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Analytic solution for the nucleolus of a three‐player cooperative game

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  • Mingming Leng
  • Mahmut Parlar

Abstract

The nucleolus solution for cooperative games in characteristic function form is usually computed numerically by solving a sequence of linear programing (LP) problems, or by solving a single, but very large‐scale, LP problem. This article proposes an algebraic method to compute the nucleolus solution analytically (i.e., in closed‐form) for a three‐player cooperative game in characteristic function form. We first consider cooperative games with empty core and derive a formula to compute the nucleolus solution. Next, we examine cooperative games with nonempty core and calculate the nucleolus solution analytically for five possible cases arising from the relationship among the value functions of different coalitions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

Suggested Citation

  • Mingming Leng & Mahmut Parlar, 2010. "Analytic solution for the nucleolus of a three‐player cooperative game," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(7), pages 667-672, October.
    • Handle: RePEc:wly:navres:v:57:y:2010:i:7:p:667-672
    DOI: 10.1002/nav.20429
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    1. S. H. Gow & L. C. Thomas, 1998. "Interchange fees for bank ATM networks," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(4), pages 407-417, June.
    2. Solymosi, Tamas & Raghavan, Tirukkannamangai E S, 1994. "An Algorithm for Finding the Nucleolus of Asignment Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(2), pages 119-143.
    3. M. Maschler & B. Peleg & L. S. Shapley, 1979. "Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 303-338, November.
    4. Fromen, Bastian, 1997. "Reducing the number of linear programs needed for solving the nucleolus problem of n-person game theory," European Journal of Operational Research, Elsevier, vol. 98(3), pages 626-636, May.
    5. Alois Behringer, Fred, 1981. "A simplex based algorithm for the lexicographically extended linear maxmin problem," European Journal of Operational Research, Elsevier, vol. 7(3), pages 274-283, July.
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    Cited by:

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    2. Luo, Chunlin & Zhou, Xiaoyang & Lev, Benjamin, 2022. "Core, shapley value, nucleolus and nash bargaining solution: A Survey of recent developments and applications in operations management," Omega, Elsevier, vol. 110(C).

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