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A Fast and Simple Algorithm for Computing Market Equilibria

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Internet and Network Economics (WINE 2008)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 5385))

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Abstract

We give a new mathematical formulation of market equilibria using an indirect utility function: the function of prices and income that gives the maximum utility achievable. The formulation is a convex program and can be solved when the indirect utility function is convex in prices. We illustrate that many economies including

  • Homogeneous utilities of degree α ∈ [0,1] in Fisher economies — this includes Linear, Leontief, Cobb-Douglas

  • Resource allocation utilities like multi-commodity flows

satisfy this condition and can be efficiently solved.

Further, we give a natural and decentralized price-adjusting algorithm in these economies. Our algorithm, mimics the natural tâtonnement dynamics for the markets as suggested by Walras: it iteratively adjusts a good’s price upward when the demand for that good under current prices exceeds its supply; and downward when its supply exceeds its demand. The algorithm computes an approximate equilibrium in a number of iterations that is independent of the number of traders and is almost linear in the number of goods. Interestingly, our algorithm applies to certain classes of utility functions that are not weak gross substitutes.

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References

  1. Arrow, K., Debreu, G.: Existence of an Equilibrium for a Competitive Economy. Econometrica 22, 265–290 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arrow, K.J., Block, H.D., Hurwicz, L.: On the stability of the competitive equilibrium, II. Econometrica 27(1), 82–109 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, N., Deng, X., Sun, X., Yao, A.: Fisher Equilibrium Price with a class of Concave Utility Functions. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 169–179. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  4. Codenotti, B., Varadarajan, K.: Computation of market equilibria by convex programming. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.) Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  5. Codenotti, B., McCune, B., Pemmaraju, S.V., Raman, R., Varadarajan, K.: An experimental study of different approaches to solve the market equilibrium problem. In: ALENEX/ANALCO, pp. 167–179 (2005)

    Google Scholar 

  6. Codenotti, B., McCune, B., Raman, R., Varadarajan, K.: Computing equilibrium prices: Does theory meet practice? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669. Springer, Heidelberg (2005)

    Google Scholar 

  7. Codenotti, B., McCune, B., Varadarajan, K.: Market equilibrium via the excess demand function. In: STOC (2005)

    Google Scholar 

  8. Cole, R., Fleischer, L.: Fast-converging tatonnement algorithms for the market problem. Technical Report TR2007-602, Dept. of Computer Science, Dartmouth College (2007), http://www.cs.dartmouth.edu/reports/

  9. Deng, X., Papadimitriou, C., Safra, S.: On the Complexity of Equilibria. In: STOC (2002)

    Google Scholar 

  10. Devanur, N., Papadimitriou, C., Saberi, A., Vazirani, V.: Market Equilibrium via a Primal-Dual-Type Algorithm. In: FOCS, pp. 389–395 (2002); Journal version to appear in the Journal of the ACM

    Google Scholar 

  11. Fleischer, L.: Approximating fractional multicommodity flow independent of the number of commodities. SIAM J. Discrete Math. 13, 505–520 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: FOCS, pp. 300–309 (1998)

    Google Scholar 

  13. Garg, R., Kapoor, S.: Auction algorithms for market equilibrium. Math. Oper. Res. 31(4), 714–729 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Garg, R., Kapoor, S.: Price roll-backs and path auctions: An approximation scheme for computing the market equilibrium. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 225–238. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  15. Garg, R., Kapoor, S., Vazirani, V.: An Auction-Based Market Equilibrium Algorithm for the Separable Gross Substitutability Case. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 128–138. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  16. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  17. Jain, K., Mahdian, M., Saberi, A.: Approximating Market Equilibrium. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 98–108. Springer, Heidelberg (2003)

    Google Scholar 

  18. Jain, K., Vazirani, V.: Eisenberg-gale markets: Algorithms and structural properties. In: STOC (2007)

    Google Scholar 

  19. Kelly, F.P., Vazirani, V.: Rate control as a market equilibrium (manuscript, 2002)

    Google Scholar 

  20. Mas-Colell, A.: The Theory of General Economic Equilibrium: A Differential Approach. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  21. Negishi, T.: A note on the stability of an economy where all goods are gross substitutes. Econometrica 26(3), 445–447 (1958)

    Article  MATH  Google Scholar 

  22. Negishi, T.: The stability of a competitive economy: A survey article. Econometrica 30(4), 635–669 (1962)

    Article  MATH  Google Scholar 

  23. Nikaido, H., Uzawa, H.: Stability and non-negativity in a Walrasian process. International Econ. Review 1, 50–59 (1960)

    Article  MATH  Google Scholar 

  24. Plotkin, S., Shmoys, D., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Math. Oper. Res. 20, 257–301 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Quah, J.K.-H.: The monotonicity of individual and market demand. Econometrica 68(4), 911–930 (2000)

    Article  MATH  Google Scholar 

  26. Samuelson, P.A.: Foundations of Economic Analysis. Harward University Press, Cambridge (1947)

    MATH  Google Scholar 

  27. Varian, H.: Microeconomic Analysis. W. W. Norton, New York (1992)

    Google Scholar 

  28. Vazirani, V.: Combinatorial algorithms for market equilibria. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V. (eds.) Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  29. Walras, L.: Elements of Pure Economics, or the Theory of Social Wealth. Lausanne, Paris (1874) (in French)

    Google Scholar 

  30. Ye, Y.: A path to the Arrow-Debreu competitive market equilibrium. Mathematical Programming (2006)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dartmouth College, USA

    Lisa Fleischer

  2. IBM T.J. Watson Research Center, USA

    Rahul Garg & Rohit Khandekar

  3. Illinois Institute of Technology, USA

    Sanjiv Kapoor

  4. Stanford University, USA

    Amin Saberi

Authors
  1. Lisa Fleischer
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  2. Rahul Garg
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  3. Sanjiv Kapoor
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  4. Rohit Khandekar
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  5. Amin Saberi
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Editor information

Editors and Affiliations

  1. Computer Science Division, University of California at Berkeley, Berkeley, CA, USA

    Christos Papadimitriou

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Shuzhong Zhang

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Fleischer, L., Garg, R., Kapoor, S., Khandekar, R., Saberi, A. (2008). A Fast and Simple Algorithm for Computing Market Equilibria. In: Papadimitriou, C., Zhang, S. (eds) Internet and Network Economics. WINE 2008. Lecture Notes in Computer Science, vol 5385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92185-1_11

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  • DOI: https://doi.org/10.1007/978-3-540-92185-1_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-92184-4

  • Online ISBN: 978-3-540-92185-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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