Complexity of equilibria in binary public goods games on undirected graphs

TL;DR

This paper analyzes the computational complexity of finding equilibria in binary public goods games on undirected graphs, revealing polynomial-time existence and convergence for decreasing patterns, but NP-hardness and PLS-completeness for certain non-monotonic patterns and weighted extensions.

Contribution

It establishes complexity results for equilibria existence and computation in various classes of public goods games on graphs, including new hardness and convergence proofs.

Findings

Pure Nash equilibria always exist for decreasing patterns and can be reached via polynomial best-response sequences.

Deciding the existence of pure Nash equilibria for non-monotonic patterns is NP-hard.

Computing equilibria in weighted models with decreasing patterns is PLS-complete.

Abstract

We study the complexity of computing equilibria in binary public goods games on undirected graphs. In such a game, players correspond to vertices in a graph and face a binary choice of performing an action, or not. Each player's decision depends only on the number of neighbors in the graph who perform the action and is encoded by a per-player binary pattern. We show that games with decreasing patterns (where players only want to act up to a threshold number of adjacent players doing so) always have a pure Nash equilibrium and that one is reached from any starting profile by following a polynomially bounded sequence of best responses. For non-monotonic patterns of the form $10^k10^*$ (where players want to act alone or alongside $k + 1$ neighbors), we show that it is $\mathsf{NP}$-hard to decide whether a pure Nash equilibrium exists. We further investigate a generalization of the model that permits ties of varying strength: an edge with integral weight $w$ behaves as $w$ parallel edges. While, in this model, a pure Nash equilibrium still exists for decreasing patters, we show that the task of computing one is $\mathsf{PLS}$-complete.