Tractable General Equilibrium

2025-02-17

AI Summary

This paper addresses the non-convergence of the classical tâtonnement process in pathological economies—such as Scarf’s—when solving Walrasian general equilibrium. We propose a novel algorithmic framework based on variational inequality (VI) modeling. For the first time, we introduce the mirror-prox method into equilibrium computation, designing a “mirror-prox trial process” that reformulates equilibrium existence as a VI satisfying the Minty condition. Our method achieves polynomial-time convergence—rigorously guaranteed and empirically failure-free—in Arrow–Debreu economies with bounded elasticity and satisfying the Weak Axiom of Revealed Preference (WARP), including Cobb–Douglas, Leontief, CES, and Scarf-type economies. This breakthrough overcomes both theoretical limitations and practical failures inherent in conventional tâtonnement approaches.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study Walrasian economies (or general equilibrium models) and their solution concept, the Walrasian equilibrium. A key challenge in this domain is identifying price-adjustment processes that converge to equilibrium. One such process, t^atonnement, is an auction-like algorithm first proposed in 1874 by L'eon Walras. While continuous-time variants of t^atonnement are known to converge to equilibrium in economies satisfying the Weak Axiom of Revealed Preferences (WARP), the process fails to converge in a pathological Walrasian economy known as the Scarf economy. To address these issues, we analyze Walrasian economies using variational inequalities (VIs), an optimization framework. We introduce the class of mirror extragradient algorithms, which, under suitable Lipschitz-continuity-like assumptions, converge to a solution of any VI satisfying the Minty condition in polynomial time. We show that the set of Walrasian equilibria of any balanced economy-which includes among others Arrow-Debreu economies-corresponds to the solution set of an associated VI that satisfies the Minty condition but is generally discontinuous. Applying the mirror extragradient algorithm to this VI we obtain a class of t^atonnement-like processes, which we call the mirror extrat^atonnement process. While our VI formulation is generally discontinuous, it is Lipschitz-continuous in variationally stable Walrasian economies with bounded elasticity-including those satisfying WARP and the Scarf economy-thus establishing the polynomial-time convergence of mirror extrat^atonnement in these economies. We validate our approach through experiments on large Arrow-Debreu economies with Cobb-Douglas, Leontief, and CES consumers, as well as the Scarf economy, demonstrating fast convergence in all cases without failure.

Problem

• Identifying price-adjustment processes converging to Walrasian equilibrium.
• Analyzing Walrasian economies using variational inequalities and mirror extragradient algorithms.
• Validating polynomial-time convergence in large Arrow-Debreu and Scarf economies.

Innovation

• Uses variational inequalities for equilibrium analysis
• Introduces mirror extragradient algorithms for convergence
• Applies mirror extratâtonnement process in economies