyond the type space is both necessary an…
Robust Mechanism Design with Anonymous Information
Shanghai University of Finance and Economics（上海财经大学）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. 米尔顿·斯图尔特工业与系统工程学院）
This paper develops tools for welfare and revenue analyses of Bayes-Nash equilibria in asymmetric auctions with single-dimensional agents. We employ these tools to derive price of anarchy results for social welfare and revenue. Our approach separates the standard smoothness framework into two distinct parts, isolating the analysis common to any auction from the analysis specific to a given auction. The first part relates a bidder's contribution to welfare in equilibrium to their contribution to welfare in the optimal auction using the price the bidder faces for additional allocation. Intuitive…
Auctions have been proposed as a way to provide economic incentives for primary users to dynamically allocate unused spectrum to other users in need of it. Previously proposed schemes do not take into account the fact that the power constraints of users might prevent them from transmitting their bid prices to the auctioneer with high precision and that transmitted bid prices must travel through a noisy channel. These schemes also have very high overheads which cannot be accommodated in wireless standards. We propose auction schemes where a central clearing authority auctions spectrum to users …
Interactions between bids to show ads online can lead to an advertiser's ad being shown to more men than women even when the advertiser does not target towards men. We design bidding strategies that advertisers can use to avoid such emergent discrimination without having to modify the auction mechanism. We mathematically analyze the strategies to determine the additional cost to the advertiser for avoiding discrimination, proving our strategies to be optimal in some settings. We use simulations to understand other settings.
We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution $D$ in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a no-regret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically: - If the buyer bids according to EX…
In quasi-proportional auctions, each bidder receives a fraction of the allocation equal to the weight of their bid divided by the sum of weights of all bids, where each bid's weight is determined by a weight function. We study the relationship between the weight function, bidders' private values, number of bidders, and the seller's revenue in equilibrium. It has been shown that if one bidder has a much higher private value than the others, then a nearly flat weight function maximizes revenue. Essentially, threatening the bidder who has the highest valuation with having to share the allocation …
Prices, Bids, Values: One ML-Powered Combinatorial Auction to Rule Them All
Department of Informatics, University of Zurich, Zurich, Switzerland（苏黎世大学信息学院）；ETH AI Center, Zurich, Switzerland（苏黎世联邦理工学院AI中心）；Department of Mathematics, ETH Zurich, Zurich, Switzerland（苏黎世联邦理工学院数学系）；Department of Statistics, University of California, Berkeley, USA（加州大学伯克利分校统计系）
We address the problem of improving bidders' strategies in prior-dependent revenue-maximizing auctions and introduce a simple and generic method to design novel bidding strategies if the seller uses past bids to optimize her mechanism. We propose a simple and agnostic strategy, independent of the distribution of the competition, that is robust to mechanism changes and local (as opposed to global) optimization of e.g. reserve prices by the seller. This strategy guarantees an increase in utility compared to the truthful strategy for any distribution of the competition. In textbook-style examples…
Most of the economic reports forecast that almost half of the worldwide market value unlocked by AI over the next decade (up to 6 trillion USD per year) will be in marketing&sales. In particular, AI will enable the optimization of more and more intricate economic settings, in which multiple different activities need to be jointly automated. This is the case of, e.g., Google Hotel Ads and Tripadvisor, where auctions are used to display ads of similar products or services together with their prices. As in classical ad auctions, the ads are ranked depending on the advertisers' bids, whereas, diff…
Generative Auto-Bidding in Large-Scale Competitive Auctions via Diffusion Completer-Aligner
Kuaishou Technology（快手科技）；Nanyang Technological University（南洋理工大学）；City University of HongKong（香港城市大学）；Chinese University of HongKong（香港中文大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
