  We show that the problem of finding an \epsilon-approximate Nash equilibrium
of an n by n two-person games can be reduced to the computation of an
(\epsilon/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
\epsilon-approximate market equilibrium in time polynomial in m, n, and
1/\epsilon, unless PPAD is not in P, We also extend the analysis of our
reduction to show, unless PPAD is not in 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/\sigma, under Gaussian or uniform perturbations with
magnitude \sigma.
