## Convergence to Approximate Nash Equilibria in Congestion Games (2007)

Venue: | In SODA ’07 |

Citations: | 54 - 0 self |

### BibTeX

@INPROCEEDINGS{Chien07convergenceto,

author = {Steve Chien and Alistair Sinclair},

title = {Convergence to Approximate Nash Equilibria in Congestion Games},

booktitle = {In SODA ’07},

year = {2007},

pages = {169--178}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

633 |
Equilibrium points in n-person games
- Nash
- 1950
(Show Context)
Citation Context ...con Valley Campus.s1 Introduction The emerging field of algorithmic game theory has led to a fundamental re-examination, from a computational perspective, of the classical concept of Nash equilibrium =-=[20]-=-. Much of this activity has focused on understanding the structure of Nash equilibria (as expressed, notably, in the "price of anarchy," see e.g. [22, 25, 24]) and the computational complexity of find... |

392 |
A class of games possessing pure-strategy Nash equilibria
- Rosenthal
- 1973
(Show Context)
Citation Context ...s have attracted a good deal of attention, partly because they capture a large class of routing and resource allocation scenarios, and not least because they are known to possess pure Nash equilibria =-=[23]-=-. Thus unlike general games, whose Nash equilibria may involve mixed (i.e., randomized) strategies for the players, congestion games always have a Nash equilibrium in which each player sticks to a sin... |

344 | Potential games - Monderer, Shapley - 1996 |

224 | The Complexity of Computing a Nash Equilibrium
- Daskalakis, Goldberg, et al.
- 2005
(Show Context)
Citation Context ...ty has focused on understanding the structure of Nash equilibria (as expressed, notably, in the "price of anarchy," see e.g. [22, 25, 24]) and the computational complexity of finding them (see, e.g., =-=[10, 7, 4]-=-). Considerably less is understood about the question of whether selfish players, acting in a decentralized fashion, actually arrive at a Nash equilibrium in a reasonable amount of time. This would se... |

196 |
How easy is local search
- Johnson, Papadimitriou, et al.
- 1988
(Show Context)
Citation Context ...tate, does the Nash dynamics converge rapidly? The work of [10] provides a devastating negative answer, even for symmetric# congestion games: the problem of finding a Nash equilibrium is PLS-complete =-=[14]-=-, and therefore as difficult as that of finding a local optimum in any local search problem with efficiently computable neighborhoods. Moreover, there are examples of games and initial strategies such... |

170 | Selfish Routing and the Price of Anarchy
- Roughgarden
- 2005
(Show Context)
Citation Context ...ive, of the classical concept of Nash equilibrium [20]. Much of this activity has focused on understanding the structure of Nash equilibria (as expressed, notably, in the "price of anarchy," see e.g. =-=[22, 25, 24]-=-) and the computational complexity of finding them (see, e.g., [10, 7, 4]). Considerably less is understood about the question of whether selfish players, acting in a decentralized fashion, actually a... |

141 | The complexity of pure Nash equilibria
- Fabrikant, Papadimitriou, et al.
- 2004
(Show Context)
Citation Context ...ty has focused on understanding the structure of Nash equilibria (as expressed, notably, in the "price of anarchy," see e.g. [22, 25, 24]) and the computational complexity of finding them (see, e.g., =-=[10, 7, 4]-=-). Considerably less is understood about the question of whether selfish players, acting in a decentralized fashion, actually arrive at a Nash equilibrium in a reasonable amount of time. This would se... |

136 |
Congestion games with player specific payoff functions
- Milchtaich
- 1996
(Show Context)
Citation Context ...ncing" games, which are restricted congestion games in which each strategy consists of just a single edge (or "machine"), but which may be 3sgeneralized to allow either player-specific cost functions =-=[17]-=- or weights on the players [8, 9, 13]. Milchtaich [17], Even-Dar et al. [8] and Goldberg [13] establish polynomial time convergence for versions of the Nash dynamics to (exact or approximate) Nash equ... |

135 | Algorithms, games and the internet
- Papadimitriou
- 2001
(Show Context)
Citation Context ...ive, of the classical concept of Nash equilibrium [20]. Much of this activity has focused on understanding the structure of Nash equilibria (as expressed, notably, in the "price of anarchy," see e.g. =-=[22, 25, 24]-=-) and the computational complexity of finding them (see, e.g., [10, 7, 4]). Considerably less is understood about the question of whether selfish players, acting in a decentralized fashion, actually a... |

117 | The price of anarchy in finite congestion games
- Christodoulou, Koutsoupias
- 2005
(Show Context)
Citation Context ...e studies of transient behavior (e.g., the "bounded relative slope" of [11] or "bounded slope" of [3]); it is also much weaker than the polynomial bounds typically used in studies of price of anarchy =-=[1, 6]-=-. Thirdly, it is questionable how much sense it makes to talk about "symmetric" congestion games without such a condition. This is because of the trick in [10] (see Section 3.1 below) for making any c... |

104 |
Settling the Complexity of 2-Player Nash-Equilibrium
- Chen, Deng
- 2005
(Show Context)
Citation Context ...ty has focused on understanding the structure of Nash equilibria (as expressed, notably, in the "price of anarchy," see e.g. [22, 25, 24]) and the computational complexity of finding them (see, e.g., =-=[10, 7, 4]-=-). Considerably less is understood about the question of whether selfish players, acting in a decentralized fashion, actually arrive at a Nash equilibrium in a reasonable amount of time. This would se... |

101 | The price of routing unsplittable flow
- Awerbuch, Azar, et al.
- 2005
(Show Context)
Citation Context ...e studies of transient behavior (e.g., the "bounded relative slope" of [11] or "bounded slope" of [3]); it is also much weaker than the polynomial bounds typically used in studies of price of anarchy =-=[1, 6]-=-. Thirdly, it is questionable how much sense it makes to talk about "symmetric" congestion games without such a condition. This is because of the trick in [10] (see Section 3.1 below) for making any c... |

99 | Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions - Vetta - 2002 |

91 | Playing large games using simple strategies
- Lipton, Markakis, et al.
- 2003
(Show Context)
Citation Context ...act converge rapidly to an "-Nash equilibrium. #A symmetric game is one in which the allowed strategies of all the players are the same. ##An alternative notion of approximate equilibrium (see, e.g., =-=[8, 10, 15, 16]-=-) is based on an additive error of ", rather than the relative error we use here. We would argue that our definition is equally natural, and indeed more in line with approximation guarantees in Comput... |

82 | Simple local search problems that are hard to solve - er, Yannakakis - 1991 |

79 | Convergence time to Nash equilibria
- Even-Dar, Kesselman, et al.
- 2003
(Show Context)
Citation Context ...act converge rapidly to an "-Nash equilibrium. #A symmetric game is one in which the allowed strategies of all the players are the same. ##An alternative notion of approximate equilibrium (see, e.g., =-=[8, 10, 15, 16]-=-) is based on an additive error of ", rather than the relative error we use here. We would argue that our definition is equally natural, and indeed more in line with approximation guarantees in Comput... |

68 | Computing Nash Equilibria: Approximation and smoothed complexity - Chen, Deng, et al. - 2006 |

65 | Sink equilibria and convergence
- Goemans, Mirrokni, et al.
- 2005
(Show Context)
Citation Context ...pling", while Blum et al. [3] give polynomial bounds when players use no-regret online learning algorithms. We mention also two recent developments for more general games. Goemans, Mirrokni and Vetta =-=[12, 18]-=- study convergence of Nash dynamics not to an (approximate) Nash equilibrium but instead to a "sink equilibrium", for wider classes of games for which pure equilibria need not exist. They quantify the... |

45 | Routing without regret: On convergence to Nash equilibria of regret-minimizing algorithms in routing games
- Blum, Even-Dar, et al.
- 2006
(Show Context)
Citation Context ...tion is a reasonable assumption in practice, and is similar to conditions imposed in other quantitative studies of transient behavior (e.g., the "bounded relative slope" of [11] or "bounded slope" of =-=[3]-=-); it is also much weaker than the polynomial bounds typically used in studies of price of anarchy [1, 6]. Thirdly, it is questionable how much sense it makes to talk about "symmetric" congestion game... |

41 | Fast Convergence to Wardrop Equilibria by Adaptive Sampling Methods
- Fischer, Räcke, et al.
- 2006
(Show Context)
Citation Context ...mes, under a "bounded influence" assumption; however, this assumption appears not to hold for the general multiple-resource congestion games we consider. Recent papers by Fischer, R"acke and V"ocking =-=[11]-=- and Blum, Even-Dar and Ligett [3] consider congestion games at a similar level of generality to ours, each with some version of a "bounded (relative) slope" assumption that is analogous to bounded ju... |

38 | Convergence issues in competitive games
- Mirrokni, Vetta
- 2004
(Show Context)
Citation Context ...pling", while Blum et al. [3] give polynomial bounds when players use no-regret online learning algorithms. We mention also two recent developments for more general games. Goemans, Mirrokni and Vetta =-=[12, 18]-=- study convergence of Nash dynamics not to an (approximate) Nash equilibrium but instead to a "sink equilibrium", for wider classes of games for which pure equilibria need not exist. They quantify the... |

25 |
Fast Convergence of Selfish Rerouting
- Even-Dar, Mansour
(Show Context)
Citation Context ...ted congestion games in which each strategy consists of just a single edge (or "machine"), but which may be 3sgeneralized to allow either player-specific cost functions [17] or weights on the players =-=[8, 9, 13]-=-. Milchtaich [17], Even-Dar et al. [8] and Goldberg [13] establish polynomial time convergence for versions of the Nash dynamics to (exact or approximate) Nash equilibria in these games, while EvenDar... |

23 | Efficient Nash computation in large population games with bounded influence
- Kearns, Mansour
(Show Context)
Citation Context ...act converge rapidly to an "-Nash equilibrium. #A symmetric game is one in which the allowed strategies of all the players are the same. ##An alternative notion of approximate equilibrium (see, e.g., =-=[8, 10, 15, 16]-=-) is based on an additive error of ", rather than the relative error we use here. We would argue that our definition is equally natural, and indeed more in line with approximation guarantees in Comput... |

20 |
Bounds for the Convergence Rate of Randomized Local Search in a Multiplayer Load-balancing Game
- Goldberg
(Show Context)
Citation Context ...ted congestion games in which each strategy consists of just a single edge (or "machine"), but which may be 3sgeneralized to allow either player-specific cost functions [17] or weights on the players =-=[8, 9, 13]-=-. Milchtaich [17], Even-Dar et al. [8] and Goldberg [13] establish polynomial time convergence for versions of the Nash dynamics to (exact or approximate) Nash equilibria in these games, while EvenDar... |

17 | Approximate local search in combinatorial optimization
- Orlin, Punnen, et al.
- 2004
(Show Context)
Citation Context ...d explicitly presented strategies was given by Lipton, Markakis and Mehta [16]. Finally we note that Theorem 1.1 is reminiscent of, and partially inspired by, recent work of Orlin, Punnen, and Schulz =-=[21]-=-, who show how to find an "-approximate local minimum for any problem in PLS. However, our setting differs from theirs in two crucial respects. Firstly, their algorithm finds an "-local minimum of the... |