Standard Auctions with Financially Constrained Bidders

We develop a methodology for analyzing the revenue and efficiency performance of auctions when buyers have private information about their willingness to pay and ability to pay. We then apply the framework to scenarios involving standard auction mechanisms. In the simplest case, where bidders face absolute spending limits, first-price auctions yield higher expected revenue and social surplus than second-price auctions. The revenue dominance of first-price auctions over second-price auctions carries over to the case where bidders have access to credit. These rankings are explained by differences in the extent to which financial constraints bind in different auction formats.     

Auctions in which losers set the price

We study auctions of a single asset among symmetric bidders with affiliated values. We show that the second-price auction minimizes revenue among all efficient auction mechanisms in which only the winner pays, and the price only depends on the losers' bids. In particular, we show that the kth price auction generates higher revenue than the second-price auction, for all k>2. If rationing is allowed, with shares of the asset rationed among the t highest bidders, then the (t+1)st price auction yields the lowest revenue among all auctions with rationing in which only the winners pay and the unit price only depends on the losers' bids. Finally, we compute bidding functions and revenue of the kth price auction, with and without rationing, for an illustrative example much used in the experimental literature to study first-price, second-price and English auctions.     

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.     

Allocating multiple units

Summary. This paper studies the allocation and rent distribution in multi-unit, combinatorial-bid auctions under complete information. We focus on the natural multi-unit analogue of the first-price auction, where buyers bid total payments, pay their bids, and where the seller allocates goods to maximize his revenue. While there are many equilibria in this auction, only efficient equilibria remain when the truthful equilibrium restriction of the menu-auction literature is used. Focusing on these equilibria we first show that the first-price auction just described is revenue and outcome equivalent to a Vickrey auction, which is the multi unit analogue of a second-price auction. Furthermore, we characterize these equilibria when valuations take a number of different forms: diminishing marginal valuations, increasing average valuations, and marginal valuations with single turning points. Received: December 23, 1999; revised version: December 10, 2001     

Ratifiability of efficient collusive mechanisms in second-price auctions with participation costs

We investigate whether efficient collusive bidding mechanisms are affected by potential information leakage from bidders' decisions to participate in them within the independent private values setting. We apply the concept of ratifiability introduced by Cramton and Palfrey [Cramton, P.C., Palfrey, T.R., 1995, Ratifiable mechanisms: Learning from disagreement, Games Econ. Behav. 10 (2), 255–283] and show that when the seller uses a second-price auction with participation costs, the standard efficient cartel mechanisms such as pre-auction knockouts analyzed in the literature will not be ratified by cartel members. A high-value bidder benefits from vetoing the cartel mechanism since doing so sends a credible signal that she has high value, which in turn discourages other bidders from participating in the seller's auction.     

Auctions with cross-shareholdings

Summary. We study the effect of cross-shareholding among two competing firms on their bidding behavior and the expected sales revenue for the seller in an auction environment. The bidders private signals are independent, and the model encompasses the private values model and a particular common value model as special cases. When cross-shareholding is symmetric, the bids decrease towards the collusive level as the degree of cross-shareholding increases. The Revenue Equivalence result no longer holds: the first-price auction generates higher expected revenue for the seller than the second-price auction.With asymmetric cross-shareholding, revenue comparisons are only possible in the common value setting. Expected revenue for the seller is again higher in the first-price than in the second price auction. Bidding behavior in the second-price auction is more sensitive to changes in cross-shareholding and the value environment than in the first-price auction.Received: 18 September 2000, Revised: 27 May 2003, JEL Classification Numbers: C72, D44.Correspondence to: Sudipto DasguptaWe thank Sugato Bhattacharyya, Paul Klemperer, Kunal Sengupta and Guofu Tan for helpful discussions, and an anonymous referee for suggestions that improved the paper. The usual disclaimer, of course, applies.     

Asymmetric information about rivals'' types in standard auctions

We study auctions in which bidders may know the types of some rival bidders but not others. This asymmetry in bidders' knowledge about rivals' types has different effects on the two standard auction formats. In a second-price auction, it is weakly dominant to bid one's valuation, so the knowledge of rivals' types has no effect, and the good is allocated efficiently. In a first-price auction, bidders refine their bidding strategies based on their knowledge of rivals' types, which yields an inefficient allocation. We show that the inefficient allocation in the first-price auction translates into a poor revenue performance. Given a standard regularity condition, the seller earns higher expected revenue from the second-price auction than from the first-price auction, whereas the bidders are better off from the latter.     

Bidder collusion

We analyze bidder collusion at first-price and second-price auctions. Our focus is on less than all-inclusive cartels and collusive mechanisms that do not rely on auction outcomes. We show that cartels that cannot control the bids of their members can eliminate all ring competition at second-price auctions, but not at first-price auctions. At first-price auctions, when the cartel cannot control members’ bids, cartel behavior involves multiple cartel bids. Cartels that can control bids of their members can suppress all ring competition at both second-price and first-price auctions; however, shill bidding reduces the profitability of collusion at first-price auctions.     

First-price auctions with resale: the case of many bidders

We study first-price auctions with resale when there are many bidders and derive existence and characterization results under the assumption that the winner of the initial auction runs a second-price auction with an optimal reserve price. The fact that symmetrization fails when there are more than two bidders has been observed before, but we also provide the direction: weaker bidders are less likely to win than stronger ones. For a special class of distributions and three bidders, we prove that the bid distributions are more symmetric with resale than without. Numerical simulations suggest that the more bidders there are, the more similar the allocation is to the case without resale, and thus, the more asymmetric the bid distributions are between strong and weak bidders. We also show in an example that the revenue advantage of first-price auctions over second-price auctions is positive, but decreasing in the number of bidders.     

Revenue ranking of first-price auctions with resale

When the price setter in post-auction resale is chosen according to exogenous probabilities, Hafalir and Krishna (2008) [2] showed that the first-price auction brings more expected revenues than the second-price auction with truth-bidding bidders. We complete their revenue ranking by proving that the first-price auction produces higher expected revenues the higher the probability the auction winner sets the resale price.     

An Asymptotic Solution for Sealed Bid Common-Value Auctions with Bidders Having Asymmetric Information

We investigate a first-price common-value auction where bidders have asymmetric information about an item of unknown value. We compute the unique Nash equilibrium when the bidders are constrained to translation-invariant bid functions. Further, this profile of bid functions is also an asymptotic Nash equilibrium (without the constraint on the bidders' strategies) as the a priori distribution of the true value becomes increasingly diffuse. All bidders have positive expected profits at equilibrium. In the second-price analogue with two bidders there is a continuum of Nash equilibria in which both bidders have positive expected profits. Journal of Economic Literature Classification Number: C7.     

A choice of auction format in seller cheating: a signaling game analysis

In this paper, we examine which auction format, first-price or second-price, a seller will choose when he can profitably cheat in a second price auction by observing all bids by possible buyers and submitting a shill bid as pretending to be a buyer. We model this choice of auction format in seller cheating as a signaling game in which the buyers may regard the selection of a second price auction by the seller as a signal that he is a shill bidder. By introducing trembling-hand perfectness as a refinement of signaling equilibrium, we find two possible strictly perfect signaling equilibria. One is a separating equilibrium in which a noncheating honest seller selects a first price auction and a cheating seller does a second price auction. In another pooling equilibrium, however, both cheating and non-cheating sellers select a second price auction. The conclusion that a seller chooses a second price auction even if he cannot cheat is in contrast to the previous literature, which focused on the case of independent values. We thank an anonymous referee for useful comments that have improved the paper. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 15310023 and (C) 18530139.     

Expected revenue of all-pay and first-price sealed-bid auctions with affiliated signals

In the general symmetric auction framework of Milgrom and Weber (1982) it is shown—as a new manifestation of thelinkage principle—that the all-pay sealed-bid auction yieldshigher expected revenue than the standard first-price sealed-bid auction. This raises the question why sealed-bid auctions of the standard first-price variety are observed in practice whereas the all-pay variety is not.     

Risk aversion and optimal reserve prices in first- and second-price auctions

We analyze the effects of buyer and seller risk aversion in first- and second-price auctions in the classic setting of symmetric and independent private values. We show that the seller's optimal reserve price decreases in his own risk aversion, and more so in the first-price auction. The reserve price also decreases in the buyers' risk aversion in the first-price auction. Thus, greater risk aversion increases ex post efficiency in both auctions - especially that of the first-price auction. At the interim stage, the first-price auction is preferred by all buyer types in a lower interval, as well as by the seller.     

A common value auction model with endogenous entry and information acquisition

Summary Much of the auction literature assumes both a fixed number of bidders and a fixed information setting. This sidesteps the important and often costly decisions a potential bidder must make prior to an auction: Should I enter and, if I do, what level of resources should I expend evaluating the good prior to bidding? We answer these questions for a stylized information model of a common value auction. The expected selling price is shown to be the expected value of the good minus the expected aggregate entry and information costs of the bidders. Thus, the seller indirectly pays for these costs to the bidders. There are auctions where the seller seemingly restricts the bidders' information expenditures. While this restriction does influence the entry decision, we demonstrate that the overall effect can be to improve the selling price. Finally, the probability of entry and the chosen accuracy of the information are never more in the second-price auction than in the first-price auction, and the seller prefers the second-price auction.We are grateful for the comments and suggestions of seminar participants at the University of British Columbia, Dartmouth College, the University of Wisconsin, Yale University, and the International Conference on Game Theory and Economics at SUNY Stony Brook.     

Revenue comparison in common-value auctions: Two examples

When players' affiliated values are symmetrically distributed, expected revenue in the second-price auction equals or exceeds that in the first-price auction (Milgrom and Weber, 1982). We provide two common-value examples where this ranking fails when players are asymmetrically informed.     

An experiment on auctions with endogenous budget constraints

We perform laboratory experiments comparing auctions with endogenous budget constraints. A principal imposes a budget limit on a bidder (an agent) in response to a principal-agent problem. In contrast to the existing literature where budget constraints are exogenous, this theory predicts that tighter constraints will be imposed in first-price auctions than in second-price auctions, tending to offset any advantages attributable to the lower bidding strategy of the first-price auction. Our experimental findings support this theory: principals are found to set significantly lower budgets in first-price auctions. The result holds robustly, whether the principal chooses a budget for human bidders or computerized bidders. We further show that the empirical revenue difference between first- and second-price formats persists with and without budget constraints.     

Sequential Procurement With Subcontracting

Two symmetric sellers are approached sequentially by fragmented buyers. Each buyer conducts a second-price auction and purchases from the seller who offers the lower price. Winning an auction affects bidding for future contracts because the sellers have nonconstant marginal costs. We assume that the sellers are completely informed, and we study the unique equilibrium that survives iterated elimination of weakly dominated strategies. If subcontracting between the sellers is impossible, the final allocation of contracts is generally inefficient. If postauction subcontracting is possible, the sellers can be worse off, ex ante , than when subcontracting is impossible.     

Large common value auctions with risk averse bidders

We analyze large symmetric auctions with conditionally i.i.d. common values and risk averse bidders. Our main result characterizes the asymptotic equilibrium price distribution for the first- and second-price auctions. As an implication, we show that with constant absolute risk aversion (CARA), the second-price auction raises significantly more revenue than the first-price auction. While this ranking seems robust in numerical analysis also outside the CARA specification, we show by counterexamples that the result does not generalize to all risk averse utility functions.     

Multi-dimensional reference-dependent preferences in sealed-bid auctions – How (most) laboratory experiments differ from the field

We study bidding behavior in first- and second-price sealed-bid auctions with loss-averse agents. Our model predicts overbidding in first-price induced-value auctions consistent with evidence from most laboratory experiments. Substantially different bidding behavior could result in commodity auctions where money and auction item are consumed along different dimensions of the consumption space. Differences also result in second-price auctions. Our study thereby indicates that transferring qualitative behavioral findings from induced-value laboratory experiments to the field may be problematic if subjects are loss-averse.