Public Goods Games in Directed Networks

TL;DR

This paper investigates the computational complexity of finding pure and mixed Nash equilibria in public goods games on directed networks, revealing NP-hardness and PPAD-hardness results, with efficient algorithms possible for networks with bounded treewidth.

Contribution

It establishes complexity dichotomies and hardness results for equilibrium computation in directed network public goods games, extending understanding beyond undirected cases.

Findings

Pure Nash equilibrium existence is NP-hard to decide.

Mixed Nash equilibria are PPAD-hard to find.

Polynomial-time algorithms exist for networks with bounded treewidth.

Abstract

Public goods games in undirected networks are generally known to have pure Nash equilibria, which are easy to find. In contrast, we prove that, in directed networks, a broad range of public goods games have intractable equilibrium problems: The existence of pure Nash equilibria is NP-hard to decide, and mixed Nash equilibria are PPAD-hard to find. We define general utility public goods games, and prove a complexity dichotomy result for finding pure equilibria, and a PPAD-completeness proof for mixed Nash equilibria. Even in the divisible goods variant of the problem, where existence is easy to prove, finding the equilibrium is PPAD-complete. Finally, when the treewidth of the directed network is appropriately bounded, we prove that polynomial-time algorithms are possible.