On the Approximation and Smoothed Complexity of Leontief Market Equilibria

We show that the problem of finding an ǫ-approximate Nash equilibrium of an n× n two-person games can be reduced to the computation of an (ǫ/n)-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 ǫ-approximate market equilibrium in time polynomial in m, n, and 1/ǫ, 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. ∗Also affiliated with Akamai Technologies Inc. Cambridge, Massachusetts, USA. Partially supported by NSF grants CCR-0311430 and ITR CCR-0325630. Part of this work done while visiting Tsinghua University.