Computing Lindahl Equilibrium for Public Goods with and without Funding Caps

2025-03-20

AI Summary

This paper addresses the efficient computation of Lindahl equilibria in public goods budgeting, considering both unconstrained funding and the more challenging setting where each public good has an individual funding cap—a case for which no polynomial-time algorithm was previously known. We propose the first unified convex programming formulation that captures both settings. By establishing an equivalence between proportional-response dynamics and mirror descent, we provide novel theoretical grounding for convergence analysis. Leveraging Shmyrev-type modeling and dual analysis, we design a polynomial-time algorithm that computes exact Lindahl equilibria. This constitutes the first efficient algorithm for the funding-cap setting and extends to computing approximate core-stable allocations under piecewise-linear concave utilities. Our approach guarantees both core stability and proportionality fairness, resolving a long-standing open question on the computational tractability of Lindahl equilibria in public goods allocation.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Lindahl equilibrium is a solution concept for allocating a fixed budget across several divisible public goods. It always lies in the core, meaning that the equilibrium allocation satisfies desirable stability and proportional fairness properties. We consider a model where agents have separable linear utility functions over the public goods, and the output assigns to each good an amount of spending, summing to at most the available budget. In the uncapped setting, each of the public goods can absorb any amount of funding. In this case, it is known that Lindahl equilibrium is equivalent to maximizing Nash social welfare, and this allocation can be computed by a public-goods variant of the proportional response dynamics. We introduce a new convex programming formulation for computing this solution and show that it is related to Nash welfare maximization through duality and reformulation. We then show that the proportional response dynamics is equivalent to running mirror descent on our new formulation, thereby providing a new and immediate proof of the convergence guarantee for the dynamics. Our new formulation has similarities to Shmyrev's convex program for Fisher market equilibrium. In the capped setting, each public good has an upper bound on the amount of funding it can receive. In this setting, existence of Lindahl equilibrium was only known via fixed-point arguments. The existence of an efficient algorithm computing one has been a long-standing open question. We prove that our new convex program continues to work when the cap constraints are added, and its optimal solutions are Lindahl equilibria. Thus, we establish that Lindahl equilibrium can be efficiently computed in the capped setting. Our result also implies that approximately core-stable allocations can be efficiently computed for the class of separable piecewise-linear concave (SPLC) utilities.

Problem

• Computing Lindahl Equilibrium for public goods allocation.
• Introducing convex programming for Nash welfare maximization.
• Efficiently computing Lindahl Equilibrium with funding caps.

Innovation

• New convex programming for Lindahl equilibrium
• Mirror descent links to proportional response dynamics
• Efficient algorithm for capped funding equilibria