scholarly journals Stable sets of weak tournaments

2004 ◽  
Vol 14 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Somdeb Lahiri
Keyword(s):  
Stable Set ◽  
Stable Sets ◽  
Empty Subset

In this paper we obtain conditions on weak tournaments, which guarantee that every non-empty subset of alternatives admits a stable set. We also show that there exists a unique stable set for each non-empty subset of alternatives which coincides with its set of best elements, if and only if, the weak tournament is quasi-transitive. A somewhat weaker version of this result, which is also established in this paper, is that there exists a unique stable set for each non-empty subset of alternatives (: which may or may not coincide with its set of best elements), if and only if the weak tournament is acyclic.

scholarly journals Coalition formation and history dependence

2020 ◽  
Vol 15 (1) ◽  
pp. 159-197 ◽  
Author(s):  
Bhaskar Dutta ◽  
Hannu Vartiainen
Keyword(s):  
Coalition Formation ◽  
Stable Set ◽  
Transferable Utility ◽  
Stable Sets ◽  
Solution Concepts ◽  
History Dependence ◽  
Von Neumann ◽  
Finite Games ◽  
A Chain ◽  
Indirect Dominance

Farsighted formulations of coalitional formation, for instance, by Harsanyi and Ray and Vohra, have typically been based on the von Neumann–Morgenstern stable set. These farsighted stable sets use a notion of indirect dominance in which an outcome can be dominated by a chain of coalitional “moves” in which each coalition that is involved in the sequence eventually stands to gain. Dutta and Vohra point out that these solution concepts do not require coalitions to make optimal moves. Hence, these solution concepts can yield unreasonable predictions. Dutta and Vohra restricted coalitions to hold common, history‐independent expectations that incorporate optimality regarding the continuation path. This paper extends the Dutta–Vohra analysis by allowing for history‐dependent expectations. The paper provides characterization results for two solution concepts that correspond to two versions of optimality. It demonstrates the power of history dependence by establishing nonemptyness results for all finite games as well as transferable utility partition function games. The paper also provides partial comparisons of the solution concepts to other solutions.

scholarly journals Maximality in the Farsighted Stable Set

Econometrica ◽  
2019 ◽  
Vol 87 (5) ◽  
pp. 1763-1779 ◽  
Author(s):  
Debraj Ray ◽  
Rajiv Vohra
Keyword(s):  
Stable Set ◽  
Stable Sets ◽  
Von Neumann

Harsanyi (1974) and Ray and Vohra (2015) extended the stable set of von Neumann and Morgenstern to impose farsighted credibility on coalitional deviations. But the resulting farsighted stable set suffers from a conceptual drawback: while coalitional moves improve on existing outcomes, coalitions might do even better by moving elsewhere. Or other coalitions might intervene to impose their favored moves. We show that every farsighted stable set satisfying some reasonable and easily verifiable properties is unaffected by the imposition of these stringent maximality constraints. The properties we describe are satisfied by many, but not all, farsighted stable sets.

scholarly journals An existence theorem for abstract stable sets

1991 ◽  
Vol 51 (3) ◽  
pp. 468-472 ◽  
Author(s):  
Chih Chang ◽  
Gerard J. Chang
Keyword(s):  
Existence Theorem ◽  
Dominance Relation ◽  
Stable Set ◽  
Stable Sets

AbstractWe provide an existence theorem for stable sets which is equivalent to Zorn's lemma and study the connections between the unique stable set for majorization and the stable sets for the dominance relation.

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

scholarly journals Greedoids on vertex sets of B-joins of graphs

2014 ◽  
Vol 30 (3) ◽  
pp. 335-344
Author(s):  
VADIM E. LEVIT ◽  
◽  
EUGEN MANDRESCU ◽  
Keyword(s):  
Bipartite Graph ◽  
Sufficient Conditions ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Vertex Sets

Let Ψ(G) be the family of all local maximum stable sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. It was shown that Ψ(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, were analyzed in [16, 17, 18, 19, 20, 24]. If G1, G2 are two disjoint graphs, and B is a bipartite graph having E(B) as an edge set and bipartition {V (G1), V (G2)}, then by B-join of G1, G2 we mean the graph B (G1, G2) whose vertex set is V (G1) ∪ V (G2) and edge set is E(G1) ∪ E(G2) ∪ E (B). In this paper we present several necessary and sufficient conditions for Ψ(B (G1, G2)) to form a greedoid, an antimatroid, and a matroid, in terms of Ψ(G1), Ψ(G2) and E (B).

scholarly journals A Conley-type decomposition of the strong chain recurrent set

2017 ◽  
Vol 39 (5) ◽  
pp. 1261-1274 ◽  
Author(s):  
OLGA BERNARDI ◽  
ANNA FLORIO
Keyword(s):  
Metric Space ◽  
Continuous Flow ◽  
Invariant Set ◽  
Stable Set ◽  
Stable Sets ◽  
Compact Metric Space ◽  
Chain Recurrent Set ◽  
Type Decomposition ◽  
Recurrent Set

For a continuous flow on a compact metric space, the aim of this paper is to prove a Conley-type decomposition of the strong chain recurrent set. We first discuss in detail the main properties of strong chain recurrent sets. We then introduce the notion of strongly stable set as a closed invariant set which is the intersection of the $\unicode[STIX]{x1D714}$-limits of a specific family of nested and definitively invariant neighborhoods of itself. This notion strengthens that of stable set; moreover, any attractor turns out to be strongly stable. We then show that strongly stable sets play the role of attractors in the decomposition of the strong chain recurrent set; indeed, we prove that the strong chain recurrent set coincides with the intersection of all strongly stable sets and their complementary.

GENERALIZED CORES AND STABLE SETS FOR FUZZY GAMES

2006 ◽  
Vol 08 (01) ◽  
pp. 95-109 ◽  
Author(s):  
S. MUTO ◽  
S. ISHIHARA ◽  
E. FUKUDA ◽  
S. H. TIJS ◽  
R. BRÂNZEI
Keyword(s):  
Stable Set ◽  
Stable Sets ◽  
Core Elements ◽  
Fuzzy Games ◽  
Fuzzy Game

Core elements (a la Aubin) of a fuzzy game can be associated with additive separable supporting functions of fuzzy games. Generalized cores whose elements consist of more general separable supporting functions of the game are introduced and studied. While the Aubin core of unanimity games can be empty, the generalized core of unanimity games is nonempty. Properties of the generalized cores and their relations to stable sets are studied. For convex fuzzy games interesting properties are found such as the fact that the generalized core is a unique generalized stable set.

scholarly journals Bifurcating attractors and Galerkin approximates

1987 ◽  
Vol 35 (3) ◽  
pp. 321-347
Author(s):  
R. Wells ◽  
J. A. Dutton
Keyword(s):  
Unstable Manifold ◽  
Linear Subspace ◽  
Original System ◽  
Navier Stokes ◽  
Stable Set ◽  
Asymptotically Stable ◽  
Stable Sets ◽  
Compact Neighborhood ◽  
Navier Stokes Equation ◽  
Finite Dimensional

Let u̇ = A0u + μA1u + J (u) be a Navier-Stokes parameterized evolution equation in a Hilbert space H and let F1 ⊂ F2 ⊂ F3 ⊂ … be an increasing sequence of finite dimensional spaces such that every Fn ⊕ ℝ contains the center-unstable linear subspace Hu ⊕ ℝ ⊂ H ⊕ ℝ of the system u̇ = A0u + μA1u + J (u), u̇ = 0. Then each Fn ⊕ ℝ determines a Galerkin approximant of the original system, with the same center-unstable linear subspace Hu ⊕ ℝ The flow on the center-unstable manifold of the original system may be identified with a parameterized flow on Hu given by x = f∞ (x,μ). The flow on the center-unstable manifold of the Galerkin approximant determined by Fn ⊕ ℝ may be identified with a parameterized flow on Hu given by ẋ = fn (x,μ). It is proved that Theorem I holds: in the Cktopology on a compact neighborhood of the origin in Hu ⊕ ℝ From this theorem it is concluded that Theorem 2 holds: If a certain priori bound holds relating f∞ and fn and an asymptotically stable set A of ẋ = fn (x,μ) near the origin, then ẋ = f∞ (x,μ) has an asymptotically stable set near the origin with the same Borsuk shape as A. Conversely, for each asymptotically stable set near the origin of ẋ = f∞(x,μ), there is one of the same Borsuk shape for ẋ = fn (x,μ) provided n is large enough. Informally, these results amount to the statement that asymptotically stable sets of the Navier-stokes equation, bifurcating from a steady solution, are recovered up to Borsuk shape by those of large enough Galerkin approximants.

scholarly journals The $t$-Stability Number of a Random Graph

10.37236/331 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Ross J. Kang ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Maximum Degree ◽  
Asymptotic Expression ◽  
Main Tool ◽  
Stable Set ◽  
Stable Sets ◽  
Stability Number ◽  
Expected Number ◽  
Maximum Order ◽  
Edge Probability

Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called $t$-stable (or $t$-dependent) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The $t$-stability number $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The theme of this paper is the typical values that this parameter takes on a random graph on $n$ vertices and edge probability equal to $p$. For any fixed $0 < p < 1$ and fixed non-negative integer $t$, we show that, with probability tending to $1$ as $n\to\infty$, the $t$-stability number takes on at most two values which we identify as functions of $t$, $p$ and $n$. The main tool we use is an asymptotic expression for the expected number of $t$-stable sets of order $k$. We derive this expression by performing a precise count of the number of graphs on $k$ vertices that have maximum degree at most $t$.

scholarly journals Linial’s Conjecture for Arc-spine Digraphs

2019 ◽  
Vol 17 (2) ◽  
Author(s):  
Lucas Rigo Yoshimura ◽  
Maycon Sambinelli ◽  
Cândida Nunes da Silva ◽  
Orlando Lee
Keyword(s):  
Positive Integer ◽  
Stable Set ◽  
Stable Sets ◽  
Directed Paths ◽  
Path Partition

A path partition P of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition P of D is defined as Sum (p in P) min{|p_i|, k}. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacentvertices of V(D). Given a positive integer k, we denote by alpha_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that pi_k(D) ≤ alpha_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial’s Conjecture for arc-spine digraphs.

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