On balanced games and games with committee control
type: Item
instance of: scholarly article
title: On balanced games and games with committee control
zbMATH Open document ID: 0678.90102
DOI: 10.1007/BF01746002
published in: OR Spektrum
publication date: 1989-01-01T00:00:00Z
review text: The authors introduce the notion of a flow game with committee control as a generalization of the notion of the well-known flow game with ownership. Let N be a finite player set whose subsets are called coalitions and let G be a directed network. The usual assumption that each arc in the network is owned by a player is now generalized by assuming that each arc is controlled by committees consisting of coalitions. For every coalition \(S\subset N\), let \(G_ s\) be the network obtained form G by keeping all the vertices but removing all arcs which are not controlled by S. The flow game v with committee control is generated by the network G in such a way that the worth v(S) of coalition S represents the value of a maximum flow in the network \(G_ s\). Indeed, the class of flow games with committee control includes the class of flow games with ownership because the ownership of an arc by a certain player may be interpreted as the control of the arcs by those coalitions containing the owner of the arc. Formally, the committee control is described by means of simple games. To be exact, with each arc \(\ell\) is associated a simple game \(w_{\ell}\) and a coalition S is said to control the arc \(\ell\) iff \(w_{\ell}(S)=1\). Player \(i\in N\) is called a veto player in the simple game \(w_{\ell}\) if \(w_{\ell}(T)=1\) implies \(i\in T\). A first main theorem provides the balancedness of flow games with committee control whenever all the controlling simple games have veto players. A second main theorem states that every nonnegative balanced game can be regarded as a so-called veto rich flow game, i.e., a flow game with committee control where all arcs are controlled by simple games with veto players. In summary, the theorems together state that the class of nonnegative balanced games is characterized as the class of veto rich flow games. The paper is concluded with two results concerning the (non)balancedness of linear production games with committee control.
Mathematics Subject Classification ID: 91A12
Mathematics Subject Classification ID: 91A40
Mathematics Subject Classification ID: 91B14
zbMATH DE Number: 4112426
zbMATH Keywords: flow game
zbMATH Keywords: committee control
zbMATH Keywords: coalitions
zbMATH Keywords: directed network
zbMATH Keywords: ownership
zbMATH Keywords: simple games
zbMATH Keywords: veto player
zbMATH Keywords: balancedness
zbMATH Keywords: linear production games
Wikidata QID: Q59868887
author: Imma J. Curiel
author: Jean Derks
author: Stef Tijs
reviewed by: Theo S. H. Driessen
MaRDI profile type: Publication
cites work: Q3801377
cites work: Totally balanced games arising from controlled programming problems
cites work: Totally Balanced Games and Games of Flow
cites work: Generalized Network Problems Yielding Totally Balanced Games
cites work: On the core of linear production games
full work available at URL: https://doi.org/10.1007/bf01746002
OpenAlex ID: W2034333234
description: scientific article; zbMATH DE number 4112426