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A Characterization of Complexity in Public Goods Games

Matan Gilboa

TL;DR

This work fully characterizes the computational complexity of finding non-trivial pure Nash equilibria in binary public goods games on graphs for all finite best-response patterns. By constructing a suite of gadgets (Clause, Negation, Copy, Force-1, Add-1) and performing reductions from ONE-IN-THREE 3SAT, the authors prove NP-completeness of NTPNE$(T)$ for every finite non-monotone BRP, solving open problems posed by prior work. The results are organized into targeted hardness proofs: the 0-Or-2-Neighbors pattern, all semi-sharp patterns, and all spiked patterns, with a unifying framework using pattern halving and double-pattern arguments. This completes the dichotomy between tractable monotone patterns and intractable non-monotone ones, and provides a solid foundation for understanding equilibrium complexity in homogeneous, strict public goods games on graphs. The findings have implications for modeling and algorithmic analysis of collective production decisions in networks, and they open directions for infinite-pattern and non-strict variants.

Abstract

We complete the characterization of the computational complexity of equilibrium in public goods games on graphs. In this model, each vertex represents an agent deciding whether to produce a public good, with utility defined by a "best-response pattern" determining the best response to any number of productive neighbors. We prove that the equilibrium problem is NP-complete for every finite non-monotone best-response pattern. This answers the open problem of [Gilboa and Nisan, 2022], and completes the answer to a question raised by [Papadimitriou and Peng, 2021], for all finite best-response patterns.

A Characterization of Complexity in Public Goods Games

TL;DR

This work fully characterizes the computational complexity of finding non-trivial pure Nash equilibria in binary public goods games on graphs for all finite best-response patterns. By constructing a suite of gadgets (Clause, Negation, Copy, Force-1, Add-1) and performing reductions from ONE-IN-THREE 3SAT, the authors prove NP-completeness of NTPNE for every finite non-monotone BRP, solving open problems posed by prior work. The results are organized into targeted hardness proofs: the 0-Or-2-Neighbors pattern, all semi-sharp patterns, and all spiked patterns, with a unifying framework using pattern halving and double-pattern arguments. This completes the dichotomy between tractable monotone patterns and intractable non-monotone ones, and provides a solid foundation for understanding equilibrium complexity in homogeneous, strict public goods games on graphs. The findings have implications for modeling and algorithmic analysis of collective production decisions in networks, and they open directions for infinite-pattern and non-strict variants.

Abstract

We complete the characterization of the computational complexity of equilibrium in public goods games on graphs. In this model, each vertex represents an agent deciding whether to produce a public good, with utility defined by a "best-response pattern" determining the best response to any number of productive neighbors. We prove that the equilibrium problem is NP-complete for every finite non-monotone best-response pattern. This answers the open problem of [Gilboa and Nisan, 2022], and completes the answer to a question raised by [Papadimitriou and Peng, 2021], for all finite best-response patterns.
Paper Structure (7 sections, 17 theorems, 19 equations, 10 figures)

This paper contains 7 sections, 17 theorems, 19 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Model and Definitions
  3. Hardness of the 0-Or-2-Neighbors Pattern
  4. Hardness of Semi-Sharp Patterns
  5. Semi-Sharp Patterns with Isolated Odd 1
  6. All Semi-Sharp Patterns
  7. Hardness of All Spiked Patterns

Key Result

theorem thmcountertheorem

Let $T$ be the 0-Or-2 Neighbors BRP. Then NTPNE($T$) is NP-complete.

Figures (10)

  • Figure 1: Outline of this paper.
  • Figure 2: PNE in cycles.
  • Figure 3: PNE in paths of lengths 2 and 5.
  • Figure 4: No PNE exists in this graph.
  • Figure 5: Clause Gadget.
  • ...and 5 more figures

Theorems & Definitions (42)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • ...and 32 more