Cores of cooperative games, superdifferentials of functions, and the Minkowski difference of sets
type: Item
instance of: scholarly article
title: Cores of cooperative games, superdifferentials of functions, and the Minkowski difference of sets
zbMATH Open document ID: 0971.91004
author: Vladimir I. Danilov
author: G. A. Koshevoy
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-11-22T00:00:00Z
review text: How can one describe a core \(C(v)\) of a cooperative game \(v\), i.e., the set of all feasible payments which cannot be improved upon by any coalition of players? For a simple game \(v_S\), with \(v_S(S')=1\) if \(S'\supseteq S\) and \(v_S(S')=0\) otherwise, the core is an easily described set. It is known that a general game can be reduced to simple games; namely, every cooperative game with a characteristic function \(v\) can be represented as a linear combination \(v=\sum \lambda_Sv_S\) of simple games \(v_S\). The values \(\lambda_S\) form a Möbius transform of \(v\). It is desirable to use this representation to describe the core \(C(v)\). It is known that if all the coefficients \(\lambda_S\) are non-negative (such games are called monotonic), then the core of \(v\) is the Minkowski sum of cores of the corresponding simple games: \(C(v)=\oplus \lambda_SC(v_S)\). A non-monotonic game can be represented as a difference between two monotonic ones: \(v=v^+-v^-\), where \(v^+\) contains all positive \(v_S\), and \(v^-\) contains all negative ones. The authors prove that \(C(v)=C(v^+)\ominus C(v^-)\), where the Minkowski difference \(A\ominus B\) is defined as \(\{z |z\oplus B\subseteq A\}\).
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 1497285
zbMATH Keywords: cooperative game
zbMATH Keywords: core
zbMATH Keywords: Möbius transform
zbMATH Keywords: monotonic games
zbMATH Keywords: Minkowski sum
reviewed by: Olga Kosheleva
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6756
OpenAlex ID: W2023842680
cites work: Q4708452
cites work: Q5592263
cites work: Q4084463
cites work: Q5599348
cites work: Integral Representation Without Additivity
cites work: Cores of convex games
cites work: Q3910346
DOI: 10.1006/JMAA.2000.6756
description: scientific article; zbMATH DE number 1497285