English auctions and the Stolper-Samuelson theorem

We prove that the English auction (with bidders that need not be ex ante identical and may have interdependent valuations) has an efficient ex post equilibrium. We establish this result for environments where it has not been previously obtained. We also prove two versions of the Stolper-Samuelson theorem, one for economies with n goods and n factors, and one for non-square economies. Similar assumptions and methods underlie these seemingly unrelated results.English auctions, Stolper-Samuelson, single crossing

Comparative Statics, English Auctions, and the Stolper-Samuelson Theorem

Changes in the parameters of an $n$-dimensional system of equations induce changes in its solutions. For a class of such systems, we determine the qualitative change in solutions given certain qualitative changes in parameters. Our methods and results are elementary yet useful. They highlight the existence of a common thread, our ``own effect'' assumption, in formally diverse areas of economics. We discuss several applications; among them, we establish the existence of efficient equilibria in English auctions with interdependent valuations, and a version of the Stolper-Samuelson Theorem for an $n \times n$ trade model.effficient auctions, international trade theory, implicit function theorem

Multidimensional Mechanism Design: Revenue Maximization and the Multiple-Good Monopoly

The seller of N distinct objects is uncertain about the buyers valuation for those objects. The sellers problem, to maximize expected revenue, consists of maximizing a linear functional over a convex set of mechanisms. A solution to the sellers problem can always be found in an extreme point of the feasible set. We identify the relevant extreme points and faces of the feasible set. With N = 1, the extreme points are easily described providing simple proofs of well-known results. The revenue-maximizing mechanism assigns the object with probability one or zero depending on the buyers report. With N > 1, extreme points often involve randomization in the assignment of goods. Virtually any extreme point of the feasible set maximizes revenue for a well-behaved distribution of buyers valuations. We provide a simple algebraic procedure to determine whether a mechanism is an extreme point

Approximate Revenue Maximization with Multiple Items

Maximizing the revenue from selling _more than one_ good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple "one-dimensional" mechanisms, such as selling the goods separately, _guarantee_ at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least $50\%$ when they are independent. For the case of $k>2$ independent goods, we show that selling them separately guarantees at least a $c/\log^2 k$ fraction of the optimal revenue; and, for independent and identically distributed goods, we show that selling them as one bundle guarantees at least a $c/\log k$ fraction of the optimal revenue. Additional results compare the revenues from the two simple mechanisms of selling the goods separately and bundled, identify situations where bundling is optimal, and extend the analysis to multiple buyers.Comment: Presented in ACM EC conference, 201

Sampling and Representation Complexity of Revenue Maximization

We consider (approximate) revenue maximization in auctions where the distribution on input valuations is given via "black box" access to samples from the distribution. We observe that the number of samples required -- the sample complexity -- is tightly related to the representation complexity of an approximately revenue-maximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities

Optimal Design of Robust Combinatorial Mechanisms for Substitutable Goods

In this paper we consider multidimensional mechanism design problem for selling discrete substitutable items to a group of buyers. Previous work on this problem mostly focus on stochastic description of valuations used by the seller. However, in certain applications, no prior information regarding buyers' preferences is known. To address this issue, we consider uncertain valuations and formulate the problem in a robust optimization framework: the objective is to minimize the maximum regret. For a special case of revenue-maximizing pricing problem we present a solution method based on mixed-integer linear programming formulation

Optimal Pricing Is Hard

We show that computing the revenue-optimal deterministic auction in unit-demand single-buyer Bayesian settings, i.e. the optimal item-pricing, is computationally hard even in single-item settings where the buyer’s value distribution is a sum of independently distributed attributes, or multi-item settings where the buyer’s values for the items are independent. We also show that it is intractable to optimally price the grand bundle of multiple items for an additive bidder whose values for the items are independent. These difficulties stem from implicit definitions of a value distribution. We provide three instances of how different properties of implicit distributions can lead to intractability: the first is a #P-hardness proof, while the remaining two are reductions from the SQRT-SUM problem of Garey, Graham, and Johnson [14]. While simple pricing schemes can oftentimes approximate the best scheme in revenue, they can have drastically different underlying structure. We argue therefore that either the specification of the input distribution must be highly restricted in format, or it is necessary for the goal to be mere approximation to the optimal scheme’s revenue instead of computing properties of the scheme itself.Microsoft Research (Fellowship)Alfred P. Sloan Foundation (Fellowship)National Science Foundation (U.S.) (CAREER Award CCF-0953960)National Science Foundation (U.S.) (Award CCF-1101491)Hertz Foundation (Daniel Stroock Fellowship

The collagenic structure of human digital skin seen by scanning electron microscopy after Ohtani maceration technique.

We performed a morphological scanning electron microscope (SEM) study to describe the fine structure and disposition of collagenous tissue in the human toe. After therapeutic amputation of a human right Leg, we applied the Othani maceration technique to the skin of three toes surgically explanted from the foot. We distinguished eight cutaneous regions and focused on some specialized collagenous structures differing in the thickness of the skin. The eight areas investigated were: the dorsal skin, the eponychium, the perionychium, the hyponychium, the region under the visible nail, the nail root, the plantar skin and finally the toe tip. Each of these areas is characterized by a distinctive collagenous surface disposition, with some peculiar features mostly related to dermal. papillae. At high magnification, we observed the spatial arrangement of the cottagen fibers constituting the top of the dermal, papillae that represents the attachment site of the proliferative basal layer of the epidermis. We also noted an impressive density of collagen fibers throughout the thickness of the dermal layer, organized in specialized structures and constituting the skeleton of dermal, thermoreceptorial corpuscles or sweat glands. A combination of SEM and Ohtani technique disclosed the three-dimensional architecture of the collagenous matrix of tarsal skin under physiologic conditions, giving a detailed description of the most reactive tissue during pathologic processes