Abstract
We present polynomial-time interior-point algorithms for solving the Fisher and Arrow–Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of \({O(n^{4}log(1/\epsilon}\))) for computing an \({\epsilon}\) -equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n 4 L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound \(O(n^{8}log(1/\epsilon\))) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow–Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method.
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References
Arrow K. and Debreu G. (1954). Existence of a competitive equilibrium for a competitive economy. Econometrica 22(3): 265–290
Atkinson D.S. and Vaidya P.M. (1992). A scaling technique for finding the weighted analytic center of a polytope.. Math. Program. 57: 163–192
Brainard, W.C., Scarf, H.E.: How to compute equilibrium prices in 1891. Cowles Foundation Discussion Paper 1270, August (2000)
Codenotti, B., Varadarajan, K.: Efficient computation of equilibrium prices for market with Leontief utilities. In: Proc. of ICALP 2004, pp. 371–381 (2004)
Codenotti, B., Pemmaraju, S., Varadarajan, K.: On the polynomial time computation of equilibria for certain exchange economies. In: Proc of SODA 2005, pp. 72–81 (2005)
Cottle, R., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem, Chap. 5.9: Interior–point methods, pp. 461–475. Academic Boston (1992)
Deng, X., Huang, S., Ye, Y.: Computation of the Arrow–Debreu equilibrium for a class of non-homogenous utility functions. Technical Report, State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing, 10084, China
Deng, X., Papadimitriou, C.H., Safra, S.: On the complexity of equilibria. In: Proc. of ACM Symposium on Theory of Computing 2002, pp. 67–71 (2002)
Devanur, N.R., Papadimitriou, C.H., Saberi, A., Vazirani, V.V.: Market equilibrium via a primal-dual-type algorithm. In: Proc. of The 43rd Annual IEEE Symposium on Foundations of Computer Science 2002, pp. 389–395; journal version on http://www.cc.gatech.edu/~saberi/ (2004)
Dirkse S.P. and Ferris M.C. (1996). A pathsearch damped Newton method for computing general equilibria. Ann. Oper. Res. 68: 211–232
Eisenberg E. and Gale D. (1959). Consensus of subjective probabilities: the pari-mutuel method. Ann. Mathe. Statist. 30: 165–168
Eaves B.C. (1976). A finite algorithm for the linear exchange model. J. Math. Econ. 3: 197–203
Eaves B.C. (1985). Finite solution of pure trade markets with Cobb-Douglas utilities. Math. Program. Study 23: 226–239
Esteban-Bravo M. (2004). Computing equilibria in general equilibrium models via interior-point methods. Comput. Econ. 23(2): 147–171
Ferris M.C. and Pang J.S. (1997). Engineering and economic applications of complementarity problems. SIAM Rev. 39(4): 669–713
Gale, D.: The Theory of Linear Economic Models. McGraw Hill, New York (1960)
Ginsburgh, V., Keyzer, M.: The Structure of Applied General Equilibrium Models. The MIT Press, Cambridge (1077)
Güler O. (1993). Existence of interior points and interior paths in nonlinear monotone complementarity problems. Math. Oper. Res. 18(1): 128–147
Jain, K.: A polynomial time algorithm for computing the Arrow–Debreu market equilibrium for linear utilities. In: Proc. of The 43rd Annual IEEE Symposium on Foundations of Computer Science 2004, pp. 286–294 (2004)
Jain, K., Mahdian, M., Saberi, A.: Approximating market equilibria. In: Proc. of APPROX 2003, pp. 98–108 (2003)
Jain, K., Vazirani, V.V., Ye, Y.: Market equilibria for homothetic, quasi-concave utilities and economies of scale in production. In: Proc. of SODA 2005, pp. 63–71 (2005)
Kojima M., Mizuno S. and Yoshise A. (1991). Math. Program. 50: 331–342
Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A unified approach for interior-point algorithms for linear complementarity problems. Lecture Notes in Computer Science, vol. 538. Springer, Berlin Heidelberg New York (1991)
Megiddo N. (1989). Pathways to the optimal set in linear programming. In: Megiddo, N. (eds) Progress in Mathematical Programming Interior Point and Related Methods, pp 131–158. Springer, Berlin Heidelberg New York
Mehrotra S. and Ye Y. (1993). On finding an interior point on the optimal face of linear programs. Math. Program. 62: 497–516
Mizuno S., Todd M.J. and Ye Y. (1993). On adaptive step primal–dual interior–point algorithms for linear programming.. Math. Oper. Res. 18: 964–981
Monteiro R.D.C. and Adler I. (1989). Interior path following primal–dual algorithms: Part I : Linear programming. Math. Program. 44: 27–41
Monteiro R.D.C. and Tsuchiya T. (2004). A new iteration-complexity bound for the MTY predictor- corrector algorithm. SIAM J. Optim. 15(2): 319–347
Negishi T. (1960). Welfare economics and the existence of an equilibrium for a competitive economy. Metroeconomica, XII: 92–97
Nenakhov E. and Primak M. (1983). About one algorithm for finding the solution of the Arrow–Debreu model.. Kibernetica 3: 127–128
Nesterov Yu.E. and Nemirovskii A.S. (1993). Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications, SIAM, Philadelphia
Primak M.E. (1993). A converging algorithm for a linear exchange model.. J. Math. Econ. 22(2): 181–187
Rutherford T.F. (1999). Applied general equilibrium modeling with MPSGE as a GAMS subsystem. Comput. Econ. 14: 1–46
Scarf, H.E.: The Computation of Economic Equilibria (with collaboration of T. Hansen, Cowles Foundation Monograph, No. 24. Yale University Press, New Haven (1973)
Todd, M.J.: The computation of fixed points and applications. (Lecture Notes in Economics and Mathematical Systems, vol. 124. Springer, Berlin Heidelberg New York (1976)
Walras, L.: Elements of Pure Economics, or the Theory of Social Wealth 1874 (1899, 4th ed.; 1926, rev ed., 1954, Engl. Transl.)
Yang Z. (1999). Computing Equilibria and Fixed Points: The Solution of Nonlinear Inequalities. Kluwer, Boston
Ye Y., Güler O., Tapia R.A. and Zhang Y. (1993). Math. Program. 59: 151–162
Ye Y. (1997). Interior Point Algorithms: Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York
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Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.
This author was supported in part by NSF Grants DMS-0306611 and DMS-0604513. The author would like to thank Curtis Eaves, Osman Güler, Kamal Jain and Mike Todd for insightful discussions on this subject, especially on their mathematical references and economic interpretations of the fixed-point model presented in this paper.