Monopoly Pricing in a Vertical Market with Demand Uncertainty

We study a vertical market with an upsteam supplier and multiple downstream retailers. Demand uncertainty falls to the supplier who acts first and sets a uniform wholesale price before the retailers observe the realized demand and engage in retail competition. Our focus is on the supplier's optimal pricing decision. We express the price elasticity of expected demand in terms of the mean residual demand (MRD) function of the demand distribution. This allows for a closed form characterization of the points of unitary elasticity that maximize the supplier's profits and the derivation of a mild unimodality condition for the supplier's objective function that generalizes the widely used increasing generalized failure rate (IGFR) condition. A direct implication is that optimal prices between different markets can be ordered if the markets can be stochastically ordered according to their MRD functions or equivalently to their elasticities. Based on this, we apply the theory of stochastic orders to study the response of the supplier's optimal price to various features of the demand distribution. Our findings challenge previously established economic insights about the effects of market size, demand transformations and demand variability on wholesale prices and indicate that the conclusions largely depend on the exact notion that will be employed. We then turn to measure market performance and derive a distribution free and tight bound on the probability of no trade between the supplier and the retailers. If trade takes place, our findings indicate that ovarall performance depends on the interplay between demand uncertainty and level of retail competition

Acyclic domains of linear orders: a survey

Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.acyclic set;alternating scheme;distributive lattice;effet Condorcet;linear order,maximal chain,permutoèdre lattice, single-peaked domain,weak Bruhat order,value restriction.

Order Analogues and Betti Polynomials

AbstractWe exhibit an order-preserving surjection from the lattice of subgroups of a finite abelianp-group of typeλonto the product of chains of lengths the parts of the partitionλ. Thereby, we establish the subgroup lattice as an order-theoretic, not just enumerative,p-analogue of the chain product. This insight underlies our study of the simplicial complexesΔS(p), whose simplices are chains of subgroups of orderspk, somek∈S. Each of these subgroup complexes is homotopy equivalent to a wedge of spheres of dimension |S|−1. The number of spheres in the wedge,βS(p), is known to have nonnegative coefficients as a polynomial inp. Our main result provides a topological explanation of this enumerative result. We use our order-preserving surjection to findβS(pmaximal simplices inΔS(pwhose deletion leaves a contractible subcomplex. This work suggests a definition of order analogue; our main result holds for any semimodular lattices that are order analogues of a semimodular lattice

A remarkable sequence of integers

A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.Comment: 20 pages, 5 figure

Temporal evolution of generalization during learning in linear networks

We study generalization in a simple framework of feedforward linear networks with n inputs and n outputs, trained from examples by gradient descent on the usual quadratic error function. We derive analytical results on the behavior of the validation function corresponding to the LMS error function calculated on a set of validation patterns. We show that the behavior of the validation function depends critically on the initial conditions and on the characteristics of the noise. Under certain simple assumptions, if the initial weights are sufficiently small, the validation function has a unique minimum corresponding to an optimal stopping time for training for which simple bounds can be calculated. There exists also situations where the validation function can have more complicated and somewhat unexpected behavior such as multiple local minima (at most n) of variable depth and long but finite plateau effects. Additional results and possible extensions are briefly discussed

Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures

Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-trivial extension of this semigroup approach that the natural analog of the Poisson maximum entropy property remains valid if the compound Poisson distributions under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. Sufficient conditions for compound distributions to be log-concave are discussed and applications to combinatorics are examined; new bounds are derived on the entropy of the cardinality of a random independent set in a claw-free graph, and a connection is drawn to Mason's conjecture for matroids. The present results are primarily motivated by the desire to provide an information-theoretic foundation for compound Poisson approximation and associated limit theorems, analogous to the corresponding developments for the central limit theorem and for Poisson approximation. Our results also demonstrate new links between some probabilistic methods and the combinatorial notions of log-concavity and ultra-log-concavity, and they add to the growing body of work exploring the applications of maximum entropy characterizations to problems in discrete mathematics.Comment: 30 pages. This submission supersedes arXiv:0805.4112v1. Changes in v2: Updated references, typos correcte

Performance bounds on optimal fixed prices

Cataloged from PDF version of article.We consider the problem of selling a fixed stock of items over a finite horizon when the buyers arrive following a Poisson process. We obtain a general lower bound on the performance of using a fixed price rather than dynamically adjusting the price. The bound is 63.21% for one unit of inventory, and it improves as the inventory increases. For the one-unit case, we also obtain tight bounds: 89.85% for the constantelasticity and 96.93% for the linear price-response functions. © 2013 Elsevier B.V. All rights reserved