Revised: July 16, 2016

Published: December 30, 2016

**Keywords:**market equilibrium, FIXP, dichotomy, non-linear complementarity

**Categories:**ecommerce, game theory, algorithmic game theory, Nash equilibrium, market equilibrium, fixed point, dichotomy, completeness, complexity theory

**ACM Classification:**F.2.0, G.0

**AMS Classification:**68Q25, 91B26, 91B50, 90C30, 90C33

**Abstract:**
[Plain Text Version]

Piecewise-linear, concave (PLC) utility functions play an important role in work done at the intersection of economics and algorithms. We prove that the problem of computing an equilibrium in Arrow-Debreu markets with PLC utilities and PLC production sets is in the class $\textsf{FIXP}$. Recently it was shown that these problems are also $\textsf{FIXP}$-hard (Garg et al., arXiv:1411.5060), hence settling the long-standing question of the complexity of this problem. Central to our proof is capturing equilibria of these markets as fixed points of a continuous function via a nonlinear complementarity problem (NCP) formulation.

Next, we provide dichotomies for equilibrium computation problems, both Nash and market. There is a striking resemblance in the dichotomies for these two problems, hence providing a unifying view. We note that in the past, dichotomies have played a key role in bringing clarity to the complexity of decision and counting problems.

A preliminary version of this paper appeared as part of a paper in STOC'14.