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On Houseswapping, the Strict Core, Segmentation, and Linear Programming

Author

Listed:
  • Thomas Quint

    (University of Nevada, Reno)

  • Jun Wake

    (Gakashuin University)

Abstract

We consider the n-player houseswapping game of Shapley-Scarf (1974), with indifferences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multivalued. We define a condition on such games called "segmentability", which means that the set of players can be partitioned into a "top trading segmentation." It generalizes GaleÌs well-known idea of the partition of players into "top trading cycles" (which is used to find the unique strict core allocation in the model with no indifference). We prove that a game has a nonempty strict core if and only if it is segmentable. We then use this result to devise an O(n3) algorithm which takes as input any houseswapping game, and returns either a strict core allocation or a report that the strict core is empty. Finally, we are also able to construct a linear inequality system whose feasible regionÌs extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothblum (1991) for the marriage game of Gale and Shapley (1962).

Suggested Citation

  • Thomas Quint & Jun Wake, 2003. "On Houseswapping, the Strict Core, Segmentation, and Linear Programming," Cowles Foundation Discussion Papers 1416, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1416
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    References listed on IDEAS

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    1. Shapley, Lloyd & Scarf, Herbert, 1974. "On cores and indivisibility," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 23-37, March.
    2. Roth, Alvin E. & Postlewaite, Andrew, 1977. "Weak versus strong domination in a market with indivisible goods," Journal of Mathematical Economics, Elsevier, vol. 4(2), pages 131-137, August.
    3. Alvin E. Roth & Uriel G. Rothblum & John H. Vande Vate, 1993. "Stable Matchings, Optimal Assignments, and Linear Programming," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 803-828, November.
    4. Wako, Jun, 1984. "A note on the strong core of a market with indivisible goods," Journal of Mathematical Economics, Elsevier, vol. 13(2), pages 189-194, October.
    5. Ma, Jinpeng, 1994. "Strategy-Proofness and the Strict Core in a Market with Indivisibilities," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(1), pages 75-83.
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    JEL classification:

    • 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
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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