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DOI: 10.4230/LIPIcs.ITCS.2022.41

On the Existence of Competitive Equilibrium with Chores

Authors: Bhaskar Ray Chaudhury, Jugal Garg, Peter McGlaughlin, and Ruta Mehta

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

We study the chore division problem in the classic Arrow-Debreu exchange setting, where a set of agents want to divide their divisible chores (bads) to minimize their disutilities (costs). We assume that agents have linear disutility functions. Like the setting with goods, a division based on competitive equilibrium is regarded as one of the best mechanisms for bads. Equilibrium existence for goods has been extensively studied, resulting in a simple, polynomial-time verifiable, necessary and sufficient condition. However, dividing bads has not received a similar extensive study even though it is as relevant as dividing goods in day-to-day life. In this paper, we show that the problem of checking whether an equilibrium exists in chore division is NP-complete, which is in sharp contrast to the case of goods. Further, we derive a simple, polynomial-time verifiable, sufficient condition for existence. Our fixed-point formulation to show existence makes novel use of both Kakutani and Brouwer fixed-point theorems, the latter nested inside the former, to avoid the undefined demand issue specific to bads.

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Bhaskar Ray Chaudhury, Jugal Garg, Peter McGlaughlin, and Ruta Mehta. On the Existence of Competitive Equilibrium with Chores. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chaudhury_et_al:LIPIcs.ITCS.2022.41,
  author =	{Chaudhury, Bhaskar Ray and Garg, Jugal and McGlaughlin, Peter and Mehta, Ruta},
  title =	{{On the Existence of Competitive Equilibrium with Chores}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{41:1--41:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.41},
  URN =		{urn:nbn:de:0030-drops-156378},
  doi =		{10.4230/LIPIcs.ITCS.2022.41},
  annote =	{Keywords: Fair Division, Competitive Equilibrium, Fixed Point Theorems}
}

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DOI: 10.4230/OASIcs.SOSA.2019.20

Approximating Maximin Share Allocations

Authors: Jugal Garg, Peter McGlaughlin, and Setareh Taki

Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Abstract

We study the problem of fair allocation of M indivisible items among N agents using the popular notion of maximin share as our measure of fairness. The maximin share of an agent is the largest value she can guarantee herself if she is allowed to choose a partition of the items into N bundles (one for each agent), on the condition that she receives her least preferred bundle. A maximin share allocation provides each agent a bundle worth at least their maximin share. While it is known that such an allocation need not exist [Procaccia and Wang, 2014; Kurokawa et al., 2016], a series of work [Procaccia and Wang, 2014; David Kurokawa et al., 2018; Amanatidis et al., 2017; Barman and Krishna Murthy, 2017] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times their maximin share. Recently, [Ghodsi et al., 2018] improved the approximation guarantee to 3/4. Prior works utilize intricate algorithms, with an exception of [Barman and Krishna Murthy, 2017] which is a simple greedy solution but relies on sophisticated analysis techniques. In this paper, we propose an alternative 2/3 maximin share approximation which offers both a simple algorithm and straightforward analysis. In contrast to other algorithms, our approach allows for a simple and intuitive understanding of why it works.

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Jugal Garg, Peter McGlaughlin, and Setareh Taki. Approximating Maximin Share Allocations. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 20:1-20:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{garg_et_al:OASIcs.SOSA.2019.20,
  author =	{Garg, Jugal and McGlaughlin, Peter and Taki, Setareh},
  title =	{{Approximating Maximin Share Allocations}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{20:1--20:11},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{69},
  editor =	{Fineman, Jeremy T. and Mitzenmacher, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.20},
  URN =		{urn:nbn:de:0030-drops-100465},
  doi =		{10.4230/OASIcs.SOSA.2019.20},
  annote =	{Keywords: Fair division, Maximin share, Approximation algorithm}
}

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Documents authored by McGlaughlin, Peter

On the Existence of Competitive Equilibrium with Chores

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Approximating Maximin Share Allocations

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