Constant Inapproximability for Fisher Markets
For theoretical computer scientists studying market equilibrium computation, this establishes a strong inapproximability result, showing the problem is hard even for constant approximations.
The paper proves that computing constant-factor approximations (better than 1/11) of market equilibria in Fisher markets with SPLC utilities is PPAD-complete, ruling out a PTAS unless PPAD=P.
We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD$=$P. In fact, we prove that computing any approximation better than $1/11$ is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility functions.