Budgets play a significant role in real-world sequential auction markets such as those implemented by internet companies. To maximize the value provided to auction participants, spending is smoothed across auctions so budgets are used for the best opportunities. Motivated by a mechanism used in practice by several companies, this paper considers a smoothing procedure that relies on {\em pacing multipliers}: on behalf of each buyer, the auction market applies a factor between 0 and 1 that uniformly scales the bids across all auctions. Reinterpreting this process as a game between buyers, we introduce the notion of {\em pacing equilibrium}, and prove that they are always guaranteed to exist. We demonstrate through examples that a market can have multiple pacing equilibria with large variations in several natural objectives. We show that pacing equilibria refine another popular solution concept, competitive equilibria, and show further connections between the two solution concepts. Although we show that computing either a social-welfare-maximizing or a revenue-maximizing pacing equilibrium is NP-hard, we present a mixed-integer program (MIP) that can be used to find equilibria optimizing several relevant objectives. We use the MIP to provide evidence that: (1) equilibrium multiplicity occurs very rarely across several families of random instances, (2) static MIP solutions can be used to improve the outcomes achieved by a dynamic pacing algorithm with instances based on a real-world auction market, and (3) for the instances we study, buyers do not have an incentive to misreport bids or budgets provided there are enough participants in the auction.
Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a pacing multiplier between zero and one, and her bid on each item is multiplicatively scaled down by the pacing multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is known as a pacing equilibrium. In this work, we show that the problem of finding an approxi…
Budget-management systems are one of the key components of modern auction markets. Internet advertising platforms typically offer advertisers the possibility to pace the rate at which their budget is depleted, through budget-pacing mechanisms. We focus on multiplicative pacing mechanisms in an online setting in which a bidder is repeatedly confronted with a series of advertising opportunities. After collecting bids, each item is then allocated through a single-item, second-price auction. If there were no budgetary constraints, bidding truthfully would be an optimal choice for the advertiser. H…
Mature internet advertising platforms offer high-level campaign management tools to help advertisers run their campaigns, often abstracting away the intricacies of how each ad is placed and focusing on aggregate metrics of interest to advertisers. On such platforms, advertisers often participate in auctions through a proxy bidder, so the standard incentive analyses that are common in the literature do not apply directly. In this paper, we take the perspective of a budget management system that surfaces aggregated incentives -- instead of individual auctions -- and compare first and second pric…
In this paper, we analyze a natural learning algorithm for uniform pacing of advertising budgets, equipped to adapt to varying ad sale platform conditions. On the demand side, advertisers face a fundamental technical challenge in automating bidding in a way that spreads their allotted budget across a given campaign subject to hidden, and potentially dynamic, cost functions. This automation and calculation must be done in runtime, implying a necessarily low computational cost for the high frequency auction rate. Advertisers are additionally expected to exhaust nearly all of their sub-interval (…
Notifications are important for the user experience in mobile apps and can influence their engagement. However, too many notifications can be disruptive for users. A typical mobile app usually has several types of notification, managed by distinct teams with objectives that are possibly conflicting with each other, or even with the overall platform objective. Therefore, there is a need for careful curation of notifications sent to users of these different types. In this work, we study a novel centralized approach for notification optimization, where we view the opportunities to send user notif…
We present a new algorithm for computing pure-strategy $\varepsilon$-Bayes-Nash equilibria ($\varepsilon$-BNEs) in combinatorial auctions with continuous value and action spaces. An essential innovation of our algorithm is to separate the algorithm's search phase (for finding the $\varepsilon$-BNE) from the verification phase (for computing the $\varepsilon$). Using this approach, we obtain an algorithm that is both very fast and provides theoretical guarantees on the $\varepsilon$ it finds. Our main technical contribution is a verification method which allows us to upper bound the $\varepsilo…
The internet advertising market is a multi-billion dollar industry, in which advertisers buy thousands of ad placements every day by repeatedly participating in auctions. An important and ubiquitous feature of these auctions is the presence of campaign budgets, which specify the maximum amount the advertisers are willing to pay over a specified time period. In this paper, we present a new model to study the equilibrium bidding strategies in standard auctions, a large class of auctions that includes first- and second-price auctions, for advertisers who satisfy budget constraints on average. Our…
This paper studies equilibrium quality of semi-separable position auctions (known as the Ad Types setting) with greedy or optimal allocation combined with generalized second-price (GSP) or Vickrey-Clarke-Groves (VCG) pricing. We make three contributions: first, we give upper and lower bounds on the Price of Anarchy (PoA) for auctions which use greedy allocation with GSP pricing, greedy allocations with VCG pricing, and optimal allocation with GSP pricing. Second, we give Bayes-Nash equilibrium characterizations for two-player, two-slot instances (for all auction formats) and show that there ex…
We study equilibria in two-buyer sequential second-price (or first-price) auctions for identical goods. Buyers have weakly decreasing incremental values, and we make a behavioural no-overbidding assumption: the buyers do not bid above their incremental values. Structurally, we show equilibria are intrinsically linked to a greedy bidding strategy. We then prove three results. First, any equilibrium consists of three phases: a competitive phase, a competition reduction phase and a monopsony phase. In particular, there is a time after which one buyer exhibits monopsonistic behaviours. Second, the…
Our paper concerns the computation of Nash equilibria of first-price auctions with correlated values. While there exist several equilibrium computation methods for auctions with independent values, the correlation of the bidders' values introduces significant complications that render existing methods unsatisfactory in practice. Our contribution is a step towards filling this gap: inspired by the seminal fictitious play process of Brown and Robinson, we present a learning heuristic-that we call fictitious bidding (FB)-for estimating Bayes-Nash equilibria of first-price auctions with correlated…
In this work we introduce a new class of mechanisms composed of a traditional Generalized Second Price (GSP) auction and a fair division scheme, in order to achieve some desired level of fairness between groups of Bayesian strategic advertisers. We propose two mechanisms, $β$-Fair GSP and GSP-EFX, that compose GSP with, respectively, an envy-free up to one item, and an envy-free up to any item fair division scheme. The payments of GSP are adjusted in order to compensate advertisers that suffer a loss of efficiency due the fair division stage. We investigate the strategic learning implications …
Applications of combinatorial auctions (CA) as market mechanisms are prevalent in practice, yet their Bayesian Nash equilibria (BNE) remain poorly understood. Analytical solutions are known only for a few cases where the problem can be reformulated as a tractable partial differential equation (PDE). In the general case, finding BNE is known to be computationally hard. Previous work on numerical computation of BNE in auctions has relied either on solving such PDEs explicitly, calculating pointwise best-responses in strategy space, or iteratively solving restricted subgames. In this study, we pr…
Bounding the inefficiency of outcomes in generalized second price auctions
通用第二价格（GSP）拍卖是用于将互联网的使用货币化的主要拍卖。众所周知，在这种拍卖中，说实话并不是一种优势策略，而且可能会出现低效率的均衡。在本文中，我们研究了空间的均衡在普惠制，并量化的效率损失，可以出现在均衡下的各种来源的不确定性，以及在充分的信息设置。传统的贝叶斯博弈模型的参与者的估值（类型）的不确定性。广义第二价格（GSP）拍卖产生了另一种形式的不确定性：质量因素的选择导致底层广告分配算法的行为不确定性。我们得到的界限适用于这两种形式的不确定性，并在某种意义上说，他们适用于各种扰动下的解决方案的概念，扩展到模型的信息不对称和有限理性的学习策略的形式是强大的。我们提出了一个恒定的界限（2.927）的因素的效率损失（\emph{price of anarchy}）的相应的游戏的贝叶斯模型的部分信息的其他参与者和广告质量因素。对于充分的信息设置，我们证明了一个令人惊讶的低上限为1.282的无政府代价价格的纯纳什均衡，几乎匹配的情况下，三个广告客户的下限为1.259。此外，我们不要求系统达到平衡，并且对于任何无悔学习结果的质量退化也给出了类似的下限。我们的结论是，在拍卖中的广告客户的数量几乎没有影响的无政府代价，GSP的效率是非常强大的信念和理性的假设强加给参与者。
We study the complexity of finding an approximate (pure) Bayesian Nash equilibrium in a first-price auction with common priors when the tie-breaking rule is part of the input. We show that the problem is PPAD-complete even when the tie-breaking rule is trilateral (i.e., it specifies item allocations when no more than three bidders are in tie, and adopts the uniform tie-breaking rule otherwise). This is the first hardness result for equilibrium computation in first-price auctions with common priors. On the positive side, we give a PTAS for the problem under the uniform tie-breaking rule.
University of British Columbia（不列颠哥伦比亚大学）；Harvard University（哈佛大学）；Peking University（北京大学）
In non-truthful auctions, agents' utility for a strategy depends on the strategies of the opponents and also the prior distribution over their private types; the set of Bayes Nash equilibria generally has an intricate dependence on the prior. Using the First Price Auction as our main demonstrating example, we show that $\tilde O(n / ε^2)$ samples from the prior with $n$ agents suffice for an algorithm to learn the interim utilities for all monotone bidding strategies. As a consequence, this number of samples suffice for learning all approximate equilibria. We give almost matching (up to polylo…
