Volume 12 (2016) Article 20 pp. 1-25

Dichotomies in Equilibrium Computation and Membership of PLC Markets in FIXP

by Jugal Garg, Ruta Mehta, and Vijay V. Vazirani

Received: August 15, 2015
Revised: July 16, 2016
Published: December 30, 2016

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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.

DOI: 10.4086/toc.2016.v012a020

© 2016 Jugal Garg, Ruta Mehta, and Vijay V. Vazirani
Licensed under a Creative Commons Attribution License (CC-BY)