Testing Core Membership in Public Goods Economies

TL;DR

This paper characterizes the core in public goods economies using topological analysis and introduces an efficient algorithm for testing core membership based on convex optimization, improving computational feasibility.

Contribution

It provides a new topological characterization of the core and develops a polynomial-time algorithm for core membership testing using convex optimization techniques.

Findings

Core membership is characterized by Pareto efficiency, individual rationality, and path-connected dominance sets.

The proposed algorithm reduces core testing to solving O(n) convex optimization problems.

Core membership testing becomes feasible for utility functions with analytic expressions or via approximate convex optimization.

Abstract

This paper develops a recent line of economic theory seeking to understand public goods economies using methods of topological analysis. Our first main result is a very clean characterization of the economy's core (the standard solution concept in public goods). Specifically, we prove that a point is in the core iff it is Pareto efficient, individually rational, and the set of points it dominates is path connected. While this structural theorem has a few interesting implications in economic theory, the main focus of the second part of this paper is on a particular algorithmic application that demonstrates its utility. Since the 1960s, economists have looked for an efficient computational process that decides whether or not a given point is in the core. All known algorithms so far run in exponential time (except in some artificially restricted settings). By heavily exploiting our new structure, we propose a new algorithm for testing core membership whose computational bottleneck is the solution of $O(n)$ convex optimization problems on the utility function governing the economy. It is fairly natural to assume that convex optimization should be feasible, as it is needed even for very basic economic computational tasks such as testing Pareto efficiency. Nevertheless, even without this assumption, our work implies for the first time that core membership can be efficiently tested on (e.g.) utility functions that admit "nice" analytic expressions, or that appropriately defined $\varepsilon$-approximate versions of the problem are tractable (by using modern black-box $\varepsilon$-approximate convex optimization algorithms).