Chinese auctions are a combination between a raffle and an auction and are held in practice at charity events or festivals. In a Chinese auction, multiple players compete for several items by buying tickets, which can be used to win the items. In front of each item there is a basket, and the players can bid by placing tickets in the basket(s) corresponding to the item(s) they are trying to win. After all the players have placed their tickets, a ticket is drawn at random from each basket and the item is given to the owner of the winning ticket. While a player is never guaranteed to win an item, they can improve their chances of getting it by increasing the number of tickets for that item.
In this paper we investigate the existence of pure Nash equilibria in both the continuous and discrete settings. When the players have continuous budgets, we show that a pure Nash equilibrium may not exist for asymmetric games when some valuations are zero. In that case we prove that the auctioneer can stabilize the game by placing his own ticket in each basket. On the other hand, when all the valuations are strictly positive, a pure Nash equilibrium is guaranteed to exist, and the equilibrium strategies are symmetric when both valuations and budgets are symmetric. We also study Chinese auctions with discrete budgets, for which we give both existence results and counterexamples. While the literature on rent-seeking contests traditionally focuses on continuous costly tickets, the discrete variant is very natural and more closely models the version of the auction held in practice.
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 int…
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…
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…
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…
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 …
We study combinatorial auctions with bidders that exhibit endowment effect. In most of the previous work on cognitive biases in algorithmic game theory (e.g., [Kleinberg and Oren, EC'14] and its follow-ups) the focus was on analyzing the implications and mitigating their negative consequences. In contrast, in this paper we show how in some cases cognitive biases can be harnessed to obtain better outcomes. Specifically, we study Walrasian equilibria in combinatorial markets. It is well known that Walrasian equilibria exist only in limited settings, e.g., when all valuations are gross substitute…
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…
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…
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 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 …
Microsoft Research, New England（微软研究院新英格兰分院）；Harvard University（哈佛大学）；California Institute of Technology（加州理工学院）
Given a sample of bids from independent auctions, this paper examines the question of inference on auction fundamentals (e.g. valuation distributions, welfare measures) under weak assumptions on information structure. The question is important as it allows us to learn about the valuation distribution in a robust way, i.e., without assuming that a particular information structure holds across observations. We leverage the recent contributions of \cite{Bergemann2013} in the robust mechanism design literature that exploit the link between Bayesian Correlated Equilibria and Bayesian Nash Equilibri…
Information Asymmetries in Pay-Per-Bid Auctions: How Swoopo Makes Bank
创新的拍卖方法可以用来增加利润，Shubik著名的“美元拍卖”可能是最广为人知的例子。最近，一些主流的电子商务网站显然在更大的范围内实现了同样的目标，通过使用“按出价付费”拍卖来出售从视频游戏到金条的物品。在这些拍卖中，投标人除了（有时代替）获胜者的最终购买成本之外，还需要支付每次投标的费用。因此，即使获胜者的购买成本是物品内在价值的一小部分，拍卖商仍然可以从投标费中获得可观的利润。我们的工作为这些拍卖提供了新的分析，基于建模和数据集来自拍卖在Swoopo.com，领先的按次付费拍卖网站。虽然以前的建模工作预测无利润的均衡，我们广泛地分析了信息不对称的影响，以及Swoopo功能，如bidpacks和Swoop It Now选项，以量化这些拍卖中不完美信息的影响。我们发现，即使是小的不对称的球员（更便宜的出价，更好地估计其他球员的意图，不同的估价项目，承诺的球员愿意玩“鸡”）可以增加拍卖持续时间远远超过预测以前的工作，从而扭曲拍卖师的利润不成比例。最后，我们在Swoopo上观察到的数千次现场拍卖的数据集的背景下讨论了我们的发现，这使我们能够研究行为因素，例如积极出价的力量。最终，我们的研究结果表明，即使是完全理性的参与者，如果参与者忽视或不知道这些因素中的任何一个，结果是为每次出价支付拍卖商带来巨额利润。
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…
Kakade, Kearns, and Ortiz (KKO) introduce a graph-theoretic generalization of the classic Arrow--Debreu (AD) exchange economy. Despite its appeal as a networked version of AD, we argue that the KKO model is too local, in the sense that goods cannot travel more than one hop through the network. We introduce an alternative model in which agents may purchase goods on credit in order to resell them. In contrast to KKO, our model allows for long-range trade, and yields equilibria in more settings than KKO, including sparse endowments. Our model smoothly interpolates between the KKO and AD equilibri…
