## 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:IN→{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.

Original language | American English |
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Title of host publication | Algorithmic Game Theory - 15th International Symposium, SAGT 2022, Proceedings |

Editors | Panagiotis Kanellopoulos, Maria Kyropoulou, Alexandros Voudouris |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 151-168 |

Number of pages | 18 |

ISBN (Print) | 9783031157134 |

DOIs | |

State | Published - 2022 |

Event | 15th International Symposium on Algorithmic Game Theory, SAGT 2022 - Colchester, United Kingdom Duration: 12 Sep 2022 → 15 Sep 2022 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 13584 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 15th International Symposium on Algorithmic Game Theory, SAGT 2022 |
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Country/Territory | United Kingdom |

City | Colchester |

Period | 12/09/22 → 15/09/22 |

### Bibliographical note

Publisher Copyright:© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

## Keywords

- Computational Complexity
- Nash Equilibrium
- Public Goods