Computing Lindahl Equilibrium with and without Funding Caps

Christian Kroer, Dominik Peters

Keywords: public goods, participatory budgeting, Fisher market, Lindahl equilibrium

TL;DR: New convex programs for computing Lindahl equilibrium

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.

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 very immediate proof of the convergence guarantee for the dynamics.

In the capped setting, each public good has an upper bound on the amount of funding it can receive, which is a type of constraint that appears in fractional committee selection and participatory budgeting. 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.

Submission Number: 15