Optimal collusion-proof auctions

We study an optimal collusion-proof auction in an environment where subsets of bidders may collude not just on their bids but also on their participation. Despite their ability to collude on participation, informational asymmetry facing the potential colluders can be exploited significantly to weaken their collusive power. The second-best auction — i.e., the optimal auction in a collusion-free environment — can be made collusion-proof, if at least one bidder is not collusive, or there are multiple bidding cartels, or the second-best outcome involves a non-trivial probability of the object not being sold. Regardless, optimal collusion-proof auction prescribes non-trivial exclusion of collusive bidders, i.e., a refusal to sell to any collusive bidder with positive probability.     

Selection effects in auctions for monopoly rights

We demonstrate that auctioning market licenses may result in higher market prices than assigning them via more random allocation mechanisms. When future market profit is uncertain, winning an auction is like winning a lottery ticket. If firms differ in risk attitudes, auctions select the least risk-averse firm, which, in turn, set a higher price (or a higher quantity, in case quantity is the decision variable) in the marketplace than an average firm.     

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.     

Asymmetric English auctions

This paper studies equilibria of the n-bidder single-object English, or open ascending price, auction in a setting with interdependent values and asymmetric bidders. Maskin (in: H. Siebert (Ed.), Privatization, Institut fur Weltwirtschaften der Universität Kiel, Kiel, 1992, pp. 115-136.) showed that if the values satisfy a “single crossing” condition, then the two-bidder English auction has an efficient equilibrium. In this paper, two extensions of the single crossing condition, the “average crossing” condition and the “cyclical crossing” condition, are introduced. The main result is that under either of these conditions the n-bidder English auction has an efficient equilibrium.     

Best-of-three all-pay auctions

We study a best-of-three all-pay auction. It is shown that with values of winning and without values of losing, this auction is less productive (the players’ total expected effort is smaller) than the one-stage all-pay auction. However, with different values of losing over the contest’s stages it may be more productive than the one-stage all-pay auction.     

Private information revelation in common-value auctions

When a seller has information that could help bidders to estimate asset value, a dictum of auction theory has been that all such information should be publicly announced to bidders. The possibility of privately revealing this information to one or more bidders is introduced. Seller in some circumstances may attain higher expected revenue through revealing his information privately. Examples show that the role of private revelation is more complex than simply generating bidder asymmetry.     

Endogenous bid rotation in repeated auctions

I study collusion between two bidders in a general symmetric IPV repeated auction, without communication, side transfers, or public randomization. I construct a collusive scheme, endogenous bid rotation, that generates a payoff larger than the bid rotation payoff.     

Mixed bundling auctions

We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.     

Optimal search auctions

We study the design of profit maximizing single unit auctions under the assumption that the seller needs to incur costs to contact prospective bidders and inform them about the auction. With independent bidders’ types and possibly interdependent valuations, the seller's problem can be reduced to a search problem in which the surplus is measured in terms of virtual utilities minus search costs. Compared to the socially efficient mechanism, the optimal mechanism features fewer participants, longer search conditional on the same set of participants, and inefficient sequence of entry.     

Equilibrium reserve prices in sequential ascending auctions

We study a model where bidders have perfectly correlated valuations for two goods sold sequentially in two ascending-price auctions. The seller sets a reserve price before the beginning of each auction. Despite the lack of commitment by the seller, we characterize an equilibrium and study its properties. Strategic non-disclosure of information takes the form of non-participation in the early auction by low-valuation bidders, while high-valuation bidders bid up to their true valuations. Some buyers who would profitably buy at the reserve price refrain from participating in order to decrease the second-auction reserve price.     

Rates of information aggregation in common value auctions

We study the rates at which transaction prices aggregate information in common value auctions under the different information structures in Wilson (Rev. Econ. Stud. 44 (1977) 511) and Pesendorfer and Swinkels (Econometrica 65 (1997) 1247). We consider uniform-price auctions in which k identical objects of unknown value are auctioned to n bidders, where both n and k are allowed to diverge to infinity, and k/n converges to a number in [0,1). The Wilson assumptions lead to information aggregation at a rate proportional to , but the price aggregates information at a rate proportional to in the PS setting. We also consider English auctions, and investigate whether the extra information revealed in equilibrium improves convergence rates in these auctions.     

Tacit collusion in repeated auctions

This paper considers the question of tacit collusion in repeated auctions with independent private values and with limited public monitoring. McAfee and McMillan show that the extent of collusion is tied to the availability of transfers. Monetary transfers allow cartels to extract full surplus. A folk theorem proved by Fudenberg et al. (Econometrica 62 (1994) 997-1039) shows that transfers of future payoffs are almost as good if players are patient and communicate before auctions. We ask how the scope of collusion is affected if players dispense with explicit communication and their monitoring is limited. Collusion better than bid rotation is still feasible, but full surplus cannot be extracted. This constraint becomes less severe with more players and large cartels can become asymptotically efficient even with very limited monitoring.     

Revenue equivalence in asymmetric auctions

The Revenue Equivalence Theorem is generalized to the case of asymmetric auctions in which each player's valuation is drawn independently from a common support according to his/her distribution function.     

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.     

Input Prices as Signals of Costs to a Downstream Rival and Customer

We consider a dual distribution channel in which a vertically integrated manufacturer competes with a downstream rival in a retail market and also sells an input to the rival. We use a signalling model with a continuum of types to examine a situation in which the manufacturer has private information on the production cost of its retail product. We show that in a separating equilibrium under Cournot (Bertrand) retail competition, the manufacturer signals the uncompetitiveness (competitiveness) of its firm by charging a smaller input price than the optimal price under complete information.     

Asymmetric first price auctions

A new approach to asymmetric first price auctions is proposed which circumvents the need to examine bidding strategies directly. Specifically, the ratio of bidders' (endogenous) payoffs is analyzed and compared to the ratio of the (exogenous) distribution functions that describe beliefs. Most of the results are inferred from this comparison. In the existing theoretical literature, assumptions of first order stochastic dominance or stronger imply that the latter ratio has very specific properties, but no such assumptions are imposed here. It is proven that first order stochastic dominance is necessary for bidding strategies not to cross. When this assumption is relaxed in the numerical literature it is done in a manner that leads to exactly one crossing. However, it is straightforward to construct examples with several crossings. Finally, bid distributions will cross in auctions with two bidders whenever second order (but not first order) stochastic dominance applies.     

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.     

Multi-unit auctions with uniform prices

Summary. Auctions in which individuals can purchase more than one unit of the good being sold differ in striking ways from multi-unit auctions in which individuals may purchase only one unit. The uniform price auction in particular frequently yields Nash equilibria in which bidders underbid for their second unit and therefore pay very low prices for the good. This paper characterizes equilibria for the uniform price auction. Received: July 31, 1995; revised version: May 28, 1997     

Asymmetric first-price menu auctions under intricate uncertainty

This paper studies asymmetric first-price menu auctions in the procurement environment where the buyer does not commit to a decision rule and asymmetric sellers have interdependent costs and statistically affiliated signals. Sellers compete in bidding a menu of contracts, where a contract specifies a vector of characteristics and a payment required from the buyer for delivering these characteristics. The buyer does not commit ex-ante to a decision rule but rather upon observing all the menus offered by sellers chooses the best contract. This paper establishes the existence of a continuum of separating monotone equilibria in this game bounded above by the jointly ex-post efficient outcome and below by the jointly interim efficient outcome. It shows that the jointly ex-post efficient equilibrium outcome is the only ex-post renegotiation-proof outcome and it is also ex-ante robust to all continuation equilibria.     

Information structures in optimal auctions

A seller wishes to sell an object to one of multiple bidders. The valuations of the bidders are privately known. We consider the joint design problem in which the seller can decide the accuracy by which bidders learn their valuation and to whom to sell at what price. We establish that optimal information structures in an optimal auction exhibit a number of properties: (i) information structures can be represented by monotone partitions, (ii) the cardinality of each partition is finite, (iii) the partitions are asymmetric across agents. We show that an optimal information structure exists.