A Simple Proof of Inequalities of Integrals of Composite Functions

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a Banach vector space generated by $m$ $L_\mu^p$-spaces for $1\leq p<+\infty.$ Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by $m$ $L_\mu^\infty$-spaces instead

Second-Price Auctions with Different Participation Costs

This paper studies equilibria of second price auctions in independent private value envi- ronments 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 sub- mitting a bid, and non-monotonic 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 non-increasing reverse hazard rates. There exist non- monotonic 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

Second-Price Auctions with Different Participation Costs

This paper studies equilibria of second price auctions in independent private value envi- ronments 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 sub- mitting a bid, and non-monotonic 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 non-increasing reverse hazard rates. There exist non- monotonic 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

Second-price Auction with Two-Dimensional Private Information on Values and Participation Costs

This paper studies equilibria of second price auctions when values and participation costs are both privation information and are drawn from general distribution functions. We consider the existence and uniqueness of equilibrium. It is shown that there always exists an equilibrium for this general economy, and further there exists a unique symmetric equilibrium when all bidders are ex ante homogenous. Moreover, we identify a sufficient condition under which we have a unique equilibrium in a heterogenous economy with two bidders. Our general framework covers many relevant models in the literature as special cases

Second-price Auction with Two-Dimensional Private Information on Values and Participation Costs

This paper studies equilibria of second price auctions when values and participation costs are both privation information and are drawn from general distribution functions. We consider the existence and uniqueness of equilibrium. It is shown that there always exists an equilibrium for this general economy, and further there exists a unique symmetric equilibrium when all bidders are ex ante homogenous. Moreover, we identify a sufficient condition under which we have a unique equilibrium in a heterogenous economy with two bidders. Our general framework covers many relevant models in the literature as special cases

Second Price Auctions with Valuations and Participation Costs Both Privately Informed

This paper studies equilibria of second price auctions when valuations and participation costs are both private information with general distribution functions. We consider the existence and uniqueness of equilibrium in this general framework of two-dimensional types. It is shown that there always exists an equilibrium, and further there exists a unique symmetric equilibrium when all bidders are ex ante homogeneous. Moreover, we identify a sufficient condition under which there is a unique equilibrium in a heterogeneous economy with two bidders. Our general result includes many existing results as special cases

On Ratifiability of Efficient Cartel Mechanisms in First-Price Auctions with Participation Costs and Information Leakage

This paper investigates whether an efficient all-inclusive cartel mechanism studied by McAfee and McMillan (1992) can still preserve its efficiency when bidders can update their information through the cartel’s collusive mechanism and there is a cost to participate in the seller’s auction within the independent private values setting. It is shown that, when the seller uses the first-price auction, the usual efficient cartel mechanisms will no longer be ratifiable in the presence of both participation costs and potential information leakage. The bidder with the highest value in the cartel will have incentive to betray, sending a credible signal of his high value and thus discouraging other bidders from participating in the seller’s auction. However, the cartel mechanisms would still be efficient if either participation cost or information leakage, but not both, is present