Equilibria in first price auctions with participation costs

This paper characterizes the equilibria of first price auctions with participation costs in the independent private values environment. Bidders use cutoff strategies to decide whether they will participate in the auction. It is shown that, when bidders are homogeneous, there always exists a unique symmetric equilibrium, and further, there is no other equilibrium when valuation distribution functions are inelastic. When distribution functions are elastic at the symmetric equilibrium, there exists an asymmetric equilibrium. Inelasticity/elasticity includes concavity/convexity of distribution functions as a special case. We find similar results when bidders are heterogeneous.     

Credit contracting and bidding under wealth constraints

Summary. We model credit contracting and bidding in a first-price sealed-bid auction when bidder valuation and wealth are private information. An efficient separating equilibrium exists only if the wealth levels of both bidder types are sufficiently different. If not, high-valuation bidders signal by borrowing more and using less of their wealth – this is inefficient as wealth is a cheaper source of funds. An increase in the amount of borrowing required to signal does not necessarily decrease seller expected revenue. In contrast to separating equilibria, high-valuation bidders adopt pure strategy bids in pooling equilibria. Conditions are identified under which the lower bound on winning bids is higher in pooling than separating equilibria. Received: January 22, 2001; revised version: August 28, 2001     

Multidimensional private value auctions

We consider parametric examples of symmetric two-bidder private value auctions in which each bidder observes her own private valuation as well as noisy signals about her opponent's private valuation. We show that, in such environments, the revenue equivalence between the first and second price auctions (SPAs) breaks down and there is no definite revenue ranking; while the SPA is always efficient allocatively, the first price auction (FPA) may be inefficient; equilibria may fail to exist for the FPA. We also show that auction mechanisms provide different incentives for bidders to acquire costly information about opponents’ valuation.     

Almost common values auctions revisited

In almost common-value auctions one bidder (the advantaged bidder) has a valuation advantage over all other (regular) bidders. It is well known that in second-price auctions with two bidders, even a slight private-value advantage can have an explosive effect on auction outcomes as the advantaged bidder wins all the time and auction revenue is substantially lower than in a pure second-price common-value auction. We explore the robustness of these results to the addition of more regular bidders in second-price auctions, and the extent to which these results generalize to ascending-price English auctions in an effort to provide insight into when and why one ought to be concerned about such slight asymmetries.     

The effectiveness of English auctions

We study the performance of the English auction under different assumptions about the seller's degree of “Bayesian sophistication.” We define the effectiveness of an auction as the ratio between the expected revenue it generates for the seller and the expected valuation of the object to the bidder with the highest valuation (total surplus). We identify tight lower bounds on the effectiveness of the English auction for general private-values environments, and for private-values environments where bidders' valuations are non-negatively correlated. For example, when the seller faces 12 bidders who the seller believes have non-negatively correlated valuations whose expectations are at least as high as 60% of the maximal possible valuation, an English auction with no reserve price generates an expected price that is more than 80% of the value of the object to the bidder with the highest valuation.     

Beyond chopsticks: Symmetric equilibria in majority auction games

Majority auction games are simultaneous sealed-bid auctions of identical objects among identical bidders who each want to win a specified fraction (more than a half) of the objects. Each bidder receives no benefit from winning less than the specified fraction and no additional benefit from winning more than it. Symmetric equilibria having simple, intuitive forms are shown to exist in first-price, second-price and all-pay versions of such games when the number of bidders is sufficiently large. This contrasts with earlier results for the two-bidder “pure chopstick” majority auction games where the only known equilibria are more complicated.     

All equilibria of the multi-unit Vickrey auction

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price.     

Monotonicity and Rationalizability in a Large First Price Auction

This paper proves that the monotonicity of bidding strategies together with the rationality of bidders implies that the winning bid in a first price auction converges to the competitive equilibrium price as the number of bidders increases ( Wilson, 1977 ). Instead of analysing the symmetric Nash equilibrium, we examine rationalizable strategies ( Bernheim (1984) , Pearce (1984) ) among the set of monotonic bidding strategies to prove that any monotonic rationalizable bidding strategy must be within a small neighbourhood of the „truthful” valuation of the object, conditioned on the signal received by the bidder. We obtain an information aggregation result similar to that of Wilson (1977) , while dispensing with almost all symmetric assumptions and using a milder solution concept than the Nash equilibrium. In particular, if every bidder is ex ante identical, then any rationalizable bidding strategy must be within a small neighbourhood of the symmetric Nash equilibrium. In a symmetric first price auction, the symmetry of outcomes is implied rather than assumed.     

Potential heterogeneity in target’s value and jump bidding in takeover auctions

This ticle analyses the pre-emptive jump bidding equilibrium in takeover auctions when bidders’ valuations of the target firm follow truncated normal distribution. It shows that potential heterogeneity of the targets’ value, measured by the standard deviation of the bidders’ valuation function, is especially important when it is small and, for extremely small values, the second bidder is almost always pre-empted. It also shows that, contrary to regular clock-style auctions, the increase in standard deviation may negatively affect the expected profit of the first bidder.     

Strategic ignorance in a second-price auction

Suppose that bidders may publicly choose not to learn their values prior to a second-price auction with costly bidding. All equilibria with truthful bidding exhibit bidder ignorance when the number of bidders is sufficiently small. Ignorance considerations also affect the optimal reserve price.     

Non-bidding equilibrium in an ascending core-selecting auction

This paper investigates the perfect Bayesian equilibrium in an ascending-price core-selecting auction, which is recently used in some countries? spectrum license auctions. We suppose that there are two identical items, two small bidders, and one large bidder. The small bidders demand only one unit of the item, whereas the large bidder wants both units. Package bidding ensures that the large bidder faces no exposure problem and behaves truthfully. However, one of the small bidders stops bidding at the beginning in the equilibrium. Although small bidders generally face the free-rider problem and have incentives to underbid, once a bidder is the only small one remaining, he bids truthfully. Stopping early induces the remaining bidder to behave truthfully. Hence, each small bidder wants to be the first to stop bidding. The free-rider problem is considerably mitigated when there are many small bidders.     

Bidding behavior in competing auctions: Evidence from eBay

Much of the existing auction literature treats auctions as running independently of one another, with each bidder choosing to participate in only one auction. However, in many online auctions, a number of substitutable goods are auctioned concurrently and bidders can bid on several auctions at the same time. Recent theoretical research shows how bidders can gain from the existence of competing auctions, the current paper providing the first empirical evidence in support of competing auctions theory using online auctions data from eBay. Our results indicate that a significant proportion of bidders do bid across competing auctions and that bidders tend to submit bids on auctions with the lowest standing bid, as the theory predicts. The paper also shows that winning bidders who cross-bid pay lower prices on average than winning bidders who do not.     

Researching Preferences,Valuation and Hypothetical Bias

A number of recent papers in environmental economics have focused on the process of researching preferences – agents are uncertain about preferences but with effort may narrow their uncertainty. This issue has arisen in formulating bids in contingent valuation (CV) as well as the debate over the divergence between WTP and WTA. In the context of CV, it has been suggested that the hypothetical nature of the preference elicitation process biases responses. This paper provides both a theoretical model and experimental evidence to contribute to this debate. The model is a model of competitive bidding for a private good with two components that are particularly relevant to the debate. The first component is that bidders are unsure of their own value for the private good but may purchase information about their own value (researching preferences). The second component is that there is a probability that the auction is hypothetical – that the winning bidder will not get the private good and will not pay the winning bid. The experiment tests this theoretical model of bidding equilibrium and analyzes the effects of variations in the parameters (hypotheticalness, information costs and number of agents) on the endogenous variables (such as the proportion of bidders who become informed and the winning bid). Experimental results suggest that an increase in the hypotheticalness of an auction tends to decrease the likelihood that bidders pay for information on their valuation with an ambiguous effect on the winning bid.      

Collusion in one-shot second-price auctions

Summary. Most of the literature on collusive behavior in auctions ignores two important issues that make collusion difficult to sustain at least in one-shot interactions: the detection of cheating and the verification of bids. Colluding bidders may deceive each other by using shill bidders. Also, if the identities of the bidders and their bids are not published then it would be difficult to verify the bid of a colluding bidder. This paper addresses these problems in one shot second price auctions where one bidder offers another bidder a side payment in exchange for not participating in the auction, while the number of other bidders is stochastic. In spite of the barriers to collusion mentioned above, a simple side payment mechanism which depends only on the auction price is introduced. It induces a successful collusion, eliminates the verification problem, provides no incentive for the use of shill bidders and guarantees that the proponent obtains ex-post non-negative payoff. The colluding bidders are ex-ante strictly better off compared with the competitive case, irrespective of their types.Received: 27 November 2002, Revised: 28 January 2005, JEL Classification Numbers: C72, D44, D82.Yair Tauman: Correspondence toWe would like to thank an anonymous referee for very valuable comments and suggestions that significantly improved the paper. We thank Shmuel Zamir for a helpful discussion.     

Procurement auctions with ex-ante endogenous bribery

Corruption is a prevalent phenomenon in various procurement auctions. This paper explores a pattern of bribery between an auctioneer and a favored bidder, and also investigates the regulation scheme of buyer. In the model, the favored bidder is allowed to submit two bids simultaneously with the advantageous one to be announced; auctioneer decides the share of the difference between two bids which is the bribe transfer. The analysis shows that, the favored bidder does not participate in the corruption if his cost exceeds a threshold; otherwise he submits two bids whose difference is decreasing in the share. The corruption benefits both the auctioneer and the favored bidder but harms other bidders. The bribery endogenously leads to allocation inefficiency with a probability decreasing in the bribe share. Specifically, with two uniformly distributed bidders, we examine how the auctioneer optimizes the bribe share and how the buyer regulates the corruption. We find that, by driving the auctioneer to charge a higher bribe share that is less attractive for the favored bidder, severer regulation tends to reduce the probability of corruption. A buyer who adopts extremely severe regulation can exclude the corruption and achieve maximum social welfare, while a buyer who aims to maximize his own profit should tolerate some degree of the corruption.     

The value of an informed bidder in common value auctions

I compare two information structures in a common value first-price auction with two bidders: In one, each of the two bidders knows only his own signal about the value of the object, and in the other, one of the bidders learns his opponent's signal as well. Gaining the additional information in the second information structure makes the informed bidder worse off if the value is submodular in the bidders' signals and better off if it is supermodular. If the value is supermodular, then the seller's revenue tends to be lower with the informed bidder than without.     

The value of information for auctioneers

An auctioneer wants to sell an indivisible object to one of multiple bidders, who have private information about their valuations of the object. A bidder's information structure determines the accuracy with which the bidder knows her private valuation. The main result of the paper is that the auctioneer's revenue is a convex function of bidders' information structures. One implication is that assigning asymmetric information structures instead of symmetric information structures to bidders is always revenue-enhancing. This paper generalizes a result of Bergemann and Pesendorfer [D. Bergemann, M. Pesendorfer, Information structures in optimal auctions, J. Econ. Theory 137 (2007) 580–609], who show that revenue-maximizing information structures are asymmetric.     

Efficiency and information aggregation in large uniform-price auctions

We prove that the equilibria of a large interdependent-value, uniform-price auction model where bidders have arbitrary preferences for multiple units can be approximated by a nonatomic exchange economy. We show that the uniform-price auction is approximately efficient with a large number of participants and asymptotically aggregates idiosyncratic bidder information into the market price. More generally our analysis framework provides conditions justifying the use of nonatomic limit model approximations to analyze the large-market behavior of game-theoretic models. We demonstrate continuity requirements on the economic primitives sufficient for the equilibrium strategies of the two models to converge as the number of participants in the finite game approaches infinity.     

我国网上拍卖中竞买者出价行为的实证分析

本文从竞买者出价频率与出价水平特征、出价时间过程的自相似性、出价水平的动态变化,以及出价水平的影响因素效应等四个方面,对我国网上拍卖中竞买者的出价行为进行了实证分析,结果发现:与eBay网上竞买者的出价行为相比,我国网上竞买者具有很高的倾向在后期出价,潜在影响因素易于在拍卖后期对竞买者的出价水平产生较大的影响;竞买者具有较强的投机心理,其出价行为存在较大的个体差异,缺乏规范、典型的模式。     

Bidding in a possibly common-value auction

We analyze a second-price auction with two bidders in which only one of the bidders is informed as to whether the object is valued commonly. We show that any equilibrium strategy of the bidder who is uninformed must be part of an equilibrium when both bidders instead know that the auction is not common value, regardless of the way in which the values are different. We derive sufficient conditions for equilibrium existence.