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On the Existence of Competitive Equilibrium with Chores

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

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Author Details

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

Funding

  • Garg, Jugal: NSF Grant CCF-1942321 (CAREER).
  • Mehta, Ruta: NSF Grant CCF-1750436 (CAREER).

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITCS.2022.41

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.

Subject Classification

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

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References

  1. Kenneth Arrow and Gerard Debreu. Existence of an equilibrium for a competitive economy. Econometrica, 22(3):265-290, 1954. Google Scholar
  2. Yaron Azrieli and Eran Shmaya. Rental harmony with roommates. J. Economic Theory, 153:128-137, 2014. Google Scholar
  3. Anna Bogomolnaia, Hervé Moulin, Fedor Sandomirskiy, and Elena Yanovskaia. Competitive division of a mixed manna. Econometrica, 85(6):1847-1871, 2017. Google Scholar
  4. Anna Bogomolnaia, Hervé Moulin, Fedor Sandomirskiy, and Elena Yanovskaia. Dividing bads under additive utilities. Social Choice and Welfare, 52(3):395-417, 2019. Google Scholar
  5. Shant Boodaghians, Bhaskar Ray Chaudhury, and Ruta Mehta. Polynomial time algorithms to find an approximate competitive equilibrium for chores. CoRR, abs/2107.06649, 2021. Google Scholar
  6. Steven J. Brams and Alan D. Taylor. Fair division - from cake-cutting to dispute resolution. Cambridge University Press, 1996. Google Scholar
  7. Simina Branzei and Fedor Sandomirskiy. Algorithms for competitive division of chores. arXiv:1907.01766, 2019. Google Scholar
  8. Luitzen Egbertus Jan Brouwer. Über abbildung von mannigfaltigkeiten. Mathematische annalen, 71(1):97-115, 1911. Google Scholar
  9. Bhaskar Ray Chaudhury, Jugal Garg, Peter McGlaughlin, and Ruta Mehta. Competitive allocation of a mixed manna. In Proc. 32nd Symp. Discrete Algorithms (SODA), 2021. Google Scholar
  10. Richard Cole, Nikhil Devanur, Vasilis Gkatzelis, Kamal Jain, Tung Mai, Vijay Vazirani, and Sadra Yazdanbod. Convex program duality, Fisher markets, and Nash social welfare. In Proc. 18th Conf. Economics and Computation (EC), 2017. Google Scholar
  11. Nikhil Devanur, Jugal Garg, and László Végh. A rational convex program for linear Arrow-Debreu markets. ACM Trans. Econom. Comput., 5(1):6:1-6:13, 2016. Google Scholar
  12. Edmund Eisenberg and David Gale. Consensus of subjective probabilities: The Pari-Mutuel method. Ann. Math. Stat., 30(1):165-168, 1959. Google Scholar
  13. David Gale. The linear exchange model. Journal of Mathematical Economics, 3(2):205-209, l976. Google Scholar
  14. Jugal Garg and Peter McGlaughlin. Computing competitive equilibria with mixed manna. In Proc. 19th Conf. Auton. Agents and Multi-Agent Systems (AAMAS), pages 420-428, 2020. Google Scholar
  15. Kamal Jain. A polynomial time algorithm for computing the Arrow-Debreu market equilibrium for linear utilities. SIAM Journal on Computing, 37(1):306-318, 2007. Google Scholar
  16. Shizuo Kakutani. A generalization of Brouwer’s fixed point theorem. Duke mathematical journal, 8(3):457-459, 1941. Google Scholar
  17. Robert Maxfield. General equilibrium and the theory of directed graphs. J. Math. Econom., 27(1):23-51, 1997. Google Scholar
  18. Lionel McKenzie. On equilibrium in Graham’s model of world trade and other competitive systems. Econometrica, 22(2):147-161, 1954. Google Scholar
  19. Lionel W. McKenzie. On the existence of general equilibrium for a competitive market. Econometrica, 27(1):54-71, 1959. Google Scholar
  20. Herve Moulin. Fair Division and Collective Welfare. MIT Press, 2003. Google Scholar
  21. Herve Moulin. Fair division in the Internet age. Annual Review of Economics, 11, 2019. Google Scholar
  22. John Nash. Non-cooperative games. Ann. Math., 54(2):286-295, 1951. Google Scholar
  23. E I Nenakov and M E Primak. One algorithm for finding solutions of the Arrow-Debreu model. Kibernetica, 3:127-128, 1983. Google Scholar
  24. J. Robertson and W. Webb. Cake-Cutting Algorithms: Be Fair If You Can. AK Peters, MA, 1998. Google Scholar
  25. W. Shafer and H. Sonnenschein. Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics, 2(3):345-348, 1975. Google Scholar
  26. F. E. Su. Rental harmony: Sperner’s lemma in fair division. The American Mathematical Monthly, 106(10):930-942, 1999. Google Scholar
  27. Hal Varian. Equity, envy and efficiency. J. Econom. Theory, 29(2):217-244, 1974. Google Scholar
  28. Vijay Vazirani and Mihalis Yannakakis. Market equilibrium under separable, piecewise-linear, concave utilities. Journal of the ACM, 58(3):10, 2011. Google Scholar
  29. Yinyu Ye. A path to the Arrow-Debreu competitive market equilibrium. Math. Prog., 111(1-2):315-348, 2008. Google Scholar
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