Spending is not Easier than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

TL;DR

This paper demonstrates that computing approximate market equilibria in Fisher's model is PPAD-complete, establishing computational equivalence with Arrow-Debreu equilibria and resolving a long-standing open question.

Contribution

It proves PPAD-completeness for Fisher equilibria with additively separable, piecewise-linear, concave utilities, a novel result in computational market theory.

Findings

Fisher equilibrium computation is PPAD-complete.

Resolves open question on the complexity of Fisher equilibria.

First PPAD-completeness proof for Fisher's model.

Abstract

It is a common belief that computing a market equilibrium in Fisher's spending model is easier than computing a market equilibrium in Arrow-Debreu's exchange model. This belief is built on the fact that we have more algorithmic success in Fisher equilibria than Arrow-Debreu equilibria. For example, a Fisher equilibrium in a Leontief market can be found in polynomial time, while it is PPAD-hard to compute an approximate Arrow-Debreu equilibrium in a Leontief market. In this paper, we show that even when all the utilities are additively separable, piecewise-linear, and concave functions, finding an approximate equilibrium in Fisher's model is complete in PPAD. Our result solves a long-term open question on the complexity of market equilibria. To the best of our knowledge, this is the first PPAD-completeness result for Fisher's model.