On the Existence of Competitive Equilibrium with Chores

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

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Bhaskar Ray Chaudhury
  • University of Illinois at Urbana Champaign, IL, USA
Jugal Garg
  • University of Illinois at Urbana Champaign, IL, USA
Peter McGlaughlin
  • University of Illinois at Urbana Champaign, IL, USA
Ruta Mehta
  • University of Illinois at Urbana Champaign, IL, USA

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Exact and approximate computation of equilibria
  • Fair Division
  • Competitive Equilibrium
  • Fixed Point Theorems


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