### Abstract

As a result of some important works [4–6, 10, 15], the complexity of 2-player Nash equilibrium is by now well understood, even when equilibria with special properties are desired and when the game is symmetric. However, for multi-player games, when equilibria with special properties are desired, the only result known is due to Schaefer and Štefankovič [18]: that checking whether a 3-player NE (3-Nash) instance has an equilibrium in a ball of radius half in l∞-norm is ETR-complete, where ETR is the class Existential Theory of Reals. Building on their work, we show that the following decision versions of 3-Nash are also ETR-complete: checking whether (i) there are two or more equilibria, (ii) there exists an equilibrium in which each player gets at least h payoff, where h is a rational number, (iii) a given set of strategies are played with non-zero probability, and (iv) all the played strategies belong to a given set. Next, we give a reduction from 3-Nash to symmetric 3-Nash, hence resolving an open problem of Papadimitriou [14]. This yields ETRcompleteness for symmetric 3-Nash for the last two problems stated above as well as completeness for the class FIXP_{a}, a variant of FIXP for strong approximation. All our results extend to k-Nash, for any constant k ≥ 3.

Original language | English (US) |
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Title of host publication | Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings |

Editors | Magnus M. Halldorsson, Naoki Kobayashi, Bettina Speckmann, Kazuo Iwama |

Publisher | Springer-Verlag |

Pages | 554-566 |

Number of pages | 13 |

ISBN (Print) | 9783662476710 |

DOIs | |

State | Published - Jan 1 2015 |

Externally published | Yes |

Event | 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 - Kyoto, Japan Duration: Jul 6 2015 → Jul 10 2015 |

### 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 | 9134 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 |
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Country | Japan |

City | Kyoto |

Period | 7/6/15 → 7/10/15 |

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

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## Cite this

*Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings*(pp. 554-566). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9134). Springer-Verlag. https://doi.org/10.1007/978-3-662-47672-7_45