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The stable fixtures problem with payments

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  • Biró, Péter
  • Kern, Walter
  • Paulusma, Daniël
  • Wojuteczky, Péter

Abstract

We consider multiple partners matching games (G,b,w), where G is a graph with an integer vertex capacity function b and an edge weighting w. If G is bipartite, these games are called multiple partners assignment games. We give a polynomial-time algorithm that either finds that a given multiple partners matching game has no stable solution, or obtains a stable solution. We characterize the set of stable solutions of a multiple partners matching game in two different ways and show how this leads to simple proofs for a number of results of Sotomayor (1992, 1999, 2007) for multiple partners assignment games and to generalizations of some of these results to multiple partners matching games. We also perform a study on the core of multiple partners matching games. We prove that the problem of deciding if an allocation belongs to the core jumps from being polynomial-time solvable for b≤2 to NP-complete for b≡3.

Suggested Citation

  • Biró, Péter & Kern, Walter & Paulusma, Daniël & Wojuteczky, Péter, 2018. "The stable fixtures problem with payments," Games and Economic Behavior, Elsevier, vol. 108(C), pages 245-268.
    • Handle: RePEc:eee:gamebe:v:108:y:2018:i:c:p:245-268
    DOI: 10.1016/j.geb.2017.02.002
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    1. Peter Biro & Walter Kern & Daniel Paulusma & Peter Wojuteczky, 2015. "The Stable Fixtures Problem with Payments," CERS-IE WORKING PAPERS 1545, Institute of Economics, Centre for Economic and Regional Studies.
    2. Johan Karlander & Kimmo Eriksson, 2001. "Stable outcomes of the roommate game with transferable utility," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 555-569.
    3. Paul Milgrom, 2000. "Putting Auction Theory to Work: The Simultaneous Ascending Auction," Journal of Political Economy, University of Chicago Press, vol. 108(2), pages 245-272, April.
    4. Sotomayor, Marilda, 2007. "Connecting the cooperative and competitive structures of the multiple-partners assignment game," Journal of Economic Theory, Elsevier, vol. 134(1), pages 155-174, May.
    5. Marilda Sotomayor, 1992. "The Multiple Partners Game," Palgrave Macmillan Books, in: Mukul Majumdar (ed.), Equilibrium and Dynamics, chapter 17, pages 322-354, Palgrave Macmillan.
    6. Toda, Manabu, 2005. "Axiomatization of the core of assignment games," Games and Economic Behavior, Elsevier, vol. 53(2), pages 248-261, November.
    7. Marilda Sotomayor, 1999. "The lattice structure of the set of stable outcomes of the multiple partners assignment game," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 567-583.
    8. Walter Kern & Daniël Paulusma, 2003. "Matching Games: The Least Core and the Nucleolus," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 294-308, May.
    9. Péter Biró & Walter Kern & Daniël Paulusma, 2012. "Computing solutions for matching games," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(1), pages 75-90, February.
    10. Sasaki, Hiroo, 1995. "Consistency and Monotonicity in Assignment Problems," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(4), pages 373-397.
    11. T. E. S. Raghavan & Tamás Solymosi, 2001. "Assignment games with stable core," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(2), pages 177-185.
    12. Nimrod Megiddo, 1979. "Combinatorial Optimization with Rational Objective Functions," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 414-424, November.
    13. Xiaotie Deng & Toshihide Ibaraki & Hiroshi Nagamochi, 1999. "Algorithmic Aspects of the Core of Combinatorial Optimization Games," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 751-766, August.
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    Cited by:

    1. Han Xiao & Tianhang Lu & Qizhi Fang, 2021. "Approximate Core Allocations for Multiple Partners Matching Games," Papers 2107.01442, arXiv.org, revised Oct 2021.

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    More about this item

    Keywords

    Stable solutions; Cooperative game; Core;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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