Complexity of Public Goods Games on Graphs

TL;DR

This paper investigates the computational complexity of finding pure Nash equilibria in public goods games on networks, proving NP-completeness for some patterns and providing polynomial algorithms for others.

Contribution

It establishes NP-completeness for certain simple patterns and introduces a polynomial-time algorithm for specific cases, advancing understanding of game complexity on graphs.

Findings

NP-completeness for some patterns of the game

Polynomial-time algorithm for specific pattern T

Open problem for full characterization of complexity

Abstract

We study the computational complexity of "public goods games on networks". In this model, each vertex in a graph is an agent that needs to take a binary decision of whether to "produce a good" or not. Each agent's utility depends on the number of its neighbors in the graph that produce the good, as well as on its own action. This dependence can be captured by a "pattern" $T:{\rm I\!N}\rightarrow\{0,1\}$ that describes an agent's best response to every possible number of neighbors that produce the good. Answering a question of [Papadimitriou and Peng, 2021], we prove that for some simple pattern $T$ the problem of determining whether a non-trivial pure Nash equilibrium exists is NP-complete. We extend our result to a wide class of such $T$, but also find a new polynomial time algorithm for some specific simple pattern $T$. We leave open the goal of characterizing the complexity for all patterns.