## Abstract

One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ε-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ε-approximate equilibria are contained inside a small ball of radius Δ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximation-stable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ε,Δ) approximation-stable games must have an ε-equilibrium of support , yielding an immediate -time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that Δ and ε are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.

Original language | English |
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Title of host publication | Algorithmic Game Theory - Third International Symposium, SAGT 2010, Proceedings |

Pages | 78-89 |

Number of pages | 12 |

Edition | M4D |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

Event | 3rd International Symposium on Algorithmic Game Theory, SAGT 2010 - Athens, Greece Duration: 18 Oct 2010 → 20 Oct 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | M4D |

Volume | 6386 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 3rd International Symposium on Algorithmic Game Theory, SAGT 2010 |
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Country/Territory | Greece |

City | Athens |

Period | 18/10/10 → 20/10/10 |

### Bibliographical note

Funding Information:This work was supported in part by NSF grants CCF-0830540 and CCF-0953192, ONR grant N00014-09-1-0751, and AFOSR grant FA9550-09-1-0538.

### Funding

This work was supported in part by NSF grants CCF-0830540 and CCF-0953192, ONR grant N00014-09-1-0751, and AFOSR grant FA9550-09-1-0538.

Funders | Funder number |
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National Science Foundation | CCF-0830540, CCF-0953192 |

Office of Naval Research | N00014-09-1-0751 |

Air Force Office of Scientific Research | FA9550-09-1-0538 |