Axiomatizations of the Shapley value for cooperative games on antimatroids
type: Item
instance of: scholarly article
title: Axiomatizations of the Shapley value for cooperative games on antimatroids
zbMATH Open document ID: 1060.91006
author: René van den Brink
author: Encarnación Algaba
author: Jésus Mario Bilbao
author: A. Jiménez-Losada
published in: Mathematical Methods of Operations Research
publication date: 2003-06-26T00:00:00Z
review text: The paper provides an axiomatization of the Shapley value in cooperative games in which asymmetries in allowable coalitions is described by restricting the set of possible coalitions to be {antimatroids}. Asymmetries among the players of a cooperative game have been studied in the literature in different ways. In one class of models, the possibilities of coalition formation are determined by the positions of the players in a communication graph. Another type of asymmetry is where the possible coalitions are determined by the positions of players in a hierarchical permission structure. Yet another type of asymmetry is introduced in peer group games in which there are some assumed dependencies among players which is described by a rooted tree. Games on antimatroids includes several such asymmetries as special cases. The paper provides a generalization of axioms given for the permission structures to obtain an axiomatization of the restricted Shapley value for cooperative games on antimatroids. Some details follow. Let \(( N,v) \) denote a cooperative game with \(N=\{1,\dots,n\}\) denoting a finite set of players and \(v:2^{N}\to R\) denoting a characteristic function on \(N.\) An antimatroid \(\mathcal{A}\) on \(N\) is a subset of \(2^{N}\) satisfying A1. \(\emptyset \in \mathcal{A}\), A2. (accessibility) If \(E\in \mathcal{A}\), \(E\neq \emptyset \) then there exists \(i\in E\) \(E\backslash \{i\}\in \mathcal{A}\), A3. (closed under union) If \(E,F\in \mathcal{A}\) then \(E\cup F\in \mathcal{A}\). The paper confines attention to the case of normality, i.e., A.4. For every \(i\in N\) there exists an \(E\in \mathcal{A}\) s.t. \(i\in E.\) The interior operator \(\text{int}_{\mathcal{A}}: 2^{N}\to \mathcal{A}\) is defined by \(\text{int}_{\mathcal{ A}}( E) =\bigcup _{F\subset E,F\in \mathcal{A}}F\in \mathcal{A}\) for all \(E\subset N\). The restricted game \(v_{\mathcal{A}}\) on \(\mathcal{A}\) is defined by \(v_{\mathcal{A}}( E) =v( \text{int}_{\mathcal{A} }( E) )\). Th…
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 1940552
zbMATH Keywords: cooperative game
zbMATH Keywords: permission structure
MaRDI profile type: Publication
full work available at URL: s001860200250
OpenAlex ID: W1525213644
DOI: 10.1007/S001860200250
description: scientific article; zbMATH DE number 1940552