The least core, nucleolus, and kernel of homogeneous weighted majority games
type: Item
instance of: scholarly article
title: The least core, nucleolus, and kernel of homogeneous weighted majority games
zbMATH Open document ID: 0765.90096
DOI: 10.1016/0899-8256(92)90039-U
published in: Games and Economic Behavior
publication date: 1993-01-12T00:00:00Z
review text: A (generalized) weighted majority game on a set of players \(N=\{1,\dots,n\}\) is defined as follows: for every coalition \(S\subseteq N\), its weight \(M(S)\) is defined, and a minimal weight \(\lambda\) is given so that a coalition \(S\) is winning iff \(M(S)\geq \lambda\). For example, in simple voting, \(M(S)=| S|\), and \(\lambda=\lfloor n/2\rfloor+1\). In general, \(M(S)\) is not necessarily additive. Such a game is called homogeneous if, whenever \(M(S)>\lambda\), there exists a \(T\subset S\) such that \(M(T)=\lambda\). For some games, a simpler function \(M'\) leads to exactly the same set of winning coalitions. It is known that for every such game, there exists a representation in which the values \(M(S)\) are all integers, and among these, there exists a unique representation (called the minimal representation) for which \(M(N)\) is minimal. For constant-sum superadditive games, Peleg gave a complete description of the kernel and the nucleolus. These results are generalized to arbitrary homogeneous weighted majority games ``without steps'' (crudely speaking, a step is a player that cannot be substituted by several ``weaker'' ones). The least core, kernel, and nucleolus turn out to be directly related to the minimal representation.
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 85736
zbMATH Keywords: weighted majority game
zbMATH Keywords: winning coalitions
zbMATH Keywords: minimal representation
zbMATH Keywords: least core
zbMATH Keywords: kernel
zbMATH Keywords: nucleolus
author: Bezalel Peleg
author: Joachim Rosenmüller
reviewed by: Olga Kosheleva
MaRDI profile type: Publication
cites work: A CLASS OF MAJORITY GAMES
cites work: A class of simple games
cites work: On the Enumeration of Majority Games
cites work: On the Nucleolus of a Characteristic Function Game
cites work: Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts
cites work: On the minimal representation of homogeneous games
cites work: On the kernel of constant-sum simple games with homogeneous weights
cites work: On Weights of Constant-Sum Majority Games
cites work: Homogeneous Games: Recursive Structure and Computation
cites work: The rôle of nondegeneracy and homogeneity in n-person game theory: An equivalence theorem
cites work: Homogeneous games with countably many players
cites work: Extreme convex set functions with finite carrier: General theory
cites work: Homogeneous games as anti step functions
cites work: Q5844986
full work available at URL: 0899-8256(92)90039-u
OpenAlex ID: W2068980672
description: scientific article; zbMATH DE number 85736