Abstract
We introduce a new technique for proving membership of problems in FIXP – the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a “pseudogate” which can be used as a black box when building FIXP circuits. This pseudogate, which we term the “OPT-gate”, can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets. In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.
Highlights
Equilibria, i.e., stable states of some dynamic process or environment [Yannakakis, 2009], appear in several classic applications in economics and computer science
The class has been successful in that regard, with interesting problems related to game theory [Etessami and Yannakakis, 2010] and competitive markets [Etessami and Yannakakis, 2010; Garg et al, 2017; Chen et al, 2017] among others, being either members of FIXP, or complete for the class
Another example comes from competitive markets, where the market equilibrium notion includes convex optimization programs for maximizing the utilities of consumers and producers given a set of prices
Summary
Equilibria, i.e., stable states of some dynamic process or environment [Yannakakis, 2009], appear in several classic applications in economics and computer science. In the case of FIXP the aforementioned challenges are much more pronounced; for an existence proof to be used as a basis for a membership result, it has to display several characteristics It has to go via Brouwer’s fixed point theorem, and more importantly, it has to avoid using any “discontinuous” components, precluding the use of several types of discrete steps and limit arguments. Another example comes from competitive markets, where the market equilibrium notion includes convex optimization programs for maximizing the utilities of consumers and producers given a set of prices This offers a possible explanation as to why the literature has fallen short of producing a systematic and unified approach for proving FIXP-membership results: Up until now, it was not known how to compute these optimization programs in FIXP, or how to incorporate these programs as part of a FIXP circuit, as required for a membership result.
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