EQUILIBRIA EXISTENCE AND CHARACTERIZATION IN AUCTIONS: ACHIEVEMENTS AND OPEN QUESTIONS

Abstract Establishing existence and characterizing equilibria are both important achievements in the study of auctions. However, we recognize that equilibria existence results form the basis for well accepted characterizations. In this survey, we review the landmark results and highlight open questions regarding equilibria existence and characterizations in auctions. In addition, we review the standard assumptions underlying these results, and discuss the suitability of the Nash equilibrium solution concept. We focus our review on single‐object auctions, but also review results in multi‐unit, divisible, combinatorial and double auctions.     

NONPARAMETRIC ESTIMATION OF ENTRY COST IN FIRST‐PRICE PROCUREMENT AUCTIONS

In this paper, I investigate Samuelson's low‐price auction model with entry costs. The model's equilibrium implies that the distribution of bids is truncated at the threshold for participation. I use the model to estimate the cost of participation in Michigan highway procurement auctions. The null hypothesis of zero entry costs is rejected. Using my empirical results, I then construct an estimate of the optimal auction, which employs regular policy tools such as entry fees. Finally, I demonstrate the savings that the Michigan government could have made on payments if optimal auctions had been employed. Copyright © 2012 John Wiley & Sons, Ltd.     

Auctions with endogenous participation

We study endogenous-participation auctions where bidders only know the number of potential participants. After seeing their values for the object, potential participants decide whether or not to enter the auction. They may not want to enter the auction since they have to pay participation costs. We characterize equilibrium bidding strategies and entry decisions for both first- and second-price sealed-bid auctions when participation is endogenous. We show that there is a pure strategy entry equilibrium where only bidders with values greater than a certain cut-off point actually bid. In this context, both types of auctions generate the same expected revenue. We also show that, contrary to the predictions of the fixed number of bidders literature, the seller's expected revenue may decrease when the number of potential participants increases. In addition, we show that it is optimal for the seller to charge an entry fee, which contrasts with results from the existing literature on auctions with entry. As in the fixed-n literature, we show that first-price auctions generate more expected revenue than second-price auctions when buyers are risk-averse. Finally, we characterize the optimal auction – the auction that maximizes the seller's expected revenue – by using a direct revelation mechanism. The optimal auction involves a reserve price larger than the optimal reserve price in the fixed-n literature. The winner's payment is the second highest bid less the participation cost and losers receive a subsidy equal to the participation cost. Received: 17 August 1998 / 21 September 1999     

Identification and estimation in sequential,asymmetric, English auctions

Within the independent private-values paradigm, we derive the data-generating process of the winning bid for the last unit sold at multi-unit, sequential, asymmetric, English auctions. When the identity of the winner and the number of units won by each bidder in previous stages of the auction are observed, we demonstrate nonparametric identification and propose a semi-nonparametric estimation strategy based on orthogonal polynomials. We apply our estimator to daily data from fish auctions in Denmark. For single-unit supply, we use our estimates to compare the revenues a seller could expect to earn were a Dutch auction employed instead.     

Second‐Price Auctions with Different Participation Costs

This paper studies equilibria of second‐price auctions in independent private value environments with different participation costs. Two types of equilibria are identified: monotonic equilibria in which a bidder with a lower participation cost results in a lower cutoff for submitting a bid, and nonmonotonic equilibria in which a lower participation cost results in a higher cutoff. We show that there always exists a monotonic equilibrium, and further, that the monotonic equilibrium is unique for either concave distribution functions or strictly convex distribution functions with nonincreasing reverse hazard rates. There exist nonmonotonic equilibria when the distribution functions are strictly convex and the difference of the participation costs is sufficiently small. We also provide comparative static analysis and study the limiting properties of equilibria when the difference in bidders’ participation costs approaches zero.     

Learning through Simultaneous Play: Evidence from Penny Auctions

This paper contributes to the emerging empirical literature on penny auctions, a particular type of all‐pay auctions. We focus on the potential learning effects that bidders may experience over time but also (and particularly) across auctions as a result of their auction participation. Using detailed bid‐level information, we find that, similarly to earlier literature, bidders suffer from a sunk cost fallacy, whereby their probability of dropping out of an auction is decreasing in the number of bids they have already placed in that auction. Although we do find that learning through repeated participation alleviates the sunk cost fallacy, participation in simultaneous penny auctions emerges as a much more effective learning mechanism, ultimately contributing toward bidders earning higher individual surpluses.     

Asymmetry in first‐price auctions with affiliated private values

Collusion and heterogeneity across firms may introduce asymmetry in bidding games. A major difficulty in asymmetric auctions is that the Bayesian Nash equilibrium strategies are solutions of an intractable system of differential equations. We propose a simple method for estimating asymmetric first‐price auctions with affiliated private values. Considering two types of bidders, we show that these differential equations can be rewritten using the observed bid distribution. We establish the identification of the model, characterize its theoretical restrictions, and propose a two‐step non‐parametric estimation procedure for estimating the private value distributions. An empirical analysis of joint bidding in OCS auctions is provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Financial contracts as coordination device

We study the use of financial contracts as bid‐coordinating device in multi‐unit uniform price auctions. Coordination is required whenever firms face a volunteer's dilemma in pricing strategies: one firm (the “volunteer") is needed to increase the market clearing price. Volunteering, however, is costly, as inframarginal suppliers sell their entire capacity whereas the volunteer only sells residual demand. We identify conditions under which signing financial contracts solves this dilemma. We test our framework exploiting data on contract positions by large producers in the New York power market. Using a Monte Carlo simulation, we show that the contracting strategy is payoff dominant and provide estimates of the benefits of such strategy.     

He who must not be named

Landsberger et al. have studied a sealed bid first price auction with two players in which the ranking of the valuations is known. They argue that such a situation can arise in a sequential auction where only the name of the winner is revealed. In this paper we consider sequential auctions where two identical goods are sold sequentially to N players who are interested in both objects. In sealed bid auctions, no information is a priori revealed by the mechanism, but the seller can in principle reveal whatever he wants. We restrict our attention to the case where only the name of the winner is revealed to be in the context of Landsberger et al. for the second auction. The aim of the paper is to compare such a sequential auction with a simultaneous auction where both goods are sold as a bundle or equivalently with a sequential auction where no information is revealed. We first show that there exists an equilibrium of the sequential game in pure and monotone strategies. Then, the comparison of the seller's expected revenue in the two cases allows us to conclude that contrary to Landsberger et al.'s predictions, the seller can not use the information to increase his revenue. This result is obtained using simulations for a large class of distribution functions. The seller must not reveal the name of the winner between the two auctions and instead sell both goods using a simultaneous auction.Received: 31 July 2001, Accepted: 5 February 2003, JEL Classification: B44I wish to thank Laurent Linnemer, Thomas Ricke, Michael Visser and Shmuel Zamir for helpful comments and suggestions. I am very grateful to the referees and the associated editors in charge of my paper.     

Multiple Testing for No Cointegration under Nonstationary Volatility

With cointegration tests often being oversized under time‐varying error variance, it is possible, if not likely, to confuse error variance non‐stationarity with cointegration. This paper takes an instrumental variable (IV) approach to establish individual‐unit test statistics for no cointegration that are robust to variance non‐stationarity. The sign of a fitted departure from long‐run equilibrium is used as an instrument when estimating an error‐correction model. The resulting IV‐based test is shown to follow a chi‐square limiting null distribution irrespective of the variance pattern of the data‐generating process. In spite of this, the test proposed here has, unlike previous work relying on instrumental variables, competitive local power against sequences of local alternatives in 1/T‐neighbourhoods of the null. The standard limiting null distribution motivates, using the single‐unit tests in a multiple testing approach for cointegration in multi‐country data sets by combining P‐values from individual units. Simulations suggest good performance of the single‐unit and multiple testing procedures under various plausible designs of cross‐sectional correlation and cross‐unit cointegration in the data. An application to the equilibrium relationship between short‐ and long‐term interest rates illustrates the dramatic differences between results of robust and non‐robust tests.     

On second-price auctions and imperfect competition

Consider two sellers each of whom has one unit of an indivisible good and two buyers each of whom is interested in buying one unit. The sellers simultaneously set reserve prices and use second-price auctions as rationing device. An equilibrium in pure strategies where each seller has a regular customer is characterized. The result is applied in order to demonstrate that not allowing sellers to use second-price auctions may enhance total surplus.     

Auctions with resale and bargaining power

We show that when the weak bidder’s bargaining power in the resale market is weakened, the auctioneer’s revenue from the first-price auction with resale is lower. Using the idea of Coase Theorem, we show that when the resale market is a sequential bargaining model with no commitment, the auctioneer’s revenue is substantially reduced, and the ranking is the opposite of Hafalir and Krishna (2009). We establish a version of the Coase Theorem in the context of the auctions with resale. When Coase Theorem holds, we show that the revenue of the auction with resale is lower than the revenue of the same auction without resale. We also provide the existence and uniqueness of equilibrium for our model of auctions with resale.     

Anticipation of Auction Fever: Entry Decision,Reserve Price and the Choice of Auction Design

This paper analyzes the entry decision of rational bidders who expect to experience auction fever in English auctions. It shows that nonparticipation decision reduces seller's expected profit and this effect may outweigh the positive effect of the auction fever. We analyze the choice between English and second‐price sealed‐bid auctions and the optimal reserve price in English auctions. We show that it might be optimal for the seller to set a reserve price below her own valuation of the object. Finally, we show that the order in which bidders place their bids matters and the first bidder always has an advantage. Copyright © 2011 John Wiley & Sons, Ltd.     

Auctions with asymmetric common-values: The first-price format

We study the performance of the first-price format in auctions with asymmetric common-values. We show that, contrary to the result for second-price auctions, a small advantage for one player translates only to small changes in bidders’ strategies, and the equilibrium remains close to the first-price equilibrium of the original game. We characterize the equilibrium bidding strategies and their behavior as the degree of asymmetry increases. Finally, we compare the revenues at the optimal auction, the first-price auction and the second-price auction.     

On the lowest-winning-bid and the highest-losing-bid auctions

Theoretical models of multi-unit, uniform-price auctions assume that the price is given by the highest losing bid. In practice, however, the price is usually given by the lowest winning bid. We derive the equilibrium bidding function of the lowest-winning-bid auction when there are k objects for sale and n bidders with unit demand, and prove that it converges to the bidding function of the highest-losing-bid auction if and only if the number of losers n−k gets large. When the number of losers grows large, the bidding functions converge at a linear rate and the prices in the two auctions converge in probability to the expected value of an object to the marginal winner.     

基于信息不对称房价博弈模型的优化均衡研究

目前我国住宅市场存在严重的信息不对称,如何进行规制引起多方关注。首先利用房价博弈模型研究我国住宅市场信息不对称下的非理性均衡和精炼优化的序贯均衡,进而探求政府对市场房价经济优化规制后的理性序贯均衡,以此提出政府对住宅产业的规制方式和对策建议。     

Wars of attrition and all-pay auctions with stochastic competition

We extend the war of attrition and all-pay auction analysis of Krishna and Morgan (1997) to a stochastic competition setting. We determine the existence of equilibrium bidding strategies and discuss the potential shape of these strategies. Results for the war of attrition contrast with the characterization of the bidding equilibrium strategies in the first-price all-pay auction as well as the winner-pay auctions. Furthermore we investigate the expected revenue comparisons among the war of attrition, the all-pay auction and the winner-pay auctions and discuss the linkage principle as well. Our findings are applicable to future works on contests and charity auctions.     

Selling goods of unknown quality: forward versus spot auctions

We consider an environment where the sale can take place so early that both the seller and potential buyers have the same uncertainty about the quality of the good. We present a simple model that allows the seller to offer the good for sale before or after this uncertainty is resolved, namely via forward auction or spot auction, respectively. We solve for the equilibrium of these two auctions and then compare the resulting expected revenues. We also consider the revenue implications of insurance in forward auctions.     

Existence of optimal auctions in general environments

We provide a unified approach to the problem of existence of optimal auctions for a wide variety of auction environments. We accomplish this by first establishing a general existence result for a particular Stackelberg revelation game. By systematically specializing our revelation game to cover various types of auctions, we are able to deduce the existence of optimal Bayesian auction mechanisms for single and multiple unit auctions, as well as for contract auctions with moral hazard and adverse selection. In all cases, we allow for risk aversion and multidimensional, stochastically dependent types.     

On Markov error‐correction models,with an application to stock prices and dividends

This paper considers Markov error‐correction (MEC) models in which deviations from the long‐run equilibrium are characterized by different rates of adjustment. To motivate our analysis and illustrate the various issues involved, our discussion is structured around the analysis of the long‐run properties of US stock prices and dividends. It is shown that the MEC model is flexible enough to account for situations where deviations from the long‐run equilibrium are nonstationary in one of the states of nature and allows us to test for such a possibility. An empirical specification procedure to establish the existence of MEC adjustment in practice is also presented. This is based on a multi‐step test procedure that exploits the differences between the global and local characteristics of systems with MEC adjustment. Copyright © 2004 John Wiley & Sons, Ltd.