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Walrasian Equilibrium: Hardness, Approximations and Tractable Instances

  • Ning Chen
  • Atri Rudra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3828)

Abstract

We study the complexity issues for Walrasian equilibrium in a special case of combinatorial auction, called single-minded auction, in which every participant is interested in only one subset of commodities. Chen et al. [5] showed that it is NP-hard to decide the existence of a Walrasian equilibrium for a single-minded auction and proposed a notion of approximate Walrasian equilibrium called relaxed Walrasian equilibrium. We show that every single-minded auction has a \(\frac{2}{3}\)-relaxed Walrasian equilibrium proving a conjecture posed in [5]. Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show it is strongly NP-complete to determine the existence of δ-weak Walrasian equilibrium, for any 0<δ≤ 1.

In search of positive results, we restrict our attention to the tollbooth problem [15], where every participant is interested in a single path in some underlying graph. We give a polynomial time algorithm to determine the existence of a Walrasian equilibrium and compute one (if it exists), when the graph is a tree. However, the problem is still hard for general graphs.

Keywords

Polynomial Time Optimal Allocation Polynomial Time Algorithm General Graph Market Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Archer, A., Papadimitriou, C.H., Talwar, K., Tardos, E.: An Approximate Truthful Mechanism For Combinatorial Auctions with Single Parameter Agents. In: SODA 2003, pp. 205–214 (2003)Google Scholar
  2. 2.
    Archer, A., Tardos, E.: Truthful Mechanisms for One-Parameter Agents. In: FOCS 2001, pp. 482–491 (2001)Google Scholar
  3. 3.
    Arrow, K.K., Debreu, G.: Existence of An Equilibrium for a Competitive Economy. Econometrica 22, 265–290 (1954)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bikhchandani, S., Mamer, J.W.: Competitive Equilibrium in an Economy with Indivisibilities. Journal of Economic Theory 74, 385–413 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, N., Deng, X., Sun, X.: On Complexity of Single-Minded Auction. Journal of Computer and System Sciences 69(4), 675–687 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Conen, W., Sandholm, T.: Coherent Pricing of Efficient Allocations in Combinatorial Economies. In: AAAI 2002, Workshop on Game Theoretic and Decision Theoretic Agents, GTDT (2002)Google Scholar
  7. 7.
    Cramton, P., Shoham, Y., Steinberg, R. (eds.): Combinatorial Auctions. MIT Press, Cambridge (2005)Google Scholar
  8. 8.
    Deng, X., Papadimitriou, C.H., Safra, S.: On the Complexity of Equilibria. In: STOC 2002, pp. 67–71 (2002); Full version appeared in Journal of Computer and System Sciences 67(2), 311–324 (2003)Google Scholar
  9. 9.
    Devanur, N., Papadimitriou, C.H., Saberi, A., Vazirani, V.V.: Market Equilibrium via a Primal-Dual-Type Algorithm. In: FOCS 2002, pp. 389–395 (2002)Google Scholar
  10. 10.
    Galil, Z., Micali, S., Gabow, H.: An O(EV logV) Algorithm for Finding a Maximal Weighted Matching in General Graphs. SIAM Journal on Computing 15, 120–130 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: a Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  12. 12.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees. Algorithmica 18, 3–20 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Goldberg, A.V., Hartline, J.D.: Collusion-Resistant Mechanisms for Single-Parameter Agents. In: SODA 2005, pp. 620–629 (2005)Google Scholar
  14. 14.
    Gul, F., Stacchetti, E.: Walrasian Equilibrium with Gross Substitutes. Journal of Economic Theory 87, 95–124 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., McSherry, F.: On Profit-Maximizing Envy-Free Pricing. In: SODA 2005, pp. 1164–1173 (2005)Google Scholar
  16. 16.
    Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS Publishing Company (1997)Google Scholar
  17. 17.
    Huang, S.L., Li, M.: Approximation of Walrasian Equilibrium in Single-Minded Auctions (submitted)Google Scholar
  18. 18.
    Jain, K.: A Polynomial Time Algorithm for Computing the Arrow-Debreu Market Equilibrium for Linear Utilities. In: FOCS 2004, pp. 286–294 (2004)Google Scholar
  19. 19.
    Kelso, A.S., Crawford, V.P.: Job Matching, Coalition Formation, and Gross Substitutes. Econometrica 50, 1483–1504 (1982)zbMATHCrossRefGoogle Scholar
  20. 20.
    Lehmann, D., O’Callaghan, L.I., Shoham, Y.: Truth Revelation in Approximately Efficient Combinatorial Auctions. In: ACM Conference on E-Commerce 1999, pp. 96–102 (1999); Full version appeared in JACM 49(5), 577-602 (2002)Google Scholar
  21. 21.
    Leonard, H.B.: Elicitation of Honest Preferences for the Assignment of Individual to Positions. Journal of Political Economy 91(3), 461–479 (1983)CrossRefGoogle Scholar
  22. 22.
    Mas-Collel, A., Whinston, W., Green, J.: Microeconomic Theory. Oxford University Press, Oxford (1995)Google Scholar
  23. 23.
    Mu’alem, A., Nisan, N.: Truthful Approximation Mechanisms for Restricted Combinatorial Auctions. In: AAAI 2002, pp. 379–384 (2002)Google Scholar
  24. 24.
    Papadimitriou, C.H., Roughgarden, T.: Computing Equilibria in Multi-Player Games. In: SODA 2005, pp. 82–91 (2005)Google Scholar
  25. 25.
    Tarjan, R.E.: Decomposition by clique separators. Discrete Math 55, 221–231 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    de Vries, S., Vohra, R.: Combinatorial Auctions: A Survey. INFORMS Journal on Computing 15(3), 284–309 (2003)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Walras, L.: Elements d’economie politique pure; ou, Theorie de la richesse sociale (Elements of Pure Economics, or the Theory of Social Wealth), Lausanne, Paris (1874); Translated by William Jaffé, Irwin (1954)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ning Chen
    • 1
  • Atri Rudra
    • 1
  1. 1.Department of Computer Science & EngineeringUniversity of WashingtonSeattleUSA

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