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Paths to stable allocations

Author

Listed:
  • Ágnes Cseh

    (Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences)

  • Martin Skutella

    (TU Berlin, Institut für Mathematik)

Abstract

The stable allocation problem is one of the broadest extensions of the well-known stable marriage problem. In an allocation problem, edges of a bipartite graph have capacities and vertices have quotas to fill. Here we investigate the case of uncoordinated processes in stable allocation instances. In this setting, a feasible allocation is given and the aim is to reach a stable allocation by raising the value of the allocation along blocking edges and reducing it on worse edges if needed. Do such myopic changes lead to a stable solution? In our present work, we analyze both better and best response dynamics from an algorithmic point of view. With the help of two deterministic algorithms we show that random procedures reach a stable solution with probability one for all rational input data in both cases. Surprisingly, while there is a polynomial path to stability when better response strategies are played (even for irrational input data), the more intuitive best response steps may require exponential time. We also study the special case of correlated markets. There, random best response strategies lead to a stable allocation in expected polynomial time.

Suggested Citation

  • Ágnes Cseh & Martin Skutella, 2019. "Paths to stable allocations," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 835-862, September.
    • Handle: RePEc:spr:jogath:v:48:y:2019:i:3:d:10.1007_s00182-019-00664-6
    DOI: 10.1007/s00182-019-00664-6
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    References listed on IDEAS

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    1. Klaus, Bettina & Klijn, Flip, 2007. "Paths to stability for matching markets with couples," Games and Economic Behavior, Elsevier, vol. 58(1), pages 154-171, January.
    2. Chen, Bo & Fujishige, Satoru & Yang, Zaifu, 2016. "Random decentralized market processes for stable job matchings with competitive salaries," Journal of Economic Theory, Elsevier, vol. 165(C), pages 25-36.
    3. Diamantoudi, Effrosyni & Miyagawa, Eiichi & Xue, Licun, 2004. "Random paths to stability in the roommate problem," Games and Economic Behavior, Elsevier, vol. 48(1), pages 18-28, July.
    4. Blum, Yosef & Roth, Alvin E. & Rothblum, Uriel G., 1997. "Vacancy Chains and Equilibration in Senior-Level Labor Markets," Journal of Economic Theory, Elsevier, vol. 76(2), pages 362-411, October.
    5. Bo Chen & Satoru Fujishige & Zaifu Yang, 2010. "Decentralized Market Processes to Stable Job Matchings with Competitive Salaries," KIER Working Papers 749, Kyoto University, Institute of Economic Research.
    6. Blum, Yosef & Rothblum, Uriel G., 2002. ""Timing Is Everything" and Marital Bliss," Journal of Economic Theory, Elsevier, vol. 103(2), pages 429-443, April.
    7. Roth, Alvin E & Vande Vate, John H, 1990. "Random Paths to Stability in Two-Sided Matching," Econometrica, Econometric Society, vol. 58(6), pages 1475-1480, November.
    8. Beatriz Millán & Eliana Pepa Risma, 2018. "Random path to stability in a decentralized market with contracts," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(1), pages 79-103, June.
    9. Mourad Baïou & Michel Balinski, 2002. "The Stable Allocation (or Ordinal Transportation) Problem," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 485-503, August.
    10. Mourad Baïou & Michel Balinski, 2002. "Erratum: The Stable Allocation (or Ordinal Transportation) Problem," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 662-680, November.
    11. Fuhito Kojima & M. Ünver, 2008. "Random paths to pairwise stability in many-to-many matching problems: a study on market equilibration," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(3), pages 473-488, March.
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    2. Joseph E. Duggan, 2020. "Subjective Homophily and the Fixtures Problem," Games, MDPI, vol. 11(1), pages 1-13, February.

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