Optimal bidding in auctions of mixed populations of bidders

A mixed population of bidders consists of two asymmetric groups. Members of the first group are game-theoretic players, who maximize their expected profit and incorrectly believe that their opponents act similarly. The second group of bidders adopts an irrational strategy: they either choose their bids randomly following a given probability distribution, in a “naïve” form of bidding, or follow a decision-theoretic approach, maximizing their expected profit under the assumption that all other bids are random. In a sealed bid private-value procurement auction we examine the optimal strategy of a new player, who has perfect knowledge of the structure of the mixed bidder population and enters the auction. The optimal bid of the new bidder is derived when the cost and mark-up follow a uniform distribution in [0, 1]. The effect of the relative size of the group of game-theoretic bidders and the population size on the optimal bid price is established.     

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.     

Auctions with bidder-determined allowable combinations

Combinatorial auctions are desirable as they enable bidders to express the synergistic values of a group of assets and thus may lead to better allocations. Compared to other types of auctions, they keep bidders from being exposed to risks (of receiving only parts of combinations that would be valuable to them) or from being overly cautious (in order to minimize such risks). However, computation time needed to determine the set of optimal winning combinations in a general combinatorial auction may grow exponentially as the auction size increases, and this is sometimes given as a reason for not using combinatorial auctions. To determine the winning allocation in a reasonable time, a bid taker might try to limit the kinds of allowable combinations, but bidders may disagree on what combinations should be allowed, and this may make limiting the allowable combinations politically infeasible.This paper proposes and tests successfully a new approach to managing the computational complexity of determining the set of winning combinations. The main idea is to let bidders themselves determine and prioritize the allowable combinations. Using bidder-determined combinations has two nice properties. First, by delegating the decision on what is biddable to the bidders who know what combinations are important to them, the bid taker is able to be (and appear) fair. Second, since bidders know their economics and have the incentive to get important combinations included, bidder prioritization of combinations will tend to assure that the most economically-important combinations are included in determining the winning set of bids if the bid taker is not able to consider all of the combinations submitted by bidders. The proposed auction process is useful in situations, such as government auctions, in which the bid taker is reluctant to limit the allowable combinations.     

On being balanced in an unbalanced world

This paper examines the case of a procurement auction for a single project, in which the breakdown of the winning bid into its component items determines the value of payments subsequently made to bidder as the work progresses. Unbalanced bidding, or bid skewing, involves the uneven distribution of mark-up among the component items in such a way as to attempt to derive increased benefit to the unbalancer but without involving any change in the total bid. One form of unbalanced bidding for example, termed Front Loading (FL), is thought to be widespread in practice. This involves overpricing the work items that occur early in the project and underpricing the work items that occur later in the project in order to enhance the bidder's cash flow. Naturally, auctioners attempt to protect themselves from the effects of unbalancing—typically reserving the right to reject a bid that has been detected as unbalanced. As a result, models have been developed to both unbalance bids and detect unbalanced bids but virtually nothing is known of their use, success or otherwise. This is of particular concern for the detection methods as, without testing, there is no way of knowing the extent to which unbalanced bids are remaining undetected or balanced bids are being falsely detected as unbalanced. This paper reports on a simulation study aimed at demonstrating the likely effects of unbalanced bid detection models in a deterministic environment involving FL unbalancing in a Texas DOT detection setting, in which bids are deemed to be unbalanced if an item exceeds a maximum (or fails to reach a minimum) ‘cut-off’ value determined by the Texas method. A proportion of bids are automatically and maximally unbalanced over a long series of simulated contract projects and the profits and detection rates of both the balancers and unbalancers are compared. The results show that, as expected, the balanced bids are often incorrectly detected as unbalanced, with the rate of (mis)detection increasing with the proportion of FL bidders in the auction. It is also shown that, while the profit for balanced bidders remains the same irrespective of the number of FL bidders involved, the FL bidder's profit increases with the greater proportion of FL bidders present in the auction. Sensitivity tests show the results to be generally robust, with (mis)detection rates increasing further when there are fewer bidders in the auction and when more data are averaged to determine the baseline value, but being smaller or larger with increased cut-off values and increased cost and estimate variability depending on the number of FL bidders involved. The FL bidder's expected benefit from unbalancing, on the other hand, increases, when there are fewer bidders in the auction. It also increases when the cut-off rate and discount rate is increased, when there is less variability in the costs and their estimates, and when less data are used in setting the baseline values.     

The buy price in auctions with discrete type distributions

This paper considers second-price, sealed-bid auctions with a buy price where bidders’ types are discretely distributed. We characterize all equilibria in which bidders whose types are less than the buy price bid their own valuations. Budish and Takeyama (2001) analyze the two-bidder, two-type framework. They show that if bidders are risk-averse, then the seller can obtain a higher expected revenue from the auction with a certain buy price than from the auction without a buy price. We extend their revenue improvement result to the n-bidder, two-type framework. In case of three or more types, however, bidders’ risk aversion is not a sufficient condition for a revenue improvement. We point out that even if bidders are risk-averse, the seller cannot always obtain a higher expected revenue from the auctions with a buy price.     

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.     

Non-linear anonymous pricing combinatorial auctions

In combinatorial auctions the pricing problem is of main concern since it is the means by which the auctioneer signals the result of the auction to the participants. In order for the auction to be regarded as fair among the various participants the price signals should be such that a participant that has won a subset of items knows why his bid was a winning bid and that agents that have not acquired any item easily can detect why they lost. The problem in the combinatorial auction setting is that the winner determination problem is a hard integer programming problem and hence a linear pricing scheme supporting the optimal allocation might not exist.     

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.     

Using bid data for the management of sequential,multi-unit,online auctions with uniformly distributed bidder valuations

Internet auctions for consumers’ goods are an increasingly popular selling venue. We have observed that many sellers, instead of offering their entire inventory in a single auction, split it into sequential auctions of smaller lots, thereby reducing the negative market impact of larger lots. Information technology also makes it possible to collect and analyze detailed bid data from online auctions. In this paper, we develop and test a new model of sequential online auctions to explore the potential benefits of using real bid data from earlier auctions to improve the management of future auctions. Assuming a typical truth-revealing auction model, we quantify the effect of the lot size on the closing price and derive a closed-form solution for the problem of allocating inventory across multiple auctions when bidder valuation distributions are known. We also develop a decision methodology for allocating inventory across multiple auctions that dynamically incorporates the results of previous auctions as feedback into the management of subsequent auctions, and updating the lot size and number of auctions. We demonstrate how information signals from previous auctions can be used to update the auctioneer’s beliefs about the customers’ valuation distribution, and then to significantly increase the seller’s profit potential. We use several examples to reveal the benefits of using detailed transaction data for the management of sequential, multi-unit, online auctions and we demonstrate how these benefits are influenced by the inventory holding costs, the number of bidders, and the dispersion of consumers’ valuations.     

多物品拍卖的捆绑销售策略

拍卖商在拍卖多个不同物品的过程中,面临着是捆绑拍卖还是分开拍卖的问题.本文讨论在第二价格密封拍卖(维克里拍卖)方式下,拍卖商拍卖多个不同物品时,其所采取的最优策略(捆绑拍卖或是分开拍卖)与投标人的数量以及投标人对物品的估价的关系.     

Home-grown values and incentive compatible auction design

The behavioral properties of several auctions designed to elicit individual valuations for an object are studied using controlled laboratory experiments. Our experiments lead us to conclude that there are some behavioral differences between alternative incentive-compatible institutions for eliciting home-grown values, contrary to the theoretical expectation that these institutions are isomorphic. These results are consistent with earlier experimental results using induced values. The most important finding is that English auctions appear to elicit lower bids than Vickrey auctions, after controlling for observable socio-economic characteristics. Moreover, English auction bids also exhibit significantly less residual variance and may be sensitive to the number of rival bidders. It appears that the real-time learning allowed in the English auction significantly affects subject behavior. We also find that values elicited with the Becker, DeGroot and Marshak institution differ from those in both English and Vickrey auctions. Received November 1993/Final version May 1995     

国债招标拍卖的最优机制:数量与价格歧视

过去已有相当多的文献讨论国债拍卖,米勒和弗雷德曼认为统一价格比歧视价格拍卖有比较优势.研究国债拍卖最优分配方式的选择问题,采用最优机制设计方法分析国债拍卖活动,内生地得到最优拍卖机制.然而结果表明,如果使用价格歧视及数量歧视(定量分配),那么政府的销售收益会得到改善,这是一个最优拍卖机制,不同于米勒和弗雷德曼的观点.     

Design of discrete Dutch auctions with an uncertain number of bidders

The focus of this paper is on Dutch auctions where the bidding prices are restricted to a finite set of values and the number of bidders follows a Poisson distribution. The goal is to determine what the discrete bid levels should be to maximize the auctioneer’s expected revenue, which is the same as the average selling price of the object under consideration. We take a new approach to the problem by formulating the descending-price competitive bidding process as a nonlinear program. The optimal solution indicates that the interval between two successive bids should be wider as the Dutch auction progresses. Moreover, the auctioneer’s maximum expected revenue increases with the number of bid levels to be set as well as the expected number of bidders. Numerical examples are provided to illustrate the key results from this study and their managerial implications are discussed.     

Choice of item pricing feedback schemes for multiple unit reverse combinatorial auctions

Recently, interest in combinatorial auctions has extended to include trade in multiple units of heterogeneous items. Combinatorial bidding is complex and iterative auctions are used to allow bidders to sequentially express their preferences with the aid of auction market information provided in the form of price feedbacks. There are different competing designs for the provision of item price feedbacks; however, most of these have not been thoroughly studied for multiple unit combinatorial auctions. This paper focuses on addressing this gap by evaluating several feedback schemes or algorithms in the context of multiple unit auctions. We numerically evaluate these algorithms under different scenarios that vary in bidder package selection strategies and in the degree of competition. We observe that auction outcomes are best when bidders use a naïve bidding strategy and competition is strong. Performance deteriorates significantly when bidders strategically select packages to maximize their profit. Finally, the performances of some algorithms are more sensitive to strategic bidding than others.     

基于数据包络分析的混合多属性第二分值拍卖机制

多属性拍卖在传统价格拍卖的基础上纳入了价格和质量多个属性,已广泛应用于许多领域。本文针对输入几乎没有先验结构且投标人数量足够大的情况,在以往有关多属性拍卖研究的基础上将单个投入扩展为多个投入,将物品的多个属性(价格、完成时间、劳动力数量、质量等因素)合理划分为投入或产出,设计了运行有效的基于数据包络分析的多属性第二分值拍卖机制。与其他方法相比,数据包络分析方法可以有效解决多投入多产出问题,结合多目标规划方法,可以帮助采购方在最大化自身利益的同时,找到整体表现更好的供应商。该机制满足个人理性和激励相容,与传统第二分值拍卖机制相比,在吸引投标人的前提下能够最大化拍卖人的利益。     

Allocative efficiency in simulated multiple-unit combinatorial auctions with quantity support

We study the behavior of the Quantity Support Mechanism, a support tool, which suggests new bids for bidders in semi-sealed-bid combinatorial auctions. The support tool gives bidders a shortlist of provisionally winning bids (price–quantity combinations) they can choose from. We conducted a series of simulations to test the efficiency of the final allocations in the auctions. Our results indicate that quantity support auctions are more efficient than auctions without it, although the theoretical optimum is not always reached. Also, in our experiments, quantity support auctions led to a lower total cost to the buyer than non-combinatorial auctions, where the items were auctioned individually. The simulation results also show that the QSM cannot entirely overcome the threshold problem and what we call the “puzzle problem”.     

A new pricing scheme based on DEA for iterative multi-unit combinatorial auctions

Iterative Multi-unit Combinatorial Auctions are auctions in which: bidders can express bids in successive rounds until a stopping rule is held; there are multiple units of each item; and bids are bundles of items. Data Envelopment Analysis (DEA) is a nonparametric method for measuring the relative efficiency of a set of homogeneous units. In this work, we present an algorithm for solving an iterative multi-unit combinatorial auction in which the auctioneer computes at each round a linear anonymous price for each item by using a DEA model and pushes bidders to express bids according to them. A computational study is carried out in order to check the performance of the proposed design.      

基于树型结构的采购组合拍卖获胜概率计算方法研究

在基于树型结构的采购组合拍卖模型的研究基础上,建立了这一类采购组合拍卖的投标获胜概率的数学模型,给出了投标获胜概率的求解方法.通过算例,详细分析了具体计算方法的规律性和可行性.投标获胜概率的计算方法本身具有重要意义,同时能够为投标者提供决策依据和行为参考;也为进一步研究采购组合拍卖的投标期望收益、均衡策略和机制设计奠定了必要的理论基础.     

常数相对风险厌恶多单位拍卖的比较静态与定价结构

考虑竞价者具有常数相对风险厌恶时的多单位拍卖问题,讨论均衡出价与风险偏好以及价值分布之间的关系.在竞价者具有相同的风险偏好的对称拍卖情形,利用风险中性等价导出了比较静态的充要条件,这一充要条件综合了风险偏好和价值分布的变化.在竞价者具有不同的风险偏好的非对称拍卖中,导出了均衡出价关于风险偏好的线性定价结构,依据这一结构,对任意给定出价,竞价者要求的剩余是其风险厌恶参数的线性函数,并且可以分解为竞争性剩余和风险厌恶剩余.     

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．