Dichotomies in equilibrium computation and membership of PLC markets in FIXP

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 FIXP. Recently it was shown that these problems are also 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.