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Reforming an Envy-Free Matching

Author

Listed:
  • Takehiro Ito
  • Yuni Iwamasa
  • Naonori Kakimura
  • Naoyuki Kamiyama
  • Yusuke Kobayashi
  • Yuta Nozaki
  • Yoshio Okamoto
  • Kenta Ozeki

Abstract

We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.

Suggested Citation

  • Takehiro Ito & Yuni Iwamasa & Naonori Kakimura & Naoyuki Kamiyama & Yusuke Kobayashi & Yuta Nozaki & Yoshio Okamoto & Kenta Ozeki, 2022. "Reforming an Envy-Free Matching," Papers 2207.02641, arXiv.org.
    • Handle: RePEc:arx:papers:2207.02641
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    References listed on IDEAS

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    1. Wu, Qingyun & Roth, Alvin E., 2018. "The lattice of envy-free matchings," Games and Economic Behavior, Elsevier, vol. 109(C), pages 201-211.
    2. Gan, Jiarui & Suksompong, Warut & Voudouris, Alexandros A., 2019. "Envy-freeness in house allocation problems," Mathematical Social Sciences, Elsevier, vol. 101(C), pages 104-106.
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