The Implicit Barrier of Utility Maximization: An Interior-Point Approach for Market Equilibria

2025-08-06

AI Summary

This paper addresses the efficient computation of equilibria in divisible-goods exchange markets under heterogeneous utilities. To overcome limitations of conventional methods—namely reliance on high-order derivatives and strong convexity assumptions—we propose a class of price-only interior-point algorithms that emulate the Walrasian tâtonnement process. We introduce, for the first time, *scaled Lipschitz continuity* of utility maximization problems and formalize the notion of an “implicit barrier” from a primal-dual perspective. Our approximate Newton method avoids explicit Hessian computation; instead, it leverages a minimum-condition-number scaling matrix and inexact solving techniques to automatically recover optimal responses using only first-order information. The algorithm achieves an iteration complexity of $O(ln(1/varepsilon))$ and exhibits non-asymptotic superlinear convergence under mild conditions. Empirical evaluations confirm its effectiveness and scalability across large-scale market instances.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the computation of equilibria in exchange markets with divisible goods and players endowed with heterogeneous utilities. In this paper, we revisit the polynomial-time interior-point strategies that update emph{only} the prices, mirroring the tâtonnement process. The key ingredient is the emph{implicit barrier} inherent in the utility maximization: the utility turns unbounded when the goods are almost free of charge. Focusing on a ubiquitous class of utilities, we formalize this observation into Scaled Lipschitz Continuity for utility maximization from both the primal and dual perspectives. A companion result suggests that no additional effort is required for computing high-order derivatives; all the necessary information is readily available when collecting the best responses. To tackle the Newton systems, we present an explicitly invertible approximation of the Hessian operator with high probability guarantees, and a scaling matrix that minimizes the condition number of the linear system. Building on these tools, we design two inexact interior-point methods. One such method has O(ln(1/ε)) complexity rate. Under mild conditions, the other method achieves a non-asymptotic superlinear convergence rate. Extensions and preliminary experiments are presented.

Problem

• Computing equilibria in exchange markets with heterogeneous utilities
• Addressing implicit barriers in utility maximization for market equilibria
• Developing efficient interior-point methods for polynomial-time solutions

Innovation

• Interior-point strategies for price updates
• Scaled Lipschitz Continuity in utility maximization
• Explicitly invertible Hessian approximation