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Path cooperative games

Authors:
Qizhi Fang at Ocean University of China
Qizhi Fang
Bo Li at Macau University of Science and Technology
Bo Li
Xiaohan Shan
Xiaohan Shan
Xiaoming Sun at Institute of Computing Techonology, Chinese Academy of Sciences
Xiaoming Sun
  • Institute of Computing Techonology, Chinese Academy of Sciences
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Abstract and Figures

Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if they establish a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the core, CS-core, least-core and nucleolus of path cooperative games, which implies all of these solution concepts are polynomial-time solvable for path cooperative games.
The EPC-game with characteristic function defined in Example 1
The EPC-game with characteristic function defined in Example 1
… 
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Transforming a network D into DV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{V}$$\end{document}, where vertex v1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{1}$$\end{document} is transformed into a new edge ev1=(v1′,v1′′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_{v_{1}}=(v'_{1},v''_{1})$$\end{document} and vertex v2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{2}$$\end{document} is transformed into a new edge ev2=(v2′,v2′′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_{v_{2}}=(v'_{2},v''_{2})$$\end{document}
… 
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J Comb Optim (2018) 36:211–229
https://doi.org/10.1007/s10878-018-0296-4
Path cooperative games
Qizhi Fang1·Bo Li2·Xiaohan Shan3·
Xiaoming Sun3
Published online: 3 May 2018
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Cooperative games provide an appropriate framework for fair and stable
profit distribution in multiagent systems. In this paper, we study the algorithmic issues
on path cooperative games that arise from the situations where some commodity flows
through a network. In these games, a coalition of edges or vertices is successful if they
establish a path from the source to the sink in the network, and lose otherwise. Based on
dual theory of linear programming and the relationship with flow games, we provide
the characterizations on the core, CS-core, least-core and nucleolus of path cooperative
games, which implies all of these solution concepts are polynomial-time solvable for
path cooperative games.
An extended abstract of this paper appears at the The 21st Annual International Computing and
Combinatorics Conference (COCOON’15) Fang et al. (2015). The first author is supported by the
National Natural Science Foundation of China (NSFC) (No. 11271341). The fourth author is supported in
part by the National Natural Science Foundation of China Grant 61433014, 61502449, 61602440, the 973
Program of China Grants No. 2016YFB1000201. Finally, we would like to acknowledge our editors and a
superb set of anonymous referees for their excellent suggestions.
BXiaohan Shan
shanxiaohan@ict.ac.cn
Qizhi Fang
qfang@ouc.edu.cn
Bo Li
boli2@cs.stonybrook.edu
Xiaoming Sun
sunxiaoming@ict.ac.cn
1School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
2Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA
3CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology,
University of Chinese Academy of Sciences, Beijing 10010, China
123
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Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if it enables a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the CS-core, least-core and nucleolus of path cooperative games. Furthermore, we show that the least-core and nucleolus are polynomially solvable for path cooperative games defined on both directed and undirected network.
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