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Almost Envy-Free Allocations with Connected Bundles

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Listed:
  • Vittorio Bil`o
  • Ioannis Caragiannis
  • Michele Flammini
  • Ayumi Igarashi
  • Gianpiero Monaco
  • Dominik Peters
  • Cosimo Vinci
  • William S. Zwicker

Abstract

We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph. If the graph is a path and the utility functions are monotonic over bundles, we show the existence of EF1 allocations for at most four agents, and the existence of EF2 allocations for any number of agents; our proofs involve discrete analogues of the Stromquist's moving-knife protocol and the Su--Simmons argument based on Sperner's lemma. For identical utilities, we provide a polynomial-time algorithm that computes an EF1 allocation for any number of agents. For the case of two agents, we characterize the class of graphs that guarantee the existence of EF1 allocations as those whose biconnected components are arranged in a path; this property can be checked in linear time.

Suggested Citation

  • Vittorio Bil`o & Ioannis Caragiannis & Michele Flammini & Ayumi Igarashi & Gianpiero Monaco & Dominik Peters & Cosimo Vinci & William S. Zwicker, 2018. "Almost Envy-Free Allocations with Connected Bundles," Papers 1808.09406, arXiv.org, revised May 2022.
    • Handle: RePEc:arx:papers:1808.09406
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    References listed on IDEAS

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    1. Anna Bogomolnaia & Hervé Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2017. "Competitive Division of a Mixed Manna," Econometrica, Econometric Society, vol. 85(6), pages 1847-1871, November.
    2. Anna Bogomolnaia & Herve Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2016. "Dividing Goods and Bads Under Additive Utilities," HSE Working papers WP BRP 153/EC/2016, National Research University Higher School of Economics.
    3. Xiaotie Deng & Qi Qi & Amin Saberi, 2012. "Algorithmic Solutions for Envy-Free Cake Cutting," Operations Research, INFORMS, vol. 60(6), pages 1461-1476, December.
    4. Anna Bogomolnaia & Herve Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2016. "Dividing Goods or Bads Under Additive Utilities," HSE Working papers WP BRP 147/EC/2016, National Research University Higher School of Economics.
    5. Barbanel, Julius B. & Brams, Steven J., 2004. "Cake division with minimal cuts: envy-free procedures for three persons, four persons, and beyond," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 251-269, November.
    6. Eric Budish, 2011. "The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes," Journal of Political Economy, University of Chicago Press, vol. 119(6), pages 1061-1103.
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    Cited by:

    1. Steven J. Brams & D. Marc Kilgour & Christian Klamler, 2022. "Two-Person Fair Division of Indivisible Items when Envy-Freeness is Impossible," SN Operations Research Forum, Springer, vol. 3(2), pages 1-23, June.

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