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On computational complexity of membership test in flow games and linear production games

Author

Listed:
  • Shanfeng Zhu

    (Department of Computer Science, City University of Hong Kong, Hong Kong, P. R. China)

  • Xiaotie Deng

    (Department of Computer Science, City University of Hong Kong, Hong Kong, P. R. China)

  • Maocheng Cai

    (Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China)

  • Qizhi Fang

    (Department of Mathematics, Ocean University of Qingdao, Qingdao 266003, P. R. China)

Abstract

Let \equiv(N,v) be a cooperative game with the player set N and characteristic function v: 2N ->R. An imputation of the game is in the core if no subset of players could gain advantage by splitting from the grand coalition of all players. It is well known that, for the flow game (and equivalently, for the linear production game), the core is always non-empty and a solution in the core can be found in polynomial time. In this paper, we show that, given an imputation x, it is NP-complete to decide x is not a member of the core, for the flow game. And because of the specific reduction we constructed, the result also holds for the linear production game.

Suggested Citation

  • Shanfeng Zhu & Xiaotie Deng & Maocheng Cai & Qizhi Fang, 2002. "On computational complexity of membership test in flow games and linear production games," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 39-45.
    • Handle: RePEc:spr:jogath:v:31:y:2002:i:1:p:39-45
    Note: Received: October 2000/Final version: March 2002
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    Citations

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    Cited by:

    1. Qizhi Fang & Bo Li & Xiaohan Shan & Xiaoming Sun, 2018. "Path cooperative games," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 211-229, July.
    2. Thomas Demuynck, 2014. "The computational complexity of rationalizing Pareto optimal choice behavior," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(3), pages 529-549, March.
    3. Potters, Jos & Reijnierse, Hans & Biswas, Amit, 2006. "The nucleolus of balanced simple flow networks," Games and Economic Behavior, Elsevier, vol. 54(1), pages 205-225, January.
    4. Kumabe, Masahiro & Mihara, H. Reiju, 2008. "Computability of simple games: A characterization and application to the core," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 348-366, February.
    5. Fabrice Talla Nobibon & Laurens Cherchye & Yves Crama & Thomas Demuynck & Bram De Rock & Frits C. R. Spieksma, 2016. "Revealed Preference Tests of Collectively Rational Consumption Behavior: Formulations and Algorithms," Operations Research, INFORMS, vol. 64(6), pages 1197-1216, December.
    6. Mohammaditabar, Davood & Ghodsypour, Seyed Hassan & Hafezalkotob, Ashkan, 2016. "A game theoretic analysis in capacity-constrained supplier-selection and cooperation by considering the total supply chain inventory costs," International Journal of Production Economics, Elsevier, vol. 181(PA), pages 87-97.
    7. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Computability of simple games: A complete investigation of the sixty-four possibilities," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 150-158, March.
    8. Drechsel, J. & Kimms, A., 2010. "Computing core allocations in cooperative games with an application to cooperative procurement," International Journal of Production Economics, Elsevier, vol. 128(1), pages 310-321, November.
    9. Nicholas G. Hall & Zhixin Liu, 2010. "Capacity Allocation and Scheduling in Supply Chains," Operations Research, INFORMS, vol. 58(6), pages 1711-1725, December.
    10. Luis A. Guardiola & Ana Meca & Justo Puerto, 2021. "Enforcing fair cooperation in production-inventory settings with heterogeneous agents," Annals of Operations Research, Springer, vol. 305(1), pages 59-80, October.
    11. Xiaotie Deng & Qizhi Fang & Xiaoxun Sun, 2009. "Finding nucleolus of flow game," Journal of Combinatorial Optimization, Springer, vol. 18(1), pages 64-86, July.
    12. Luis Guardiola & Ana Meca & Justo Puerto, 2020. "Quid Pro Quo allocations in Production-Inventory games," Papers 2002.00953, arXiv.org.
    13. Demuynck, Thomas, 2011. "The computational complexity of rationalizing boundedly rational choice behavior," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 425-433.
    14. Walter Kern & Daniël Paulusma, 2009. "On the Core and f -Nucleolus of Flow Games," Mathematics of Operations Research, INFORMS, vol. 34(4), pages 981-991, November.
    15. Vijay V. Vazirani, 2023. "LP-Duality Theory and the Cores of Games," Papers 2302.07627, arXiv.org, revised Mar 2023.

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