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Sharing Supermodular Costs

Author

Listed:
  • Andreas S. Schulz

    (Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Nelson A. Uhan

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907)

Abstract

We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations; in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron---a problem that often arises in combinatorial optimization---has supermodular optimal costs. In addition, we examine the computational complexity of the least core and least core value of supermodular cost cooperative games. We show that the problem of computing the least core value of these games is strongly NP-hard and, in fact, is inapproximable within a factor strictly less than 17/16 unless P = NP. For a particular class of supermodular cost cooperative games that arises from a scheduling problem, we show that the Shapley value---which, in this case, is computable in polynomial time---is in the least core, while computing the least core value is NP-hard.

Suggested Citation

  • Andreas S. Schulz & Nelson A. Uhan, 2010. "Sharing Supermodular Costs," Operations Research, INFORMS, vol. 58(4-part-2), pages 1051-1056, August.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:4-part-2:p:1051-1056
    DOI: 10.1287/opre.1100.0841
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    References listed on IDEAS

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    Cited by:

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    2. Michela Chessa & Nobuyuki Hanaki & Aymeric Lardon & Takashi Yamada, 2022. "An experiment on the Nash program: Comparing two strategic mechanisms implementing the Shapley value," ISER Discussion Paper 1175, Institute of Social and Economic Research, Osaka University.
    3. Chessa, Michela & Hanaki, Nobuyuki & Lardon, Aymeric & Yamada, Takashi, 2023. "An experiment on the Nash program: A comparison of two strategic mechanisms implementing the Shapley value," Games and Economic Behavior, Elsevier, vol. 141(C), pages 88-104.
    4. Simai He & Jiawei Zhang & Shuzhong Zhang, 2012. "Polymatroid Optimization, Submodularity, and Joint Replenishment Games," Operations Research, INFORMS, vol. 60(1), pages 128-137, February.
    5. Kenneth Judd & Garrett van Ryzin, 2010. "Preface to the Special Issue on Computational Economics," Operations Research, INFORMS, vol. 58(4-part-2), pages 1035-1036, August.
    6. Lindong Liu & Xiangtong Qi & Zhou Xu, 2016. "Computing Near-Optimal Stable Cost Allocations for Cooperative Games by Lagrangian Relaxation," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 687-702, November.
    7. Uhan, Nelson A., 2015. "Stochastic linear programming games with concave preferences," European Journal of Operational Research, Elsevier, vol. 243(2), pages 637-646.
    8. Nguyen, Tri-Dung & Thomas, Lyn, 2016. "Finding the nucleoli of large cooperative games," European Journal of Operational Research, Elsevier, vol. 248(3), pages 1078-1092.
    9. Lindong Liu & Yuqian Zhou & Zikang Li, 2022. "Lagrangian heuristic for simultaneous subsidization and penalization: implementations on rooted travelling salesman games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 81-99, February.

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