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


  • Thomas Quint

    (University of Nevada, Reno)

  • Jun Wako

    (Gakushuin University)


We consider the n-player houseswapping game of Shapley-Scarf (1974), with indfferences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multi-valued. 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 Gale's 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 and O(n^3) 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 region's extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothbum (1991) for the marriage game of Gale and Shapley (1962).

Suggested Citation

  • Thomas Quint & Jun Wako, 2004. "On Houseswapping, the Strict Core, Segmentation, and Linear Programming," Yale School of Management Working Papers ysm373, Yale School of Management.
  • Handle: RePEc:ysm:somwrk:ysm373

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    Cited by:

    1. Alvin E. Roth & Tayfun Sönmez & M. Utku Ünver, 2005. "Efficient Kidney Exchange: Coincidence of Wants in a Structured Market," Boston College Working Papers in Economics 621, Boston College Department of Economics.
    2. Subiza Begoña & Peris Josep E., 2014. "A Solution for General Exchange Markets with Indivisible Goods when Indifferences are Allowed," Mathematical Economics Letters, De Gruyter, vol. 2(3-4), pages 1-5, November.
    3. Alcalde-Unzu, Jorge & Molis, Elena, 2011. "Exchange of indivisible goods and indifferences: The Top Trading Absorbing Sets mechanisms," Games and Economic Behavior, Elsevier, vol. 73(1), pages 1-16, September.
    4. Jaramillo, Paula & Manjunath, Vikram, 2012. "The difference indifference makes in strategy-proof allocation of objects," Journal of Economic Theory, Elsevier, vol. 147(5), pages 1913-1946.
    5. Péter Biró & Flip Klijn & Xenia Klimentova & Ana Viana, 2021. "Shapley-Scarf Housing Markets: Respecting Improvement, Integer Programming, and Kidney Exchange," Working Papers 1235, Barcelona Graduate School of Economics.
    6. Han, Weibin & Van Deemen, Adrian, 2016. "On the solution of w-stable sets," Mathematical Social Sciences, Elsevier, vol. 84(C), pages 87-92.
    7. Bando, Keisuke, 2014. "On the existence of a strictly strong Nash equilibrium under the student-optimal deferred acceptance algorithm," Games and Economic Behavior, Elsevier, vol. 87(C), pages 269-287.

    More about this item


    Shapley-Scarf Economy; Strict Core; Linear Inequality System; Extreme Points;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games


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