109,299 research outputs found
Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)
We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is defined as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution differs from the Myerson-kind value defined in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game. 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 by using its Möbius inversion.Core;polytope;building set;nested set complex;Möbius inversion;permutations;normal fan;average tree solution;Myerson value
Local cohomology and stratification
We outline an algorithm to recover the canonical (or, coarsest)
stratification of a given finite-dimensional regular CW complex into cohomology
manifolds, each of which is a union of cells. The construction proceeds by
iteratively localizing the poset of cells about a family of subposets; these
subposets are in turn determined by a collection of cosheaves which capture
variations in cohomology of cellular neighborhoods across the underlying
complex. The result is a nested sequence of categories, each containing all the
cells as its set of objects, with the property that two cells are isomorphic in
the last category if and only if they lie in the same canonical stratum. The
entire process is amenable to efficient distributed computation.Comment: Final version, published in Foundations of Computational Mathematic
The theorem of the complement for nested subpfaffian sets
Let R be an o-minimal expansion of the real field, and let
L(R) be the language consisting of all nested Rolle leaves over R. We call a
set nested subpfaffian over R if it is the projection of a boolean combination
of definable sets and nested Rolle leaves over R. Assuming that R admits
analytic cell decomposition, we prove that the complement of a nested
subpfaffian set over R is again a nested subpfaffian set over R. As a
corollary, we obtain that if R admits analytic cell decomposition, then the
pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves
over R, a one-stage process, and that P(R) is model complete in the language
L(R).Comment: final version before publicatio
All graphs have tree-decompositions displaying their topological ends
We show that every connected graph has a spanning tree that displays all its
topological ends. This proves a 1964 conjecture of Halin in corrected form, and
settles a problem of Diestel from 1992
Which nestohedra are removahedra?
A removahedron is a polytope obtained by deleting inequalities from the facet
description of the classical permutahedron. Relevant examples range from the
associahedra to the permutahedron itself, which raises the natural question to
characterize which nestohedra can be realized as removahedra. In this note, we
show that the nested complex of any connected building set closed under
intersection can be realized as a removahedron. We present two different
complementary proofs: one based on the building trees and the nested fan, and
the other based on Minkowski sums of dilated faces of the standard simplex. In
general, this closure condition is sufficient but not necessary to obtain
removahedra. However, we show that it is also necessary to obtain removahedra
from graphical building sets, and that it is equivalent to the corresponding
graph being chordful (i.e. any cycle induces a clique).Comment: 13 pages, 4 figures; Version 2: new Remark 2
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