Upper and lower bounds of the kernel and nucleolus
type: Item
instance of: scholarly article
title: Upper and lower bounds of the kernel and nucleolus
zbMATH Open document ID: 0648.90094
DOI: 10.1007/BF01770980
author: S. H. Smith
published in: International Journal of Game Theory
publication date: 1986-01-01T00:00:00Z
review text: The author investigates the classical n-person monotonic game (N,v). Let \(m(i)=\min \{v(S\cup \{i\})-v(S):\) \(i\not\in S\), \(S\neq \emptyset \}\); \(z_ i=v(N)-v(N-\{i\})+n^{-1}(v(N)-\sum_{j\in N}(v(N)-v(N-\{\) \(j\})\))); \(L=\{x:\) \(x_ i\geq m(i)\), \(i\in N\}\); \(U=\{x:\) \(x_ i\leq m(i)\), \(i\in N\}.\) Many interesting results are proved about the relationships between the sets L, U, C (core), K (kernel) and N (nucleolus). For instance, it is proved that \(K\subset L\cup int(C)\), i.e., the vector \(\{\) m(i)\(\}\) is a lower bound of the kernel outside the interior of the core; \(U\subset C\); if \(z_ t\leq m(i)\), then \(K\cap U=\{z\}\); if \(z_ t\leq m(i)\), then \(N=z\), i.e., in this case the vector \(\{\) m(i)\(\}\) is an upper bound of the nucleolus.
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 4059156
zbMATH Keywords: n-person monotonic game
zbMATH Keywords: core
zbMATH Keywords: kernel
zbMATH Keywords: nucleolus
zbMATH Keywords: lower bound
zbMATH Keywords: upper bound
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01770980
OpenAlex ID: W2046154452
description: scientific article; zbMATH DE number 4059156