An Improved Approximation Scheme for Computing Arrow-Debreu Prices for the Linear Case

We prove that the problem of computing an Arrow-Debreu market equilibrium is PPAD-complete even when all traders use additively separable, piecewise-linear and concave utility functions. In fact, our proof shows that this market-equilibrium problem does not have a fully polynomial-time approximation scheme, unless every problem in PPAD is solvable in polynomial time.

[1]  E. Sperner.  Neuer beweis für die invarianz der dimensionszahl und des gebietes , 1928 .

[2]  S. Kakutani.  A generalization of Brouwer’s fixed point theorem , 1941 .

[3]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[4]  E. Eisenberg,et al.  CONSENSUS OF SUBJECTIVE PROBABILITIES: THE PARI-MUTUEL METHOD, , 1959 .

[5]  I. Fisher,et al.  Ten economic studies in the tradition of Irving Fisher , 1967 .

[6]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[7]  B. Curtis Eaves,et al.  Homotopies for computation of fixed points , 1972, Math. Program..

[8]  H. Scarf.  The computation of equilibrium prices , 1973 .

[9]  Francesco Mallegni,et al.  The Computation of Economic Equilibria , 1973 .

[10]  Léon Walras.  Éléments d'économie politique pure, ou, Théorie de la richesse sociale , 1976 .

[11]  Christos H. Papadimitriou,et al.  Exponential lower bounds for finding Brouwer fixed points , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[12]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[13]  R. Maxfield.  General equilibrium and the theory of directed graphs , 1997 .

[14]  P. Herings,et al.  The computation of equilibrium prices , 1998 .

[15]  M. Richter,et al.  Non-computability of competitive equilibrium , 1999 .

[16]  Xiaotie Deng,et al.  On the complexity of equilibria , 2002, STOC '02.

[17]  Nikhil R. Devanur,et al.  Market equilibrium via a primal-dual-type algorithm , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[18]  J. Geanakoplos.  Nash and Walras equilibrium via Brouwer , 2003 .

[19]  Nikhil R. Devanur,et al.  An Improved Approximation Scheme for Computing Arrow-Debreu Prices for the Linear Case , 2003, FSTTCS.

[20]  Amin Saberi,et al.  Approximating Market Equilibria , 2003, RANDOM-APPROX.

[21]  Kamal Jain,et al.  A Polynomial Time Algorithm for Computing the Arrow-Debreu Market Equilibrium for Linear Utilities , 2004, FOCS.

[22]  Rahul Garg,et al.  Auction algorithms for market equilibrium , 2004, STOC '04.

[23]  Nikhil R. Devanur.  The spending constraint model for market equilibrium: algorithmic, existence and uniqueness results , 2004, STOC '04.

[24]  Ning Chen,et al.  Fisher Equilibrium Price with a Class of Concave Utility Functions , 2004, ESA.

[25]  Vijay V. Vazirani,et al.  An Auction-Based Market Equilibrium Algorithm for the Separable Gross Substitutability Case , 2004, APPROX-RANDOM.

[26]  Kamal Jain,et al.  A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[27]  Sriram V. Pemmaraju,et al.  On the polynomial time computation of equilibria for certain exchange economies , 2005, SODA '05.

[28]  H. Scarf,et al.  How to Compute Equilibrium Prices in 1891 , 2005 .

[29]  Bruno Codenotti,et al.  Market equilibrium via the excess demand function , 2005, STOC '05.

[30]  Vijay V. Vazirani,et al.  Market equilibria for homothetic, quasi-concave utilities and economies of scale in production , 2005, SODA '05.

[31]  Bruno Codenotti,et al.  Market Equilibrium for CES Exchange Economies: Existence, Multiplicity, and Computation , 2005, FSTTCS.

[32]  Shang-Hua Teng,et al.  Market equilibria with hybrid linear-Leontief utilities , 2006, Theor. Comput. Sci..

[33]  Shang-Hua Teng,et al.  On the Approximation and Smoothed Complexity of Leontief Market Equilibria , 2006, FAW.

[34]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[35]  Paul W. Goldberg,et al.  Reducibility among equilibrium problems , 2006, STOC '06.

[36]  Xiaotie Deng,et al.  Sparse Games Are Hard , 2006, WINE.

[37]  Xi Chen,et al.  Computing Nash Equilibria: Approximation and Smoothed Complexity , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[38]  Amin Saberi,et al.  Leontief economies encode nonzero sum two-player games , 2006, SODA '06.

[39]  Yinyu Ye,et al.  Exchange market equilibria with Leontief's utility: Freedom of pricing leads to rationality , 2007, Theor. Comput. Sci..

[40]  Nikhil R. Devanur,et al.  Market Equilibria in Polynomial Time for Fixed Number of Goods or Agents , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[41]  Nikhil R. Devanur,et al.  Market equilibrium via a primal--dual algorithm for a convex program , 2008, JACM.

[42]  Yinyu Ye,et al.  A path to the Arrow–Debreu competitive market equilibrium , 2007, Math. Program..

[43]  Shang-Hua Teng,et al.  Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria , 2009, ISAAC.

[44]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[45]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points , 2010, SIAM J. Comput..

[46]  Vijay V. Vazirani.  Spending Constraint Utilities with Applications to the Adwords Market , 2010, Math. Oper. Res..