Second chance offers in auctions

This paper examines situations in which a seller might make a second chance (take-it-or-leave-it) offer to a non-winning bidder at a price equal to their bid at auction. This study is motivated by the take-it-or-leave-it second chance offer rules used by eBay and a number of state procurement agencies. Equilibrium bidder behavior is determined for IPV sealed bid first price, second price, English, and Vickrey auctions when a second chance offer will be made with an exogenous probability $p$ . In all but the Vickrey auction (which elicits the dominant strategy of bidding one’s value) equilibrium bids are lower than if there were no possibility of a second chance offer and higher than if a second chance offer will be made for certain. Further, the possibility of a second chance offer erodes the strategic equivalence between second price bids and English auction drop out levels. If bidders are risk averse (with CRRA preferences), this difference leads to expected revenue dominance of the second price over the English auction, both of which dominate the Vickrey auction. The first price auction is also shown to revenue dominate the Vickrey auction, and moreover, numerical results and intuition from existing literature suggest that the first price auction revenue dominates the second price auction.     

Sequential auctions with imperfect quantity commitment

This paper studies sequential second price auctions with imperfect quantity commitment in environments involving single-unit demands, independent private values, and non-decreasing marginal costs. The paper characterizes the symmetric equilibrium strategy and demonstrates that the equilibrium price sequence is conditionally non-increasing, showing a downwards drift in cases in which the marginal cost exceeds the reserve price with positive probability. The paper also argues that unlike a strong seller who sets reserve prices strictly above marginal costs, a weak seller will typically prefer to commit to such inefficiently low reserve prices.     

Equilibria in second price auctions with participation costs

We investigate equilibria of sealed-bid second price auctions with bidder participation costs in the independent private values environment. We focus on equilibria in cutoff strategies (participate and bid the valuation iff it is greater than the cutoff), since if a bidder finds it optimal to participate, she cannot do better than bidding her valuation. When bidders are symmetric, concavity (strict convexity) of the cumulative distribution function from which the valuations are drawn is a sufficient condition for uniqueness (multiplicity) within this class. We also study a special case with asymmetric bidders and show that concavity/convexity plays a similar role.     

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.     

Pre-auction offers in asymmetric first-price and second-price auctions

We consider first-price and second-price auctions with asymmetric buyers, and examine whether pre-auction offers to a subset of buyers are profitable. A single offer is never profitable prior to a second-price auction, but may be profitable prior to a first-price auction. However, a sequence of offers is profitable in either type of auction. In our model, suitably chosen pre-auction offers work because they move the assignment when bidder valuations are “near the top” closer to the optimal, revenue-maximizing assignment.     

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     

Existence of equilibrium in large double auctions

We show the existence of a pure strategy, symmetric, increasing equilibrium in double auction markets with correlated, conditionally independent private values and many participants. The equilibrium we find is arbitrarily close to fully revealing as the market size grows. Our results provide strategic foundations for price-taking behavior in large markets.     

Uniqueness of equilibrium in sealed high-bid auctions

For the case of two buyers we show that equilibrium in the sealed high-bid auction is unique when (i) buyers' reservations prices are drawn independently from distributions with finite support and positive mass at the lower endpoint; (ii) buyers have private values; and (iii) buyers' preferences are log supermodular. For more than two buyers, we obtain the same result under the additional assumptions that (iv) buyers with the same reservation price have the same preferences; (v) buyers are risk neutral or risk averse with non-increasing absolute risk aversion; and (vi) the supports of the different buyers' distributions of reservation prices have the same upper endpoint.     

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.     

Bertrand equilibrium with subadditive costs

We show here, in contrast to recent results, that if firms have different cost functions (that are strictly subadditive), such that the ‘monopoly breakeven prices’ are different, then in a homogeneous product duopoly there is always a Bertrand equilibrium (either in pure strategies or in mixed strategies).     

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.     

Existence of an equilibrium in first price auctions

Summary We study the first price auction game with an arbitrary number of bidders when the bidders' valuations are independent from each other. In technical words, we work within the independent private value model. We show that if the supports of the valuation probability distributions have the same minimum and if this minimum is not a mass point of any of these distributions, then a Nash equilibrium of the first price auction exists. We then modify the first price auction game by adding a closed interval of messages. Every bidder has to send a message with the bid he submits. These messages are used in the resolution of the ties. The winner of the auction is chosen randomly among the highest bidders with the highest value of the message among the highest bidders. In the general case, we prove the existence of a Nash equilibrium for this augmented first price auction.I wish to thank Mamoru Kaneko and a referee for their comments on an earlier draft.     

Asset price volatility in a nonconvex general equilibrium model

Summary. Asset prices and returns are known to vary significantly more than␣output or aggregate consumption growth, and an order of magnitude in excess of what is justified by innovations to fundamentals. We study excess price volatility in a lifecycle economy with two assets (claims on capital and␣a public debt bubble), heterogeneous agents, and increasing returns to financial intermediation. We show that a relatively modest nonconvexity generates a set valued equilibrium correspondence in asset prices, with two␣stable branches. Price volatility is the outcome of an equilibrium selection mechanism, which mixes adaptive learning with “noise”, and alternates stochastically between the two stable branches of the price correspondence. Received: March 19, 1998; revised version: June 2, 1998     

The value of commitment in Stackelberg games with observation costs

We study Stackelberg games in which the follower faces a cost for observing the leader's action. We show that, irrespective of the size of the cost, the leader's value of commitment is lost completely in all pure-strategy equilibria. However, there also exists a mixed-strategy equilibrium that fully preserves the first-mover advantage. In this type of equilibrium, the probability that the follower looks at the leader's action is independent of the cost of looking.     

An experimental study of commitment in Stackelberg games with observation costs

We report on experiments examining the value of commitment in Stackelberg games where the follower chooses whether to pay some cost to perfectly observe the leader's action. Várdy [Games Econ. Behav. (2004)] shows that in the unique pure-strategy subgame perfect equilibrium of this game, the value of commitment is lost completely; however, there exists a mixed-strategy subgame perfect equilibrium where the value of commitment is fully preserved. In the data, the value of commitment is largely preserved when the cost of looking is small, while it is lost when the cost is large. Nevertheless, for small observation costs, equilibrium behavior is clearly rejected. Instead, subjects persistently play non-equilibrium strategies in which the probability of the follower choosing to observe the leader's action is a decreasing function of the observation cost.     

Multi-unit auctions with private information: an indivisible unit continuous price model

We construct a model of multi-unit auctions in which I bidders bid for two indivisible units of a common value good. Using a first-order approach, we find that there are equilibria in which bidders bid the same price for both units in the discriminatory auction, but not in the uniform auction. When there are only two bidders, under certain conditions, there are linear equilibria for both the discriminatory and the uniform auction formats. In all equilibria, bidders equalize the expected marginal benefit of bidding to the marginal costs of bidding. We show that comparison of the seller??s expected revenue across auction formats depends only on the ratio of the precision of private information to the precision of public information.     

Work costs and nonconvex preferences in the estimation of labor supply models

Works costs have not been adequately handled in labor supply estimation, likely due to their complexity. We show that, if work costs are not accounted for in the budget and time constraints in a structural labor supply model, they will be subsumed into the data generating preferences. Even if underlying preferences over consumption and leisure are convex, the presence of unobservable work costs can make these preferences appear nonconvex. However, we show that, under plausible conditions, policy relevant calculations, such as estimates of the effect of tax changes on labor supply and deadweight loss measures, are not affected by the fact that estimated preferences incorporate work costs.     

Qualitative analysis of Bayes–Nash equilibrium in weak asymmetric auctions

It is now generally accepted that there is a negative relationship between inflation and central bank independence (CBI). This study finds that inflation and CBI are endogenously determined, yet the negative correlation between the two remains robust.     

Asymmetric adjustment costs in simple general equilibrium models

This paper demonstrates that the introduction of asymmetric adjustment costs in a simple general equilibrium framework establishes a meaningful link between factor price determination and output determination, breaking the analytically convenient dichotomy of the Heckscher-Ohlin-Samuelson model. The possibility of trade between seemingly similar countries that differ in their adjustment technologies is visited.     

Efficiency and nonlinear pricing in nonconvex environments with externalities: A generalization of the Lindahl equilibrium concept

It is a well-known fact that if damaging externalities occur, nonconvexities will necessarily arise in the underlying economic environment rendering questionable the relevance of standard market approaches to the allocation problem. In response to this state of affairs, we define and study a generalization of the Lindahl Equilibrium concept that allows for nonlinear pricing systems in which the marginal outlay may depend on the quantity purchased. This “Generalized Lindahl Equilibrium” attains local pareto efficiency for its associated allocations and is able to support (in the usual sense) any pareto efficient allocation that satisfies a regularity condition.