License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.41
URN: urn:nbn:de:0030-drops-156378
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Chaudhury, Bhaskar Ray ; Garg, Jugal ; McGlaughlin, Peter ; Mehta, Ruta

On the Existence of Competitive Equilibrium with Chores

LIPIcs-ITCS-2022-41.pdf (0.6 MB)


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.

BibTeX - Entry

  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 =		{},
  URN =		{urn:nbn:de:0030-drops-156378},
  doi =		{10.4230/LIPIcs.ITCS.2022.41},
  annote =	{Keywords: Fair Division, Competitive Equilibrium, Fixed Point Theorems}

Keywords: Fair Division, Competitive Equilibrium, Fixed Point Theorems
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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