Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to ... more Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of Rn.It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under ° has a non-empty intersection with the normal cone of X at the point.In case for every point in X it holds that the intersection of the image under ° with the normal cone of X at the point is either empty orcontains the origin 0n , then ° must have a zero point on X, i.e. there exists a point in X satisfying that 0n lies in the image of the point.Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point.In this paper we extend all these existence results by giving a general zero point existence theor...
In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polyt... more In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope coincides with the Minkowski sum of Lusztig polytopes arising from various reduced decompositions. Using this result, we formulate a conjecture about the crystal structures on FFLV polytopes.
Given two finite ordered sets A = {a 1 ,. .. , a m } and B = {b 1 ,. .. , b n }, introduce the se... more Given two finite ordered sets A = {a 1 ,. .. , a m } and B = {b 1 ,. .. , b n }, introduce the set of mn outcomes of the game O = {(a, b) | a ∈ A, b ∈ B} = {(a i , b j) | i ∈ I = {1,. .. , m}, j ∈ J = {1,. .. , n}. Two players, Alice and Bob, have the sets of strategies X and Y that consist of all monotone non-decreasing mappings x : A → B and y : B → A, respectively. It is easily seen that each pair (x, y) ∈ X × Y produces at least one deal, that is, an outcome (a, b) ∈ O such that x(a) = b and y(b) = a. Denote by G(x, y) ⊆ O the set of all such deals related to (x, y). The obtained mapping G = G m,n : X × Y → 2 O is a game correspondence. Choose an arbitrary deal g(x, y) ∈ G(x, y) to obtain a mapping g : X × Y → O, which is a game form. We use notation g ∈ G and show that each such game form is tight and, hence, Nash-solvable, that is, for any pair u = (u A , u B) of utility functions u A : O → R of Alice and u B : O → R of Bob, the obtained monotone bargaining game (g, u) has at least one Nash equilibrium in pure strategies. Moreover, the same equilibrium can be selected for all g ∈ G. We also obtain an efficient algorithm that determines such an equilibrium in time linear in mn, although the numbers of strategies |X| = m+n−1 m and |Y | = m+n−1 n are exponential in mn. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.
It is known that the distribution of an integrable random vector ξ in R d is uniquely determined ... more It is known that the distribution of an integrable random vector ξ in R d is uniquely determined by a (d + 1)-dimensional convex body called the lift zonoid of ξ. This concept is generalised to define the lift expectation of random convex bodies. However, the unique identification property of distributions is lost; it is shown that the lift expectation uniquely identifies only one-dimensional distributions of the support function, and so different random convex bodies may share the same lift expectation. The extent of this nonuniqueness is analysed and it is related to the identification of random convex functions using only their onedimensional marginals. Applications to construction of depth-trimmed regions and partial ordering of random convex bodies are also mentioned.
Известия Российской академии наук. Серия математическая, 2008
The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richard... more The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richardson coefficients. We define these bijections in terms of arrays and show that they coincide with analogous bijections defined in terms of discretely concave functions using the octahedron recurrence as well as with bijections defined in terms of Young tableaux. The main ingredient in the proof of their coincidence is a functional version of the Robinson-Schensted-Knuth correspondence. This paper was written with the partial financial support of the President's programme for the support of leading scientific schools (grant no. NSh-6417.2006.6) and the Russian Foundation for Basic Research (grants no.
Even if its roots are much older, random sets theory has been considered as an academic area, par... more Even if its roots are much older, random sets theory has been considered as an academic area, part of stochastic geometry, since Matheron [7]. Random sets theory was first applied in some fields related to engineering sciences like geology, image analysis and expert systems (see Goutsias et al. [4]), and recently in non-parametric statistics (Koshevoy et al. [5]) or also (see Molchanov [9]) in economic theory (for instance in finance and game theory) and in econometrics (for instance in linear models with interval-valued dependent or independent variables). We apply in this paper random sets theory to decision making. Our main result states that under a kind of vNM condition decision making for an arbitrary random set lottery reduces to decision making for a single-valued random set lottery, and the latter set is the set-valued expectation of the former random set. Through experiments in a laboratory, we observe consistency of decision making for ordering random sets with fixed act and varied random sets.
This paper is devoted to a study of mathematical structures arising from choice functions satisfy... more This paper is devoted to a study of mathematical structures arising from choice functions satisfying the path independence property (Plott functions). We broaden the notion of a choice function by allowing of empty choice. This enables us to define a lattice structure on the set of Plott functions. Moreover, this lattice is functorially dependent on its base. We introduce a natural convex structure on the set of linear orders
The paper puts forth a theory of choice functions in a neat way connecting it to a theory of exte... more The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider four classes of heritage choice functions satisfying the conditions M, N, W, and C.
A main aim of this paper is to make connections between two well-but up to now independently-deve... more 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.
We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of ... more We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids.
We consider a class of functions satisfying the gross-substitutes property (GS-functions). We sho... more 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 quasipolymatroids. 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.
Consider the tensor product of finite dimensional vector spaces V ⊗ W. We have an action of GL(V ... more Consider the tensor product of finite dimensional vector spaces V ⊗ W. We have an action of GL(V ) on V ⊗ W induced by standard action on V . Similarly the action of GL(W) on W gives us an action on V ⊗ W. These actions of GL(V ) and GL(W) on V ⊗W clearly commute with ...
We revisit the issue of existence of equilibrium in economies with indivisible goods and money, i... more We revisit the issue of existence of equilibrium in economies with indivisible goods and money, in which agents may trade many units of items. In [5] it was shown that the existence issue is related to discrete convexity. Classes of discrete convexity are characterized by the unimodularity of the allowable directions of one-dimensional demand sets. The class of graphical unimodular system can be put in relation with a nicely interpretable economic property of utility functions, the Gross Substitutability property. The question is still open as to what could be the possible, challenging economic interpretations and relevant examples of demand structures that correspond to other classes of discrete convexity. We consider here an economy populated with agents having a taste for complementarity; their utilities are generated by compounds of specific items grouped in "packages". Simple package-utilities translate in a straightforward fashion the fact that the items forming a package are complements. General package-utilities are obtained as the convolution (or aggregation) of simple packageutilities. We prove that if the collection of packages of items, that generates the utilities of agents in the economy, is unimodular then there exists a competitive equilibrium. Since any unimodular set of vectors can be implemented as a collection of 0-1 vectors ([3]), we get examples of demands for each class of discrete convexity.
For a d-variate measure a convex, compact set in R d1 , its lift zonoid, is constructed. This yie... more For a d-variate measure a convex, compact set in R d1 , its lift zonoid, is constructed. This yields an embedding of the class of d-variate measures having ®nite absolute ®rst moments into the space of convex, compact sets in R d1. 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 de®nes an ordering of random vectors that re¯ects their variability. The ordering is investigated in detail and, as an application, inequalities for random determinants are given.
We introduce a new scatter matrix functional which is a multivariate affine equivariant extension... more We introduce a new scatter matrix functional which is a multivariate affine equivariant extension of the mean deviation E(|x − Med(x)|). The estimate is constructed using the data vectors (centered with the multivariate Oja median) and their angular distances. The angular distance is based on Randles interdirections. The new estimate is called the zonoid covariance matrix (the ZCM), as it is the regular covariance matrix of the centers of the facets of the zonotope based on the data set. There is a kind of symmetry between the zonoid covariance matrix and the affine equivariant sign covariance matrix; interchanging the roles of data vectors and hyperplanes yields the sign covariance matrix as the zonoid covariance matrix. (It turns out that the symmetry relies on the zonoid of the distribution and its projection body which is also a zonoid.) The influence function and limiting distribution of the new scatter estimate, the ZCM, are derived to consider the robustness and efficiency properties of the estimate. Finite-sample efficiencies are studied in a small simulation study. The influence function of the ZCM is unbounded (linear in the radius of the contamination vector) but less influential in the tails than that of the regular covariance matrix (quadratic in the radius). The estimate is highly efficient in the multivariate normal case and performs better than the regular covariance matrix for heavy-tailed distributions.
The framework used to prove the multiplicative deformation of the algebra of Feynman-Bender diagr... more The framework used to prove the multiplicative deformation of the algebra of Feynman-Bender diagrams is a twisted shifted dual law (in fact, twisted twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameter is a perturbation of the shuffle coproduct which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.
We present a list of " local " axioms and an explicit combinatorial construction for th... more We present a list of " local " axioms and an explicit combinatorial construction for the regular B 2-crystals (crystal graphs of irreducible highest weight integrable modules over U q (sp 4)). Also a new combinatorial model for these crystals is developed.
In this paper we introduce the concept of quasi-building set that may underlie the coalitional st... more In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payoff his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some specifications of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently defined for this class. For graph games it therefore differs from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.
Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to ... more Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of Rn.It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under ° has a non-empty intersection with the normal cone of X at the point.In case for every point in X it holds that the intersection of the image under ° with the normal cone of X at the point is either empty orcontains the origin 0n , then ° must have a zero point on X, i.e. there exists a point in X satisfying that 0n lies in the image of the point.Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point.In this paper we extend all these existence results by giving a general zero point existence theor...
In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polyt... more In this paper we prove that in type $\tt A_n$, the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope coincides with the Minkowski sum of Lusztig polytopes arising from various reduced decompositions. Using this result, we formulate a conjecture about the crystal structures on FFLV polytopes.
Given two finite ordered sets A = {a 1 ,. .. , a m } and B = {b 1 ,. .. , b n }, introduce the se... more Given two finite ordered sets A = {a 1 ,. .. , a m } and B = {b 1 ,. .. , b n }, introduce the set of mn outcomes of the game O = {(a, b) | a ∈ A, b ∈ B} = {(a i , b j) | i ∈ I = {1,. .. , m}, j ∈ J = {1,. .. , n}. Two players, Alice and Bob, have the sets of strategies X and Y that consist of all monotone non-decreasing mappings x : A → B and y : B → A, respectively. It is easily seen that each pair (x, y) ∈ X × Y produces at least one deal, that is, an outcome (a, b) ∈ O such that x(a) = b and y(b) = a. Denote by G(x, y) ⊆ O the set of all such deals related to (x, y). The obtained mapping G = G m,n : X × Y → 2 O is a game correspondence. Choose an arbitrary deal g(x, y) ∈ G(x, y) to obtain a mapping g : X × Y → O, which is a game form. We use notation g ∈ G and show that each such game form is tight and, hence, Nash-solvable, that is, for any pair u = (u A , u B) of utility functions u A : O → R of Alice and u B : O → R of Bob, the obtained monotone bargaining game (g, u) has at least one Nash equilibrium in pure strategies. Moreover, the same equilibrium can be selected for all g ∈ G. We also obtain an efficient algorithm that determines such an equilibrium in time linear in mn, although the numbers of strategies |X| = m+n−1 m and |Y | = m+n−1 n are exponential in mn. Our results show that, somewhat surprising, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.
It is known that the distribution of an integrable random vector ξ in R d is uniquely determined ... more It is known that the distribution of an integrable random vector ξ in R d is uniquely determined by a (d + 1)-dimensional convex body called the lift zonoid of ξ. This concept is generalised to define the lift expectation of random convex bodies. However, the unique identification property of distributions is lost; it is shown that the lift expectation uniquely identifies only one-dimensional distributions of the support function, and so different random convex bodies may share the same lift expectation. The extent of this nonuniqueness is analysed and it is related to the identification of random convex functions using only their onedimensional marginals. Applications to construction of depth-trimmed regions and partial ordering of random convex bodies are also mentioned.
Известия Российской академии наук. Серия математическая, 2008
The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richard... more The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richardson coefficients. We define these bijections in terms of arrays and show that they coincide with analogous bijections defined in terms of discretely concave functions using the octahedron recurrence as well as with bijections defined in terms of Young tableaux. The main ingredient in the proof of their coincidence is a functional version of the Robinson-Schensted-Knuth correspondence. This paper was written with the partial financial support of the President's programme for the support of leading scientific schools (grant no. NSh-6417.2006.6) and the Russian Foundation for Basic Research (grants no.
Even if its roots are much older, random sets theory has been considered as an academic area, par... more Even if its roots are much older, random sets theory has been considered as an academic area, part of stochastic geometry, since Matheron [7]. Random sets theory was first applied in some fields related to engineering sciences like geology, image analysis and expert systems (see Goutsias et al. [4]), and recently in non-parametric statistics (Koshevoy et al. [5]) or also (see Molchanov [9]) in economic theory (for instance in finance and game theory) and in econometrics (for instance in linear models with interval-valued dependent or independent variables). We apply in this paper random sets theory to decision making. Our main result states that under a kind of vNM condition decision making for an arbitrary random set lottery reduces to decision making for a single-valued random set lottery, and the latter set is the set-valued expectation of the former random set. Through experiments in a laboratory, we observe consistency of decision making for ordering random sets with fixed act and varied random sets.
This paper is devoted to a study of mathematical structures arising from choice functions satisfy... more This paper is devoted to a study of mathematical structures arising from choice functions satisfying the path independence property (Plott functions). We broaden the notion of a choice function by allowing of empty choice. This enables us to define a lattice structure on the set of Plott functions. Moreover, this lattice is functorially dependent on its base. We introduce a natural convex structure on the set of linear orders
The paper puts forth a theory of choice functions in a neat way connecting it to a theory of exte... more The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider four classes of heritage choice functions satisfying the conditions M, N, W, and C.
A main aim of this paper is to make connections between two well-but up to now independently-deve... more 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.
We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of ... more We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids.
We consider a class of functions satisfying the gross-substitutes property (GS-functions). We sho... more 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 quasipolymatroids. 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.
Consider the tensor product of finite dimensional vector spaces V ⊗ W. We have an action of GL(V ... more Consider the tensor product of finite dimensional vector spaces V ⊗ W. We have an action of GL(V ) on V ⊗ W induced by standard action on V . Similarly the action of GL(W) on W gives us an action on V ⊗ W. These actions of GL(V ) and GL(W) on V ⊗W clearly commute with ...
We revisit the issue of existence of equilibrium in economies with indivisible goods and money, i... more We revisit the issue of existence of equilibrium in economies with indivisible goods and money, in which agents may trade many units of items. In [5] it was shown that the existence issue is related to discrete convexity. Classes of discrete convexity are characterized by the unimodularity of the allowable directions of one-dimensional demand sets. The class of graphical unimodular system can be put in relation with a nicely interpretable economic property of utility functions, the Gross Substitutability property. The question is still open as to what could be the possible, challenging economic interpretations and relevant examples of demand structures that correspond to other classes of discrete convexity. We consider here an economy populated with agents having a taste for complementarity; their utilities are generated by compounds of specific items grouped in "packages". Simple package-utilities translate in a straightforward fashion the fact that the items forming a package are complements. General package-utilities are obtained as the convolution (or aggregation) of simple packageutilities. We prove that if the collection of packages of items, that generates the utilities of agents in the economy, is unimodular then there exists a competitive equilibrium. Since any unimodular set of vectors can be implemented as a collection of 0-1 vectors ([3]), we get examples of demands for each class of discrete convexity.
For a d-variate measure a convex, compact set in R d1 , its lift zonoid, is constructed. This yie... more For a d-variate measure a convex, compact set in R d1 , its lift zonoid, is constructed. This yields an embedding of the class of d-variate measures having ®nite absolute ®rst moments into the space of convex, compact sets in R d1. 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 de®nes an ordering of random vectors that re¯ects their variability. The ordering is investigated in detail and, as an application, inequalities for random determinants are given.
We introduce a new scatter matrix functional which is a multivariate affine equivariant extension... more We introduce a new scatter matrix functional which is a multivariate affine equivariant extension of the mean deviation E(|x − Med(x)|). The estimate is constructed using the data vectors (centered with the multivariate Oja median) and their angular distances. The angular distance is based on Randles interdirections. The new estimate is called the zonoid covariance matrix (the ZCM), as it is the regular covariance matrix of the centers of the facets of the zonotope based on the data set. There is a kind of symmetry between the zonoid covariance matrix and the affine equivariant sign covariance matrix; interchanging the roles of data vectors and hyperplanes yields the sign covariance matrix as the zonoid covariance matrix. (It turns out that the symmetry relies on the zonoid of the distribution and its projection body which is also a zonoid.) The influence function and limiting distribution of the new scatter estimate, the ZCM, are derived to consider the robustness and efficiency properties of the estimate. Finite-sample efficiencies are studied in a small simulation study. The influence function of the ZCM is unbounded (linear in the radius of the contamination vector) but less influential in the tails than that of the regular covariance matrix (quadratic in the radius). The estimate is highly efficient in the multivariate normal case and performs better than the regular covariance matrix for heavy-tailed distributions.
The framework used to prove the multiplicative deformation of the algebra of Feynman-Bender diagr... more The framework used to prove the multiplicative deformation of the algebra of Feynman-Bender diagrams is a twisted shifted dual law (in fact, twisted twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameter is a perturbation of the shuffle coproduct which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.
We present a list of " local " axioms and an explicit combinatorial construction for th... more We present a list of " local " axioms and an explicit combinatorial construction for the regular B 2-crystals (crystal graphs of irreducible highest weight integrable modules over U q (sp 4)). Also a new combinatorial model for these crystals is developed.
In this paper we introduce the concept of quasi-building set that may underlie the coalitional st... more In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payoff his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some specifications of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently defined for this class. For graph games it therefore differs from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.
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Papers by Gleb Koshevoy