The bounded core for games with precedence constraints
type: Item
instance of: scholarly article
title: The bounded core for games with precedence constraints
zbMATH Open document ID: 1272.91023
author: Michel Grabisch
author: Peter Sudhölter
published in: Annals of Operations Research
publication date: 2013-04-02T00:00:00Z
review text: In this paper, the authors consider cooperative games in which there are restrictions on cooperation. These restrictions are modeled by a hierarchical structure on the set of players (cf. [\textit{U. Faigle} and \textit{W. Kern}, Int. J. Game Theory 21, No. 3, 249--266 (1992; Zbl 0779.90078)]), who are motivated by the existence of a partial order in the sense that if a coalition can be formed and a player belongs to a coalition, then, all players preceding him in the partial-order relation are also in the coalition. The classic model of cooperative games as a special case appears when all players are incomparable. It is a known fact that the core of cooperative games with transferable utility and with precedence constraints can be unbounded. Therefore, they appear in the literature various attempts to define a subset of the core that is bounded and having good properties. Although this core, which is a convex polyhedral set, can have unbounded faces, this paper considers the so-called bounded core that is defined as the union of all bounded faces. The bounded core can be interpreted as the set of core elements such that every player gets the most out of their subordinates. It coincides with the core in classic games. The paper shows that the bounded core is nonempty when the core is nonempty. To do this, they formulate a generalization of the Bondareva-Shapley theorem (cf. [\textit{O. N. Bondareva}, Probl. Kibern. 10, 119--139 (1963; Zbl 1013.91501)]) for games with precedence constraints given in [\textit{U. Faigle}, Z. Oper. Res. 33, No. 6, 405--422 (1989; Zbl 0685.90103)]. Furthermore, the bounded core is a continuous correspondence on games with coinciding precedence constraints. They generalize some well-known properties of solutions on a set of classic games: Pareto optimality, covariance under strategic equivalence, anonymity, boundedness, boundedness w.r.t. singletons, and the two-person zero-inessential game property. Furthermore, they generalize several p…
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 6149328
zbMATH Keywords: TU games
zbMATH Keywords: bounded core
zbMATH Keywords: restricted cooperation
reviewed by: Balbina Casas-Méndez
MaRDI profile type: Publication
full work available at URL: s10479-012-1228-9
OpenAlex ID: W2106730071
cites work: Q4708452
cites work: Hierarchical organization structures and constraints on coalition formation
cites work: Cores of games with restricted cooperation
cites work: The Shapley value for cooperative games under precedence constraints
cites work: Games with permission structures: The conjunctive approach
cites work: Ensuring the boundedness of the core of games with restricted cooperation
cites work: Axiomatizations of the core on the universal domain and other natural domains
cites work: The kernel and bargaining set for convex games
cites work: On the reduced game property and its converse
cites work: Introduction to the Theory of Cooperative Games
cites work: Convex Analysis
DOI: 10.1007/S10479-012-1228-9
description: scientific article; zbMATH DE number 6149328