Multiple unit auctions with strategic price-quantity decisions

Summary. I study a multiple unit auction where symmetric risk-neutral bidders choose prices and quantities endogenously. In the model, bidders (a) may place non-linear valuations on the auctioned units, and (b) bid for several units at the same price (“lumpy” bids). I characterize quantity-symmetric and strictly monotone-increasing price equilibria for discriminatory and competitive auctions, and show that (i) if quantity strategy profiles are equal across auctions revenue- equivalence holds, (ii) expected revenue is higher if bidders bid for the entire supply rather than for shares of it, and (iii) equilibrium allocations may fail to be Pareto-optimal. Received: April 14, 1995; revised version: September 3, 1997     

Auctions of Heterogeneous Objects

Simultaneous sealed bid auctions of heterogeneous objects are analyzed. Each bidder's reservation value for an object depends upon the other objects he obtains. Bidders' reservation values are common knowledge. In simultaneous first-price auctions, the set of Walrasian equilibrium allocations contains the set of pure strategy Nash equilibrium allocations which in turn contains the set of strict Walrasian equilibrium allocations. Hence, pure strategy Nash equilibria (when they exist) are efficient. Mixed strategy Nash equilibria may be inefficient. In simultaneous second-price auctions, any efficient allocation can be implemented as a pure strategy Nash equilibrium outcome if a Walrasian equilibrium exists.Journal of Economic LiteratureClassification Numbers: D44, D51.     

Multiunit Pay‐Your‐Bid Auction with One‐Dimensional Multiunit Demands*

An arbitrary number of units of a good is sold to two bidders through a discriminatory auction. The bidders are homogeneous ex ante and their demand functions are two‐step functions that depend on a single parameter. We characterize the symmetric Bayesian equilibrium and prove its existence and uniqueness. We compare this equilibrium with the equilibrium of the multiunit Vickrey auction and with the equilibria of the single‐unit first price and second price auctions. We examine the consequences of bundling all units into one package. We study the impacts that variations of the “relative” supply have on the equilibrium, on the bidders' average payoffs per unit, and on the efficiency of the equilibrium allocation.     

Collusion via Signalling in Simultaneous Ascending Bid Auctions with Heterogeneous Objects, with and without Complementarities

Collusive equilibria exist in simultaneous ascending bid auctions with multiple objects, even with large complementarities in the buyers' utility functions. The bidders collude by dividing the objects among themselves, while keeping the prices low. In the most collusive equilibrium the complementarities are never realized. The scope for collusion however narrows as the ratio between the number of bidders and the number of objects increases.     

Pure strategy equilibria of single and double auctions with interdependent values

We prove the existence of monotonic pure strategy equilibrium for many kinds of asymmetric auctions with n bidders and unitary demands, interdependent values and independent types. The assumptions require monotonicity only in the own bidder's type. The payments can be a function of all bids. Thus, we provide a new equilibrium existence result for asymmetrical double auctions and a small number of bidders. The generality of our setting requires the use of special tie-breaking rules. We present an example of a double auction with interdependent values where all equilibria are trivial, that is, they have zero probability of trade. This is related to Akerlof's “market for lemmons” example and to the “winner's curse,” establishing a connection between them. However, we are able to provide sufficient conditions for non-trivial equilibrium existence.     

Strategic Jump Bidding in English Auctions

This paper solves for equilibria of sequential bid (or English) auctions with affiliated values when jump bidding strategies may be employed to intimidate one's opponents. In these equilibria, jump bids serve as correlating devices which select asymmetric bidding functions to be played subsequently. Each possibility of jump bidding provides a Pareto improvement for the bidders from the symmetric equilibrium of a sealed bid, second-price auction. The expanded set of equilibria can approximate either first- or second-price outcomes and produce exactly the set of expected prices between those two bounds. These results contrast with standard conclusions that equate English and second-price auctions.     

Symmetric Separating Equilibria in English Auctions

We characterize the set of perfect Bayesian equilibria in symmetric separating strategies in the model of English auctions given by P. R. Milgrom and R. J. Weber (1982, Econometrica50, 1089–1122). There is a continuum of such equilibria. The equilibrium derived by Milgrom and Weber is that in which bids are maximal. Only in the case of pure private values does a restriction to weakly undominated strategies select a unique equilibrium. This has important implications for empirical studies of English auctions, particularly outside the pure private values paradigm. Journal of Economic Literature Classification Numbers: D44, D82.     

On the failure of monotonicity in uniform-price auctions

Except for well-studied special cases in which bidders have single-unit demand or bidders are risk-neutral with independent private values, equilibria of uniform-price auctions with private values need not possess familiar monotonicity properties. In particular, equilibria in weakly undominated strategies may exist in which some bidders bid strictly less on some units when they have strictly higher values for every unit.     

Asymmetric Nash equilibrium in emission rights auctions

This paper studies the problems of emission rights auctions, and presents a uniform price auction mechanism based on three assumptions, i.e., all buyers are asymmetric, every buyer submits a nonincreasing continuous demand function, and every buyer's valuation to per unit of the emission rights is common value information. It focuses on solving the asymmetric Nash equilibrium for this auction mechanism. It concludes that there exist multiple Nash equilibria in our auction mechanism, but the arbitrary low equilibrium prices cannot emerge. We also give several suggestions on how to induce the auction to a desired ideal equilibrium state in mechanism design of emission rights auctions.     

Monotone Equilibrium in Multi-Unit Auctions

In two-sided multi-unit auctions having a variety of payment rules, including uniform-price and discriminatory auctions, a monotone pure-strategy equilibrium (MPSE) exists when bidders are risk neutral with independent multi-dimensional types and interdependent values. In fact, all mixed-strategy equilibria are ex post allocation and interim expected payment equivalent to MPSE. Thus, for standard expected surplus/revenue analysis, there is no loss restricting attention to monotone strategies.     

Private monitoring in auctions

We study infinitely repeated first-price auctions in which a bidder only learns whether or not he won the object. While repetition of the stage-game equilibrium is the unique Nash equilibrium in public strategies, with patient bidders there are simple Nash equilibria in private strategies that improve on bid rotation. Sequential rationality is appropriately captured by essentially perfect Bayesian equilibrium (EPBE), which ignores behavior after irrelevant histories. Our main result is the construction of EPBEa that improve upon bid rotation. Assuming symmetry, the exclusionary schemes of Skrzypacz and Hopenhayn [Tacit collusion in repeated auctions, J. Econ. Theory 114 (2004), 153–169], including asymptotically efficient ones, are supported as EPBEa.     

Equilibrium selection in common-value second-price auctions

This note considers equilibrium selection in common-value second-price auctions with two bidders. We show that for each ex post equilibrium in continuous and undominated strategies, a sequence of “almost common-value” auctions can be constructed such that each of them possesses a unique undominated and continuous equilibrium and the corresponding sequence of equilibria converges to that ex post equilibrium. As an implication, no equilibrium selection of this model based on perturbations seems to be more convincing than others.     

Equilibrium of a sequence of auctions when bidders demand multiple items

We construct an ex post perfect equilibrium in a sequence of second-price auctions with two bidders where each bidder’s marginal values for additional units are decreasing. This equilibrium implies an increasing path of transaction prices and is ex post efficient.     

The Relevance of a Choice of Auction Format in a Competitive Environment

We examine the relevance of an auction format in a competitive environment by comparing uniform and discriminatory price auctions with many bidders in a private values setting. We show that if the number of objects for sale is small relative to the number of bidders, then all equilibria of both auctions are approximately efficient and lead to approximately the same revenue. When the number of objects for sale is proportional to the number of bidders, then the particulars of the auction format matter. All equilibria of the uniform auction are efficient, while all of the equilibria of the discriminatory auction are inefficient. The relative revenue rankings of the auction formats can go in either direction, depending on the specifics of the environment. These conclusions regarding the efficiency and revenue ranking are in contrast to the previous literature, which focused on the case of independent information across agents.     

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.     

The Evolutionary Stability of Auctions over Bargaining

This paper considers equilibrium in transaction mechanisms. In an environment with homogeneous buyers and sellers, which eliminates the advantage auctions possess of matching buyers and sellers, both auctions and bargaining are equilibria. However, only auctions are evolutionarily stable. This identifies a new advantage of auctions over bargaining, arising from the division of the gains from trade.Journal of Economic LiteratureClassification Numbers: C78, C73, D44.     

Bidding for the future: signaling in auctions with an aftermarket

This paper considers auctions where bidders compete for an advantage in future strategic interactions. When bidders wish to exaggerate their private information, equilibrium bidding functions are biased upwards as bidders attempt to signal via the winning bid. Signaling is most prominent in second-price auctions where equilibrium bids are “above value.” In English and first-price auctions, signaling is less extreme since the winner incurs the cost of her signaling choice. The opportunity to signal lowers bidders’ payoffs and raises revenue. When bidders understate their private information, separating equilibria need not exist and the auction may not be efficient.     

Multi-unit demand auctions with synergies: behavior in sealed-bid versus ascending-bid uniform-price auctions

We construct a relatively simple model of bidding with synergies and solve it for both open outcry and sealed-bid uniform-price auctions. The essential behavioral forces involved in these auctions are: (1) A demand reduction force resulting from the monopsony power that bidders with multiple-unit demands have when synergies are relatively inconsequential, and (2) Bidding above stand-alone values in order to capture significant complementarities between units. The latter creates a potential “exposure problem,” as bidders may win only parts of a package and earn negative profits. Bidding outcomes are closer to equilibrium in clock compared to sealed-bid auctions. However, there are substantial and systematic deviations from equilibrium, with patterns of out-of-equilibrium play differing systematically between the two auction formats. These patterns of out-of-equilibrium play are analyzed, along with their effects on revenue and efficiency.     

Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction

In second price Internet auctions with a fixed end time, such as those on eBay, many bidders submit their bids in the closing minutes or seconds of an auction. We propose an internet auction model, in which very late bids have a positive probability of not being successfully submitted, and show that late bidding in a fixed deadline auction can occur at equilibrium in auctions both with private values and with uncertain, dependent values. Late bidding may also arise out of equilibrium, as a best reply to incremental bidding. However, the strategic advantages of late bidding are severely attenuated in auctions that apply an automatic extension rule such as auctions conducted on Amazon. Field data show that there is more late bidding on eBay than on Amazon, and this difference grows with experience. We also study the incidence of multiple bidding, and its relation to late bidding.     

Ascending auctions for multiple objects: the case for the Japanese design

Summary. We consider two ascending auctions for multiple objects, namely, an English and a Japanese auction, and derive a perfect Bayesian equilibrium of the Japanese auction by exploiting its strategic equivalence with the survival auction, which consists of a finite sequence of sealed-bid auctions. Thus an equilibrium of a continuous time game is derived by means of backward induction in finitely many steps. We then show that all equilibria of the Japanese auction induce equilibria of the English auction, but that many collusive or signaling equilibria of the English auction do not have a counterpart in the Japanese auction.Received: 2 September 2004, Revised: 20 April 2005, JEL Classification Numbers: C72, D44.Fabrizio Germano: Correspondence toWe are indebted to Philippe Jehiel for useful discussions and to Nicolas Vieille for suggestions in the proof of Proposition 2; we also thank seminar audiences in Athens, Basel, Beer-Sheva, Berlin, Brussels, Cambridge, Edinburgh, Exeter, Lausanne, Lisbon, London, Louvain-la-Neuve, Namur and Tel Aviv. Germano acknowledges financial support from Euopean Commission, TMR Network Grant ERBFMRXCT0055, “Cooperation and Information” as well from the Spanish Ministry of Science and Technology, Grants SEC2001-0792, SEJ2004-06319, and in form of a Ramon y Cajal Fellowship. Lovo is member of GREGHEC, unité CNRS, FRE-2810. The support of the Economic and Social Research Council (ESRC) is also gratefully acknowledged. The work was part of the programme of the ESRC Research for Economic Learning and Social Evolution.