The Complexity of Non-Monotone Markets

We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We then prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utility functions, by proving that it is PPAD-complete when the Constant Elasticity of Substitution parameter \rho is any constant less than -1

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Electronic Colloquium on Computational Complexity, Report No. 31 (2006) On the Approximation and Smoothed Complexity of Leontief Market Equilibria

We show that the problem of finding an ɛ-approximate Nash equilibrium of an n × n two-person games can be reduced to the computation of an (ɛ/n) 2-approximate market equilibrium of a Leontief economy. Together with a recent result of Chen, Deng and Teng, this polynomial reduction implies that the Leontief market exchange problem does not have a fully polynomial-time approximation scheme, that is, there is no algorithm that can compute an ɛ-approximate market equilibrium in time polynomial in m, n, and 1/ɛ, unless PPAD ⊆ P, We also extend the analysis of our reduction to show, unless PPAD ⊆ RP, that the smoothed complexity of the Scarf’s general fixed-point approximation algorithm (when applying to solve the approximate Leontief market exchange problem) or of any algorithm for computing an approximate market equilibrium of Leontief economies is not polynomial in n and 1/σ, under Gaussian or uniform perturbations with magnitude σ