Leontief Exchange Markets Can Solve Multivariate Polynomial Equations, Yielding FIXP and ETR Hardness

We show FIXP-hardness of computing equilibria in Arrow-Debreu exchange markets under Leontief utility functions, and Arrow-Debreu markets under linear utility functions and Leontief production sets, thereby settling these open questions [33]. As corollaries, we obtain FIXPhardness for piecewise-linear concave (PLC) utilities and for Arrow-Debreu markets under linear utility functions and polyhedral production sets. In all cases, as required under FIXP, the set of instances mapped onto will admit equilibria, i.e., will be “yes” instances. If all instances are under consideration, then in all cases we prove that the problem of deciding if a given instance admits an equilibrium is ETR-complete, where ETR is the class Existential Theory of Reals. As a consequence of the results stated above, and the fact that membership in FIXP has been established for PLC utilities [17], the entire computational difficulty of Arrow-Debreu markets under PLC utility functions lies in the Leontief utility subcase. This is perhaps the most unexpected aspect of our result, since Leontief utilities are meant for the case that goods are perfect complements, whereas PLC utilities are very general, capturing not only the cases when goods are complements and substitutes, but also arbitrary combinations of these and much more. The main technical part of our result is the following reduction: Given a set S of simultaneous multivariate polynomial equations in which the variables are constrained to be in a closed bounded region in the positive orthant, we construct a Leontief exchange market M which has one good corresponding to each variable in S. We prove that the equilibria ofM, when projected onto prices of these latter goods, are in one-to-one correspondence with the set of solutions of the polynomials. This reduction is related to a classic result of Sonnenschein [32, 31]. ∗Supported by NSF Grants CCF-0914732 and CCF-1216019. †Max-Planck-Institut für Informatik, Saarbrücken, Germany. [email protected] ‡College of Computing, Georgia Institute of Technology, Atlanta. [email protected] §College of Computing, Georgia Institute of Technology, Atlanta. [email protected] ¶College of Computing, Georgia Institute of Technology, Atlanta. [email protected] ar X iv :1 41 1. 50 60 v1 [ cs .C C ] 1 8 N ov 2 01 4