Gleb A. Koshevoy

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.

Choice functions and abstract convex geometries

Abstract A main aim of this paper is to make connections between two well – but up to now independently – developed theories, the theory of choice functions and the theory of closure operators. It is shown that the classes of ordinally rationalizable and path independent choice functions are related to the classes of distributive and anti-exchange closures.

The Lorenz Zonoid of a Multivariate Distribution

Abstract The article extends the usual Lorenz curve and Lorenz order of univariate distributions to the multivariate case. For a given probability distribution in nonnegative d space, d ≥ 1, we define and investigate the Lorenz zonoid and the Lorenz surface, which are sets in (d + 1) space. The surface equals the usual Lorenz curve when d = 1. Included is the definition of the Lorenz surface of a finite number of vectors that may be interpreted as the endowments of economic units in d commodities. As a notion of increasing multivariate disparity, we introduce the set inclusion of Lorenz zonoids and show that it is equivalent to directional majorization.

Multivariate Lorenz majorization

The Lorenz zonotope is a multivariate generalization of the Lorenz curve. It allows to define multivariate Lorenz majorization whose properties are studied.

Lift zonoids, random convex hulls and the variability of random vectors

For a d-variate measure a convex, compact set in Rd+1, its lift zonoid, is constructed. This yields an embedding of the class of d-variate measures having finite absolute first moments into the space of convex, compact sets in [Rd+1. The embedding is continuous, positive homogeneous and additive and has useful applications to the analysis and comparison of random vectors. The left zonoid is related to random convex sets and to the convex hull of a multivariate random sample. For an arbitrary sampling distribution, bounds are derived on the expected volume of the random convex hull. The set inclusion of lift zonoids defines an ordering of random vectors that reflects their variability. The ordering is investigated in detail and, as an application, inequalities for random determinants are given.

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.

Zonoid Data Depth: Theory and Computation

A new notion of data depth in d-space is presented, called the zonoid data depth. It is affine equivariant and has useful continuity and monotonicity properties. An efficient algorithm is developed that calculates the depth of a given point with respect to a d-variate empirical distribution.

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.

A topological approach to social choice with infinite populations

Abstract We consider a topological approach to social choice theory that was initiated by Chichilnisky. We investigate the existence problem of Chichilnisky rules in the framework of infinite populations. The existence of such rules depends on a topology on the set of profiles and a group of permutations of the set of participants.

Solution Concepts for Games with General Coalitional Structure

We introduce several new solution concepts for cooperative games with arbitrary coalition structure. Of our main interest are coalitions structures being so-called building sets. A collection of sets is a building set if every singleton is a member and if the union of any two non-disjoint sets in the collection is also in the collection. For example, the vertex sets of all connected subgraphs of a connected graph form a building set. To a building set is associated a collection of strictly nested sets. These sets give rise to a polyhedral complex. Maximal strictly nested sets correspond to the vertices of this complex. For a given characteristic function on a building set a marginal payoff vector is associated to any strictly nested set. The GC-solution is defined as the average of the marginal payoff vectors over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the complex and differs for graphical building sets from the Myerson value. The HS-solution is defined as the average of marginal payoff vectors over the class of so-called half-space nested sets. The NT-solution is defined as the average of marginal payoff vectors over another specific class of maximal strictly nested sets, the class of so-called NT-nested sets. On graphical buildings the collection of NT-nested sets corresponds to the set of normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions. For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it.