On the Approximation and Smoothed Complexity of Leontief Market Equilibria

TL;DR

This paper demonstrates the computational hardness of finding approximate market equilibria in Leontief economies and analyzes the smoothed complexity of algorithms solving such problems, showing they are unlikely to be polynomial-time.

Contribution

It establishes a polynomial reduction from approximate Nash equilibria to Leontief market equilibria and analyzes the smoothed complexity of equilibrium algorithms, revealing inherent computational limitations.

Findings

No fully polynomial-time approximation scheme exists for Leontief market equilibrium unless PPAD in P.

The smoothed complexity of fixed-point algorithms for Leontief economies is not polynomial under Gaussian or uniform perturbations.

The reduction links the complexity of Nash equilibria to Leontief market equilibria, implying similar computational hardness.

Abstract

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.