All-pay auctions with risk-averse players

We study independent private-value all-pay auctions with risk-averse players. We show that: (1) Players with low values bid lower and players with high values bid higher than they would bid in the risk neutral case. (2) Players with low values bid lower and players with high values bid higher than they would bid in a first-price auction. (3) Players’ expected utilities in an all-pay auction are lower than in a first-price auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-averse players and the seller’s expected revenue. In particular, we show that in all-pay auctions the seller’s expected payoff in the risk-averse case may be either higher or lower than in the risk neutral case.     

How to bid in unified second-price auctions when requests are duplicated

In display advertising auctions, a unique display opportunity may trigger many bid requests being sent to the same buyer. Bid request duplication is an issue: programmatic bidding agents might bid against themselves. In a simplified setting of unified second-price auctions, the optimal solution for the bidder is to randomize the bid, which is quite unusual. Our results motivate the recent switch to a unified first-price auction by showing that a unified second-price auction could have been detrimental to all participants.     

All-pay auctions with pre- and post-bidding options

Motivated by the emergence of online penny or pay-to-bid auctions, in this study, we analyze the operational consequences of all-pay auctions competing with fixed list price stores. In all-pay auctions, bidders place bids, and highest bidder wins. Depending on the auction format, the winner pays either the amount of their bid or that of the second-highest bid. All losing bidders forfeit their bids, regardless of the auction format. Bidders may visit the store, both before and after bidding, and buy the item at the fixed list price. In a modified version, we consider a setting where bidders can use their sunk bid as a credit towards buying the item from the auctioneer at a fixed price (different from the list price). We characterize a symmetric equilibrium in the bidding/buying strategy and derive optimal list prices for both the seller and auctioneer to maximize expected revenue. We consider two situations: (1) one firm operating both channels (i.e. fixed list price store and all-pay auction), and (2) two competing firms, each operating one of the two channels.     

Ranking asymmetric auctions

We compare the expected revenue in first- and second-price auctions with asymmetric bidders. We consider “close to uniform” distributions with identical supports and show that in the case of identical supports the expected revenue in second-price auctions may exceed that in first-price auctions. We also show that asymmetry over lower valuations has a stronger negative impact on the expected revenue in first-price auctions than in second-price auctions. However, asymmetry over high valuations always increases the revenue in first-price auctions.     

A new end-of-auction model for curbing sniping

The online auction market has been growing at a spectacular rate. Most auctionsare open-bid auctions where all the participants know the current highest bid.This knowledge has led to a phenomenon known as sniping, whereby somebidders may wait until the last possible moment before bidding, therebydepriving other bidders of the opportunity to respond and also preventingsellers from obtaining the highest price for an item. This is especially true inthe case of the commonly used second-price, fixed-deadline auction. We considera procedure involving a randomly determined stopping time and show that thisapproach eliminates the potential benefits to a sniper. The scheme enables allbidders to compete more fairly and promotes an early bidding strategy, which islikely to increase the price received by the seller while providing adequatebidding opportunities for would-be buyers.     

Charity auctions for the happy few

Recent literature has shown that all-pay auctions raise more money for charity than either winner-pay auctions or lotteries. We demonstrate that first-price and second-price winner-pay auctions have a better revenue performance than first-price and second-price all-pay auctions when bidders are sufficiently asymmetric. Lotteries can also provide higher revenue than all-pay auctions. To prove this, we consider a framework with complete information. Complete information is helpful and may reflect events that occur, for instance, in a local service club (such as a voluntary organization) or at a show-business dinner.     

First price auctions: Monotonicity and uniqueness

I study monotonicity and uniqueness of the equilibrium strategies in a two-person first price auction with affiliated signals. Existing results establish uniqueness within the class of non-decreasing bidding strategies. I show that there is an effectively unique Nash equilibrium within the class of piecewise monotone strategies. The main result is that in equilibrium, the strategies must be strictly increasing within the support of winning bids. This result provides the missing link for the analysis of uniqueness in two-person first price auctions. The analysis applies to asymmetric environments as well and does not require risk neutrality. Received: August 1997/Last revision: March 2000     

Monotonicity in asymmetric first-price auctions with affiliation

I study monotonicity of equilibrium strategies in first-price auctions with asymmetric bidders, risk aversion, affiliated types, and interdependent values. Every mixed-strategy equilibrium is shown to be outcome-equivalent to a monotone pure-strategy equilibrium under the “priority rule” for breaking ties. This provides a missing link to establish uniqueness in the “general symmetric model” of Milgrom and Weber (Econometrica 50:1089–1122, 1982). Non-monotone equilibria can exist under the “coin-flip rule” but they are distinguishable: all non-monotone equilibria have positive probability of ties whereas all monotone equilibria have zero probability of ties. This provides a justification for the standard empirical practice of restricting attention to monotone strategies. Hendricks et al. (2003) provide an overview of recent empirical work. For a survey of experimental work, see Kagel and Levin (2002).     

预算约束下广义第二价格关键字拍卖均衡分析

在关键字拍卖中,广告主预算约束是影响投标策略均衡的重要因素。针对广泛应用的广义第二价格机制,分析了预算约束下的关键字拍卖均衡性质。按照广告主的点击估价和预算分布情况,将广告主划分为天然完全赢家、天然部分赢家和天然输家这三种类型,提出了广告主投标临界值与关键值的概念,基于关键字拍卖不断重复进行的特点,分别从长期视角与短期视角建立了静态模型与动态模型,研究结果表明:静态环境下拍卖存在唯一的纯策略纳什均衡;动态环境下当广告主采用最优反应投标策略时拍卖将收敛至唯一的纳什均衡点,并且动态环境下拍卖商收入不低于静态环境下的拍卖收入。数值算例的结果表明在动态环境下不同的初始报价会导致拍卖收敛至不同的均衡点。研究结果为拍卖商提供了拍卖收入预测和拍卖机制优化的决策依据。     

考虑全局投标者后悔心理行为的组合拍卖的投标均衡策略

许多实验研究表明投标者在拍卖过程中往往表现出预期后悔心理行为,并且投标者的预期后悔心理行为将会对投标策略产生影响,但以往大多是针对单物品拍卖研究考虑投标者后悔心理行为的投标均衡策略,而针对多物品拍卖情形的研究较少关注。本文着重研究了考虑投标者后悔心理行为的组合拍卖的投标均衡策略问题,在全局投标者存在预期后悔心理行为的假设下,依据Engelbrecht-Wiggans和Katok提出的后悔函数刻画了投标者的后悔心理行为,在此基础上,构建了组合拍卖模型,通过分析给出了全局投标者投标均衡策略需要满足的充分和必要条件。进一步地,依据构建的模型,通过数值实验分析了局部投标者人数、组合效应系数和全局投标者后悔参数对全局投标者投标策略的影响。最后,通过一个关于无线电频谱组合拍卖的算例说明了本文给出的模型及投标均衡策略确定方法的潜在应用和优越性。     

基于网上拍卖的共同价值托投标分析

基于信息不对称条件下的共同价值模型,刻画了网上拍卖过程中可能存在的托投标行为,并运用博弈理论结合竞价关系求解了竞标者的赢标概率和最优竞价策略．研究发现,在共同价值模型下,参与竞标的人数与嬴者诅咒的发生存在联系;在最高出价者赢标的概率模型基础上,分别建立了是否存在托投标行为时的竞标者收益模型,并求解了竞标者的均衡竞价策略．     

车辆牌照拍卖模型

提出多个相同物品(如车辆牌照)同时密封拍卖的模型，给出对称均衡竞标策略；证明了该拍卖方式与第一价格密封连续拍卖产生相同的预期收益；对估价为均匀分布的拍卖预期收益进行了研究。     

Investigating buyer and seller strategies in online auctions

The annual turnover of online auctions is already in tens of billions of dollars and this amount is predicted to grow substantially over the next few years. Hence, it is important to know how buyers and sellers can influence their chances of success. Therefore, data were collected from eBay auctions for three different categories of collectible items, namely those with a published guide price, those with a rough guide price and those having no easily obtainable guide price. The options available to buyers and sellers of items were then analysed. It was found that it was hard for the seller to influence an item's achieved price significantly, apart from items with no guide price where the starting price could have an effect. Most bidders bid close to the current value and so there were insufficient data to determine the consequences of timing on the placing of high bids. For low bids, delaying a bid was found to improve significantly the chances of winning for one of the data sets.     

On the robustness of non-linear personalized price combinatorial auctions

Though the VCG auction assumes a central place in the mechanism design literature, there are a number of reasons for favoring Iterative Combinatorial Auctions (ICAs). Several promising ICA formats were developed based on primal–dual and subgradient algorithms. Prices are interpreted as a feasible dual solution and the provisional allocation is interpreted as a feasible primal solution. iBundle(3), dVSV and Ascending Proxy Auction result in VCG payoffs when the coalitional value function satisfies buyer submodularity and bidders bid straightforward, which is an ex-post Nash equilibrium in this case. iBEA and CreditDebit auctions do not even require the buyer submodularity and achieve the same properties for general valuations. Often, however, one cannot assume straightforward bidding and it is not clear from the theory how these non-linear personalized price auctions (NLPPAs) perform in this case. Robustness of auctions with respect to different bidding behavior is a critical issue for any application. We conducted a large number of computational experiments to analyze the performance of NLPPAs with respect to different bidding strategies and valuation models. We compare NLPPAs with the VCG auction and with ICAs with linear prices, such as ALPS and the Combinatorial Clock Auction. While NLPPAs performed very well in case of straightforward bidding, we observe problems with revenue, efficiency, and speed of convergence when bidders deviate.     

An experimental test of alternative models of bidding in ascending auctions

The theory and behavior of the clock version of the ascending auction has been well understood for at least 20 years. The more widely used oral outcry version of the ascending auction that allows bidders to submit their own bids has been the subject of some recent controversy mostly in regard to whether or not jump bidding, i.e. bidders submitting bids higher than required by the auctioneer, should be allowed. Isaac, Salmon & Zillante (2005) shows that the standard equilibrium for the clock auction does not apply to the non-clock format and constructs an equilibrium bid function intended to match with field data on ascending auctions. In this study, we will use economic experiments to provide a direct empirical test of that model while simultaneously providing empirical evidence to resolve the policy disputes centered around the place of jump bidding in ascending auctions.Received: March 2005The authors would like to thank Florida State University for providing the funding for the experiments in this paper and Bradley Andrews for programming assistance.     

Estimating advertisers' values for paid search clickthroughs

Paid search is an important form of online advertisement. Clickthroughs from slots are bid for by advertisers. The process of formulating bids is a complex one involving bidders in competing against other advertisers in multiple auctions. It would be helpful in managing the bidding process if it were possible to determine the values placed on a clickthrough by different advertisers. The theory of two models for estimating advertiser values and associated parameters is presented. The models are applied to a set of data for searches on the term Personal Loans. The results of the model that fits the data better are evaluated. The utility of the model to practitioners is discussed. Some issues raised by the results about the role of bidding agents and the discriminatory power of Customer Relationship Management systems are considered. Ways to develop the preferred model are outlined. It is suggested that the model has implications for evaluating forecasting methods for use in paid search auctions.     

Coasian dynamics in repeated English auctions

We extend the Coase conjecture to the case of a seller with a single object, who faces n potential buyers and holds a sequence of English auctions until the object is sold. In an independent-private-values environment in which buyers and sellers share the same discount factor, we show that the (perfect Bayesian) equilibrium path of reserve prices obeys a Coasian logic. Moreover, the equilibrium reserve path lies below that for the repeated sealed-bid, second-price auctions studied by McAfee and Vincent (in Games Econ Behav 18:246–276). Nevertheless, the open (English) and sealed-bid formats are shown to be revenue equivalent.     

考虑交易费用的一级价格密封招标研究

本文在独立私人价值模型的基础上,通过引入投标成本,建立了考虑交易费用的一级价格密封招标博弈模型,给出了对称的进入点均衡和相应的对称均衡报价策略。研究表明,交易费用的出现排除了高成本类型的投标者,但也导致了竞标人数的不确定性,招标方为了降低招标无效率的可能性,必须限制潜在投标者人数;另一方面,在招投标过程中降低交易费用对交易双方来说都是有利的,招标方给予投标者投标补偿以激励其参与竞标的机制是可行的。     

Robust strategic bidding in auction-based markets

In this paper, we propose an alternative methodology for devising revenue-maximizing strategic bids under uncertainty in the competitors’ bidding strategy. We focus on markets endowed with a sealed-bid uniform-price auction with multiple divisible products. On recognizing that the bids of competitors may deviate from equilibrium and are of difficult statistical characterization, we proposed a two-stage robust optimization model with equilibrium constraints aiming to devise risk-averse strategic bids. The proposed model is a trilevel optimization problem that can be recast as a particular instance of a bilevel program with equilibrium constraints. Reformulation procedures are proposed to find a single-level equivalent formulation suitable for column-and-constraint generation (CCG) algorithm. Results show that even for the case in which an imprecision of 1% is observed on the rivals’ bids in the equilibrium point, the robust solution provides a significant risk reduction (of 79.9%) in out-of-sample tests. They also indicate that the best strategy against high levels of uncertainty on competitors’ bid approaches to a price-taker offer, i.e., bid maximum capacity at marginal cost.