### Abstract

The celebrated PPAD hardness result for finding an exact Nash equilibrium in a two-player game initiated a quest for finding approximate Nash equilibria efficiently, and is one of the major open questions in algorithmic game theory. We study the computational complexity of finding an ε-approximate Nash equilibrium with good social welfare. Hazan and Krauthgamer and subsequent improvements showed that finding an e-approximate Nash equilibrium with good social welfare in a two player game and many variants of this problem is at least as hard as finding a planted clique of size O (1ogn) in the random graph Q (n, 1/2). We show that any polynomial time algorithm that finds an ε-approximate Nash equilibrium with good social welfare refutes (the worst-case) Exponential Time Hypothesis by Impagliazzo and Paturi, confirming the recent conjecture by Aaronson, Impagliazzo and Moshkovitz. Specifically, it would imply a 2O (n^{1/2})algorithm for SAT. Our lower bound matches the quasi-polynomial time algorithm by Lipton, Markakis and Mehta for solving the problem. Our key tool is a reduction from the PCP machinery to finding Nash equilibrium via free games, the framework introduced in the recent work by Aaronson, Impagliazzo and Moshkovitz. Techniques developed in the process may be useful for replacing planted clique hardness with ETH-hardness in other applications.

Original language | English (US) |
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Title of host publication | Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |

Publisher | Association for Computing Machinery |

Pages | 970-982 |

Number of pages | 13 |

Edition | January |

ISBN (Electronic) | 9781611973747 |

State | Published - Jan 1 2015 |

Event | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States Duration: Jan 4 2015 → Jan 6 2015 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Number | January |

Volume | 2015-January |

### Other

Other | 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 |
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Country | United States |

City | San Diego |

Period | 1/4/15 → 1/6/15 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

^{o (1ogn)}-time breaks the exponential time hypothesis. In

*Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015*(January ed., pp. 970-982). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 2015-January, No. January). Association for Computing Machinery.