Fisher Meets Lindahl: A Unified Duality Framework for Market Equilibrium

2025-11-06

AI Summary

Lindahl equilibria—market equilibria for public goods—have long suffered from theoretical underdevelopment, particularly in computational tractability, dynamic mechanisms, and welfare properties, lagging significantly behind Fisher equilibria for private goods. Method: We develop the first unified dual framework that rigorously establishes structural correspondence between the two: mapping private-good allocations and prices to public-good contributions and shadow prices via duality. Our approach constructs a dual utility function from indirect utilities and integrates convex optimization, KKT conditions, proportional-response dynamics (PRD), and tâtonnement. Contributions: (1) We prove that Lindahl equilibria maximize Nash social welfare exactly under concave homogeneous utilities; (2) for non-homogeneous utilities, we derive a $(1/e)^{1/e}$-approximation guarantee for optimal Nash welfare; (3) we design novel distributed dynamic adjustment mechanisms applicable to both public-good and chore markets—thereby systematically closing longstanding theoretical gaps in Lindahl equilibrium theory.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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
• Security and Privacy: Large-scale security measurements

Abstract

The Fisher market equilibrium for private goods and the Lindahl equilibrium for public goods are classic and fundamental solution concepts for market equilibria. While Fisher market equilibria have been well-studied, the theoretical foundations for Lindahl equilibria remain substantially underdeveloped. In this work, we propose a unified duality framework for market equilibria. We show that Lindahl equilibria of a public goods market correspond to Fisher market equilibria in a dual Fisher market with dual utilities, and vice versa. The dual utility is based on the indirect utility, and the correspondence between the two equilibria works by exchanging the roles of allocations and prices. Using the duality framework, we address the gaps concerning the computation and dynamics for Lindahl equilibria and obtain new insights and developments for Fisher market equilibria. First, we leverage this duality to analyze welfare properties of Lindahl equilibria. For concave homogeneous utilities, we prove that a Lindahl equilibrium maximizes Nash Social Welfare (NSW). For concave non-homogeneous utilities, we show that a Lindahl equilibrium achieves $(1/e)^{1/e}$ approximation to the optimal NSW, and the approximation ratio is tight. Second, we apply the duality framework to market dynamics, including proportional response dynamics (PRD) and t^atonnement. We obtain new market dynamics for the Lindahl equilibria from market dynamics in the dual Fisher market. We also use duality to extend PRD to markets with total complements utilities, the dual class of gross substitutes utilities. Finally, we apply the duality framework to markets with chores. We propose a program for private chores for general convex homogeneous disutilities that avoids the"poles"issue, whose KKT points correspond to Fisher market equilibria. We also initiate the study of the Lindahl equilibrium for public chores.

Problem

• Developing theoretical foundations for Lindahl equilibria in public goods markets
• Establishing duality between Fisher and Lindahl market equilibrium concepts
• Addressing computation and dynamics gaps for public goods market equilibria

Innovation

• Unified duality framework for market equilibria
• Lindahl equilibria correspond to dual Fisher markets
• Duality enables computation and dynamics analysis