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

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Summary

Introduction

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.

Our Contribution
Related Work
Fixed Point Theorems
The Class FIXP
The OPT-gate
A motivating example
Pseudogates
The OPT-gate for Linear Programming
The OPT-gate for Convex Optimization
Compute k m
Concave n-player games
Computing an ε-proper equilibrium via systems of conditional convex constraints
Solving Systems of Conditional Convex Constraints
FIXP-membership
FIXP-hardness
The K-K-M Lemma
The rainbow K-K-M lemma and Bapat’s Brouwer fixed point generalization
Arrow-Debreu Markets
The computational Arrow-Debreu market problem
The Hylland-Zeckhauser pseudomarket mechanism
Conclusion and Future Work
A Piecewise Differentiable Concave Functions
Full Text
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