## The Structure and Complexity of Nash Equilibria for a Selfish Routing Game (2002)

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### BibTeX

@MISC{Fotakis02thestructure,

author = {Dimitris Fotakis and Spyros Kontogiannis and Elias Koutsoupias and Marios Mavronicolas and Paul Spirakis},

title = {The Structure and Complexity of Nash Equilibria for a Selfish Routing Game},

year = {2002}

}

### Years of Citing Articles

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### Abstract

In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.

### Citations

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Citation Context ... shift gears to study the computational complexity of NASH EQUILIBRIUM SOCIAL COST. We have obtained both negative and positive results here. We first show that the problem is #P-complete (see, e.g., =-=[16]-=-) in general for the case of mixed Nash equilibria (Theorem 8). On the positive side, we get around the established hardness of computing exacly the social cost of any mixed Nash equilibrium by presen... |

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Citation Context ...⎣ (X j ≥ xc j ⎤ m∑ ) ⎦ ≤ hj(x) j=1 j=1 In the following lemma, we obtain an upper bound on the social cost of P in terms of µ ∗ α. 5 We use the following standard form of the Chernoff bound (see e.g. =-=[23]-=-) with 1 + δ = x/Λ j : Let X1, X2, . . . , Xn be independent 0/1 random variables, let X = ∑n i=1 Xi, and let IE[X] denote the expectation of X. Then for any δ > 0, ( IPr[X ≥ (1 + δ)IE[X]] < e δ (1 + ... |

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Citation Context ...wide range of assumptions on the syntax of the Nash equilibria and on the parameters of the system. 1 Introduction Nash equilibrium [14] is arguably the most important solution concept in Game Theory =-=[15]-=-. It may be viewed to represent a steady state of the play of a strategic ⋆ This work has been partially supported by the IST Program of the European Union under contract numbers IST-1999-14186 (ALCOM... |

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Citation Context ...lts for these algorithmic problems. Our results span and contrast a wide range of assumptions on the syntax of the Nash equilibria and on the parameters of the system. 1 Introduction Nash equilibrium =-=[14]-=- is arguably the most important solution concept in Game Theory [15]. It may be viewed to represent a steady state of the play of a strategic ⋆ This work has been partially supported by the IST Progra... |

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Citation Context ...g a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [3]. Similar in motivation and spirit to our paper is the very recent paper by Deng et al. =-=[2]-=-, which proves complexity, approximability and inapproximability results for the problem of computing an exchange equilibrium in markets with indivisible goods. 2 Framework Most of our definitions are... |

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Citation Context ... the supports of the worst (or the best) Nash equilibrium. (These are optimization problems.) • Given a Nash equilibrium, determine its social cost. (This turns out to be a hard counting problem (cf. =-=[36]-=-).) 1Our study distinguishes between pure Nash equilibria, where each user chooses exactly one link (with probability one), and mixed Nash equilibria, where the choices of each user are modeled by a ... |

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Citation Context ...d links is in Nash equilibrium after each such iteration. ⊓⊔ (This nice observation is due to B. Monien [13].) We remark that Apure can be viewed as a variant of Graham’s Longest Processing Time (LPT =-=[4]-=-) algorithm for assigning tasks to identical machines. Nevertheless, since in our case the links may have different capacities, our algorithm instead of choosing the link that will first become idle, ... |

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Citation Context ...equilibria of general bimatrix games in either strategic or extensive form appear in [10, 19]. All known such algorithms incur exponential running time, with the seminal algorithm of Lemke and Howson =-=[8]-=- being the prime example. Issues of computational complexity for the computation of Nash equilibria in general games have been raised by Megiddo [11], Megiddo and Papadimitriou [12], and Papadimitriou... |

158 | Tight bounds for worst-case equilibria
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Citation Context ...e to selfish behavior of the users. (Later studies of the selfish routing game from the same point of view, that of performance, include the works by Koutsoupias et al. [6], and by Czumaj and Vöcking =-=[1]-=-.) Unlike these previous papers, our work considers the selfish routing game from the point of view of computational complexity and attempts to classify certain algorithmic problemssrelated to the com... |

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Citation Context ...being the prime example. Issues of computational complexity for the computation of Nash equilibria in general games have been raised by Megiddo [11], Megiddo and Papadimitriou [12], and Papadimitriou =-=[17]-=-. The N P-hardness of computing a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [3]. Similar in motivation and spirit to our paper is the ver... |

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Citation Context ... [11], Megiddo and Papadimitriou [12], and Papadimitriou [17]. The N P-hardness of computing a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel =-=[3]-=-. Similar in motivation and spirit to our paper is the very recent paper by Deng et al. [2], which proves complexity, approximability and inapproximability results for the problem of computing an exch... |

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Citation Context ...lfish routing game considered in this paper was first introduced by Koutsoupias and Papadimitriou [16] as a vehicle for the study of the price of selfishness for routing over non-cooperative networks =-=[14]-=-, like the Internet. This game was subsequently studied in the work of Mavronicolas and Spirakis [18], where fully mixed Nash equilibria were introduced and analyzed. In both works, the aim had been t... |

119 |
The price of selfish routing
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(Show Context)
Citation Context ...rithm for computing a mixed Nash equilibrium (Theorem 5). More specifically, the algorithm computes a generalized fully mixed Nash equilibrium; this is a generalization of fully mixed Nash equilibria =-=[9]-=-. We continue to establish that for the model of uniform capacities, and assuming that there are only two users, the worst mixed Nash equilibrium (with respect to social cost) is the fully mixed Nash ... |

118 | Computation of Equilibria in Finite Games
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Citation Context ...he (expected) behavior of other players and acts rationally. Despite the apparent simplicity of the concept, computation of Nash equilibria in finite games has been long observed to be difficult (cf. =-=[10, 19]-=-); in fact, it is arguably one of the few, most important algorithmic problems for which no polynomial-time algorithms are known. Indeed, Papadimitriou [18, p. 1] actively advocates the problem of com... |

104 |
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Citation Context ...ilibria. By the linearity of the expected latency cost functions we consider, the celebrated result of Nash [25] on the existence of Nash equilibria, which follows from Kakutani’s Fixed Point Theorem =-=[8]-=- 1 , assures that a mixed, but not necessarily pure, Nash equilibrium always exists. The first result (Theorem 3.1), remarked by Kurt Mehlhorn, establishes that a pure Nash equilibrium always exists. ... |

85 | Game Theory - Myerson - 1991 |

76 |
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Citation Context ... of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [5]. For other algorithmic works on the computation of Nash equilibria, see, e.g., [12, 13]. The book by Scarf =-=[33]-=- is devoted to the computation of Nash equilibria over various economic settings. A similar in motivation and spirit to our paper is the very recent paper by Deng et al. [2], which proves complexity, ... |

54 | Approximate equilibria and ball fusion
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Citation Context ... performance loss in routing due to selfish behavior of the users. (Later studies of the selfish routing game from the same point of view, that of performance, include the works by Koutsoupias et al. =-=[6]-=-, and by Czumaj and Vöcking [1].) Unlike these previous papers, our work considers the selfish routing game from the point of view of computational complexity and attempts to classify certain algorith... |

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Citation Context ...[n] and link ℓ ∈ [m]. 2.2 Algorithmic Problems In this section, we formally define several algorithmic problems related to Nash equilibria. The definitions are given in the style of Garey and Johnson =-=[4]-=-. Π1: NASH EQUILIBRIUM SUPPORTS INSTANCE: A number n of users; a number m of links; for each user i, a rational number wi > 0, called the traffic of user i; for each link j, a rational number cj > 0, ... |

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Citation Context ... max{Λ 1 , m+n−1 2C(m) }, IPr[X j ≥ ⌈ xc j⌉ ] ≥ fj(x) (7) where fj(x) is defined in (5). Using the fact that in “balls and bins” experiments, the occupancy numbers are negatively associated (see e.g. =-=[3]-=-), we obtain that ⎡ m∧ IPr[Lmax < x] = IPr⎣ (X j < ⌈ xc j⌉ ⎤ m∏ [ ) ⎦ ≤ IPr X k < ⌈ xc j⌉] j=1 ≤ ≤ j=1 m∏ (1 − fj(x)) j=1 ⎛ ⎞ m∑ exp ⎝− fj(x) ⎠ For the first inequality, we use that the random variabl... |

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Citation Context ...ve form appear in [19, 35]; see also [24, Section 3.1]. All known such algorithms incur exponential running time, with the seminal algorithm of Lemke and Howson [17] being the prime example; see also =-=[32, 34]-=- for still inefficient extensions. Issues of computational complexity for the computation of Nash equilibria in general games have been raised by Megiddo [20], Megiddo and Papadimitriou [21], and Papa... |

32 |
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Citation Context ...vially in P, since it amounts to computing the maximum. 3 Consider a counting problem Π with solution Π(x) on any instance x. An algorithm A is a fully polynomial-time randomized approximation scheme =-=[9, 10]-=- for Π, or FPRAS for short, if for each instance x, for any error parameter ε > 0, IPr (|A(x) − Π(x)| ≤ εΠ(x)) ≥ 3 1 , and the running time of A is polynomial in |x| and 4 ε . 3arbitrarily). We leave... |

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28 |
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(Show Context)
Citation Context ...ve form appear in [19, 35]; see also [24, Section 3.1]. All known such algorithms incur exponential running time, with the seminal algorithm of Lemke and Howson [17] being the prime example; see also =-=[32, 34]-=- for still inefficient extensions. Issues of computational complexity for the computation of Nash equilibria in general games have been raised by Megiddo [20], Megiddo and Papadimitriou [21], and Papa... |

24 | Finding mixed strategies with small supports in extensive form games
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Citation Context ...computing a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [5]. For other algorithmic works on the computation of Nash equilibria, see, e.g., =-=[12, 13]-=-. The book by Scarf [33] is devoted to the computation of Nash equilibria over various economic settings. A similar in motivation and spirit to our paper is the very recent paper by Deng et al. [2], w... |

17 | A note on the complexity of P-matrix LCP and computing an equilibrium. IBM Research Report 6439. Available at: http://theory.stanford.edu/ megiddo/pdf/plcp.pdf
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(Show Context)
Citation Context ...ime, with the seminal algorithm of Lemke and Howson [8] being the prime example. Issues of computational complexity for the computation of Nash equilibria in general games have been raised by Megiddo =-=[11]-=-, Megiddo and Papadimitriou [12], and Papadimitriou [17]. The N P-hardness of computing a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [3]. ... |

16 | A computer scientist looks at game theory
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(Show Context)
Citation Context ...and inapproximability results for the problem of computing an exchange equilibrium in markets with indivisible goods. A general account on overlaps between Computer Science and Game Theory appears in =-=[7]-=-. 2 Framework Most of our definitions are patterned after those in [16, Sections 1 & 2] and [18, Section 2]. 2.1 Notation Throughout, denote for any integer m ≥ 2, [m] = {1, . . . , m}. For an event E... |

7 |
On Total Functions, Existence Theorems and
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(Show Context)
Citation Context ...of Lemke and Howson [8] being the prime example. Issues of computational complexity for the computation of Nash equilibria in general games have been raised by Megiddo [11], Megiddo and Papadimitriou =-=[12]-=-, and Papadimitriou [17]. The N P-hardness of computing a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [3]. Similar in motivation and spirit... |

7 | On the Complexity of the Parity Argument and Other Inecient Proofs of Existence - Papadimitriou - 1994 |

5 | The Price of Sel Routing - Mavronicolas, Spirakis - 2001 |

1 |
Computing Equlibria for Two-Person Games
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Citation Context ...he (expected) behavior of other players and acts rationally. Despite the apparent simplicity of the concept, computation of Nash equilibria in finite games has been long observed to be difficult (cf. =-=[10, 19]-=-); in fact, it is arguably one of the few, most important algorithmic problems for which no polynomial-time algorithms are known. Indeed, Papadimitriou [18, p. 1] actively advocates the problem of com... |

1 |
Fast Algorithms for Finding Randomized Stategies in Game Trees
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Citation Context ...computing a Nash equilibrium of a general bimatrix game with maximum payoff has been established by Gilboa and Zemel [5]. For other algorithmic works on the computation of Nash equilibria, see, e.g., =-=[12, 13]-=-. The book by Scarf [33] is devoted to the computation of Nash equilibria over various economic settings. A similar in motivation and spirit to our paper is the very recent paper by Deng et al. [2], w... |

1 | http://www.cs.berkeley.edu/ christos/games/cs294.html for the course CS294: Seminar on Algorithmic Aspects of Game Theory, taught by C. H. Papadimitriou at - page - 2001 |

1 | http://www.abeautifulmind.com for the movie A Beautiful Mind - page |