Vladimir I. Danilov

Discrete convexity and equilibria in economies with indivisible goods and money

Abstract We consider a production economy with many indivisible goods and one perfectly divisible good. The aim of the paper is to provide some light on the reasons for which equilibrium exists for such an economy. It turns out, that a main reason for the existence is that supplies and demands of indivisible goods should be sets of a class of discrete convexity. The class of generalized polymatroids provides one of the most interesting classes of discrete convexity.

Measurable systems and behavioral sciences

Individual choices often depend on the order in which the decisions are made. In this paper, we expose a general theory of measurable systems (an example of which is an individual characterized by her preferences) allowing for incompatible (non-commuting) measurements. The basic concepts are illustrated in an example of non-classical rational choice. We conclude with a discussion of some of the basic properties of non-classical systems in the context of social sciences. In particular, we argue that the distinctive feature of non-classical systems translates into a formulation of bounded rationality.

Social choice mechanisms

1. Basic Concepts.- 1.1 Preferences.- 1.2 Social Choice Correspondences.- 1.3 Monotone Social Choice Correspondences.- 1.4 Social Choice Mechanisms.- 1.5 Effectivity Functions and Blockings.- 1.A1 Arrows Impossibility Theorem.- 1.A2 Non-manipulable SCFs.- 1.A3 Minimal Monotone SCCs.- Bibliographic Comments.- 2. Nash-consistent Mechanisms.- 2.1 Definitions and Examples.- 2.2 Blockings Generated by Consistent Mechanisms.- 2.3 Strongly Monotone Social Choice Correspondences.- 2.4 Nash-implementable Correspondences.- 2.5 Implementation: the Case of Two Participants.- 2.6 Acceptable Mechanisms.- 2.A A Simple Mechanism for the Implementation of Walrasian Equilibria.- Bibliographic Comments.- 3. Strategy-proof Mechanisms.- 3.1 Dominant Strategies. The Revelation Principle.- 3.2 Single-Peaked Environment.- 3.3 Linear Environment.- 3.4 The Transferable Environment. Groves Mechanisms.- 3.5 Further Properties of Groves Mechanisms.- 3.A1 The Simple Transferable Environment Case.- 3.A2 Acceptable Mechanisms in Transferable Environment.- Bibliographic Comments.- 4. Cores and Stable Blockings.- 4.1 Stable Outcomes.- 4.2 Additive Blockings.- 4.3 Convex Blockings.- 4.4 Almost Additive Blockings.- 4.5 Necessary Stability Conditions.- 4.6 Veto as a Decision-making Procedure.- 4.A1 Balanced Blockings.- 4.A2 Blockings with Infinite Number of Alternatives.- 4.A3 The Harems Lemma.- Bibliographic Comments.- 5. Strongly Consistent Mechanisms.- 5.1 Definitions and Examples.- 5.2 A Tokens Mechanism (or Veto-mechanism).- 5.3 Blockings Generated by SC-mechanisms.- 5.4 Direct Core Mechanisms.- 5.5 Laminable Blockings.- 5.6 A Necessary and Sufficient Condition of Laminability.- 5.7 Neutral Laminable Blockings.- 5.A Implementation via Strong Equilibria.- Bibliographic Comments.- References.

The structure of non-manipulable social choice rules on a tree

Abstract In this paper we give a theorem describing a structure of any non-manipulable social choice rule on a tree. In particular, any such rule is a median of dictatorial and constant rules.

Gross substitution, discrete convexity, and submodularity

We consider a class of functions satisfying the gross-substitutes property (GS-functions). We show that GS-functions are concave functions, whose parquets are constituted by quasi-polymatroids. The class of conjugate functions to GS-functions turns out to be the class of polyhedral supermodular functions. The class of polyhedral GS-functions is a proper subclass of the class of polyhedral submodular functions. PM-functions, concave functions whose parquets are constituted by g-polymatroids, form a proper subclass of the class of GS-functions. We provide an additional characterization of PM-functions.

Autowave model of localized plastic flow of solids

The features of plastic flow localization at all stages of strain hardening and at the prefracture stage were analyzed. It was shown that macroscopic localization of plastic flow at these stages can be considered as a self-organization process. At the linear hardening stage, an autowave process of flow localization occurs in the sample, which is characterized by the wavelength and propagation velocity. At the prefracture stage, the autowave process collapses with macroneck formation followed by the nucleation of a ductile crack.

Maximal Unimodular Systems of Vectors

A subset R of a vector space V(orRn) is called unimodular(or U -system) if every vector r?R has an integral representation in every basis B?R. A U -system R is calledmaximal if one cannot add a non-zero vector not colinear to vectors of R such that the new system is unimodular and spans RR. In this work, we refine assertions of Seymour7and give a description of maximal U -systems. We show that a maximal U -system can be obtained as amalgams (as 1- and 2-sums) of simplest maximal U -systems called components. A component is a maximal U -system having no 1- and 2-decompositions. It is shown that there are three types of components: the root systems An, which are graphic, cographic systems related to non-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a special system E5representing the matroid R10from7which is neither graphic nor cographic. We give conditions that are necessary and sufficient for maximality of an amalgamated U -system. We give a complete description of all 11 maximal U -systems of dimension 6.

Existence of stable matchings in some three-sided systems

Abstract We give here some restrictions on preferences sufficient for existence of stable matching in multi-sided systems.

A generalized economic equilibrium

Abstract In economic equilibrium models in a convex environment additional conditions such as Slater or resource relatedness guarantee existence of equilibrium prices. However, these conditions are rather unrealistic. Removing them we come to a generalization of price concept. It is shown that in the pure exchange setting this generalization coincides with the classical notion of ‘exchange value’. In an economy with production we call this generalization the ‘production value’. The corresponding equilibria are characterized by a stratification of agents, goods and currencies. We also give a topological version of generalized price.

Mathematics of Plott choice functions

Abstract This paper is devoted to the study of mathematical structures related to the class of choice functions satisfying the path independence property (Plott functions). The set of Plott functions has a natural lattice structure. We describe join-irreducible and meet-irreducible elements of this lattice. We introduce a convex structure on the set of simple words and show that Plott functions are in a natural one-to-one correspondence with convex subsets of simple words. In particular, this correspondence is compatible with the lattice structures on both sets. All these structures and relations are functorially dependent on a change of base sets.