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## How bad is selfish routing? (2002)

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Venue: | JOURNAL OF THE ACM |

Citations: | 651 - 27 self |

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Citation Context ...n arbitrarily large amount. This example is of particular interest because functions of the form ℓ(x) =1/(u−x) arise as the delay functions of M/M/1 queues with capacity or service rate u (see, e.=-=g., [4]-=-) and are therefore common in the networking literature [4, 26, 27, 29, 35]. However, in networks where the largest possible change in edge latency resulting from a single agent rerouting its flow is ... |

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Citation Context ...Our approach to quantifying the inefficiency inherent in a selfishly defined solution (dubbed the “price of anarchy” by Papadimitriou [37]) is inspired by recent work of Koutsoupias and Papadimitriou =-=[28]-=-. In [28], network routing was modeled as a noncooperative game (though with a different model than ours, and only for two-node networks) and the worst-case ratio of the social welfare achieved by a N... |

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Citation Context ...f our results to these assumptions. Related Work Unregulated traffic has been modeled as network flow with all flow paths between a given source-destination pair having equal latency since the 1950’s =-=[3, 52]-=- (see also Knight [24]). Beckman et al. [3], observing that such an equilibrium flow is an optimal solution to a related convex program (see also Section 2), gave existence and uniqueness results for ... |

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Citation Context ...Braess’s Paradox. The addition of an intuitively helpful link can negatively impact all of the users of a congested network. of and extensions to this traffic model have been studied (see for example =-=[1, 11, 16, 21, 32, 33, 34, 35, 41, 43, 47, 49]-=-). In the past several decades, much of the work on this traffic model has been inspired by a “paradox” first discovered by Braess [6] and later reported by Murchland [31] (see also [2] for a non-tech... |

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Citation Context ...ibrium if and only if for each i, f (i) minimizes Ci(f) givenf (j) for j �= i. We will focus on the case where for each edge e, x · ℓe(x) is a convex function; under this assumption, results of Rosen =-=[40]-=- imply that a flow at Nash equilibrium must exist and will be essentially unique. Our main result for this model is an analogue of Theorem 3.1. 19sTheorem 5.4 If f is at Nash equilibrium for the finit... |

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Citation Context ...umption, we can view network users as independent agents participating in a non-cooperative game and expect the routes chosen by users to form a Nash equilibrium in the sense of classical game theory =-=[24]-=-. In other words, we assume that each agent uses the minimumlatency path from its source to its destination, given the link congestion caused by the rest of the network users. It is well-known that Na... |

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Citation Context ...ilibrium with that of the optimal assignment of traffic to paths. Our approach to quantifying the inefficiency inherent in a selfishly defined solution (dubbed the “price of anarchy” by Papadimitriou =-=[37]-=-) is inspired by recent work of Koutsoupias and Papadimitriou [28]. In [28], network routing was modeled as a noncooperative game (though with a different model than ours, and only for two-node networ... |

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Citation Context ...Braess’s Paradox. The addition of an intuitively helpful link can negatively impact all of the users of a congested network. of and extensions to this traffic model have been studied (see for example =-=[1, 11, 16, 21, 32, 33, 34, 35, 41, 43, 47, 49]-=-). In the past several decades, much of the work on this traffic model has been inspired by a “paradox” first discovered by Braess [6] and later reported by Murchland [31] (see also [2] for a non-tech... |

213 | The price of anarchy is independent of the network topology.
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Citation Context ...e prove that if (G, r, ℓ) is an instance with linear latency functions, then ρ(G, r, ℓ) ≤ 4. The upper bound of Corollary 2.8 is not best possible for higher-degree 3 polynomials, either; Roughgarden =-=[44]-=- has recently shown that if all latency functions of an instance (G, r, ℓ) are polynomials with degree at most p, thenρ(G, r, ℓ) =O( p ln p )(see Section 6 for details). On the other hand, for each va... |

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Citation Context ...16, 17] for a survey). Since these early works, many properties 2 This approach is thus in the spirit of the analyses of online algorithms via resource augmentation given by Kalyanasundaram and Pruhs =-=[23] a-=-nd Phillips et al. [39]. 3sl (x) = x v s t l (x) = 1 w (a) Before l (x) = 1 l (x) = x l (x) = x v s l (x) = 0 t l(x) = 1 l(x) = x w (b) After l (x) = 1 Figure 1: Braess’s Paradox. The addition of an... |

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Citation Context ...3], observing that such an equilibrium flow is an optimal solution to a related convex program (see also Section 2), gave existence and uniqueness results for traffic equilibria. Dafermos and Sparrow =-=[13]-=- were perhaps the first authors interested in computing the equilibrium efficiently, and many subsequent papers gave increasingly efficient methods for computing equilibria (see [16, 17] for a survey)... |

172 | Tight bounds for worst-case equilibria
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Citation Context ...three or more parallel links; tight results have recently been obtained for parallel networks with any number of links by Mavronicolas and Spirakis [30] (for a special case) and by Czumaj and Vöcking =-=[10]-=- (for the general case). Equilibria in Other Settings Braess’s Paradox is not particular to traffic in networks; perhaps the most compelling analogue occurs in a mechanical network of strings and spri... |

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Citation Context ...network performance. For example, existing Internet protocols place little restriction on how network traffic is routed, allowing network users to make decisions in a selfish or even malicious manner =-=[3]-=-. The central question of this paper is how much does network performance suffer from this lack of regulation ? As a first step toward formalizing this question mathematically, we assume that, in the ... |

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Citation Context ...nce these early works, many properties 2 This approach is thus in the spirit of the analyses of online algorithms via resource augmentation given by Kalyanasundaram and Pruhs [23] and Phillips et al. =-=[39]-=-. 3sl (x) = x v s t l (x) = 1 w (a) Before l (x) = 1 l (x) = x l (x) = x v s l (x) = 0 t l(x) = 1 l(x) = x w (b) After l (x) = 1 Figure 1: Braess’s Paradox. The addition of an intuitively helpful link... |

132 | The price of selfish routing
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Citation Context ...works and partial results for twonode networks with three or more parallel links; tight results have recently been obtained for parallel networks with any number of links by Mavronicolas and Spirakis =-=[30]-=- (for a special case) and by Czumaj and Vöcking [10] (for the general case). Equilibria in Other Settings Braess’s Paradox is not particular to traffic in networks; perhaps the most compelling analogu... |

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Citation Context ...Braess’s Paradox. The addition of an intuitively helpful link can negatively impact all of the users of a congested network. of and extensions to this traffic model have been studied (see for example =-=[1, 11, 16, 21, 32, 33, 34, 35, 41, 43, 47, 49]-=-). In the past several decades, much of the work on this traffic model has been inspired by a “paradox” first discovered by Braess [6] and later reported by Murchland [31] (see also [2] for a non-tech... |

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Citation Context ...sumptions. Related Work Unregulated traffic has been modeled as network flow with all flow paths between a given source-destination pair having equal latency since the 1950’s [3, 52] (see also Knight =-=[24]-=-). Beckman et al. [3], observing that such an equilibrium flow is an optimal solution to a related convex program (see also Section 2), gave existence and uniqueness results for traffic equilibria. Da... |

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Citation Context .... fP Letting δ tend to 0, continuity and monotonicity of the edge latency functions give the following useful characterization of a flow at Nash equilibrium, occasionally called a Wardrop equilibrium =-=[22]-=- or Wardrop’s Principle [50, 51] in the literature, due to an influential paper of Wardrop [52]. Lemma 2.2 Aflowf feasible for instance (G, r, ℓ) is at Nash equilibrium if and only if for every i ∈{1,... |

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Citation Context ...ongestion caused by the rest of the network users. It is well-known that Nash equilibria do not in general optimize social welfare; perhaps the most famous example is that of “The Prisoner’s Dilemma” =-=[12, 28]-=-. We are then interested in comparing the total latency of a Nash equilibrium with that of the optimal assignment of traffic to paths. This line of research was recently initiated by Koutsoupias and P... |

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Citation Context ...Paradox (even in its worst-possible manifestation) is algorithmically difficult [42]. In a related model with finitely many agents, each controlling a strictly positive amount of flow, Korilis et al. =-=[26, 27]-=- studied strategies for adding new edges and/or capacity to a network that guarantee an improvement in network performance. In contrast to this previous work, we are interested in quantifying the diff... |

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Citation Context ...work performance [18, 51], discovered new types of “paradoxes” [12, 15, 20, 48, 50], and proved that detecting Braess’s Paradox (even in its worst-possible manifestation) is algorithmically difficult =-=[42]-=-. In a related model with finitely many agents, each controlling a strictly positive amount of flow, Korilis et al. [26, 27] studied strategies for adding new edges and/or capacity to a network that g... |

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Citation Context ...Paradox (even in its worst-possible manifestation) is algorithmically difficult [42]. In a related model with finitely many agents, each controlling a strictly positive amount of flow, Korilis et al. =-=[26, 27]-=- studied strategies for adding new edges and/or capacity to a network that guarantee an improvement in network performance. In contrast to this previous work, we are interested in quantifying the diff... |

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Citation Context ...ntly, and many subsequent papers gave increasingly efficient methods for computing equilibria (see [10] for a survey); others have extended these results to more sophisticated models (see for example =-=[1, 6, 10, 13, 21, 22, 27, 29, 30]). In the -=-past several decades, much of the work on this model has been inspired by a "paradox" first discovered by Braess [4] and later reported by Murchland [20]. The essence of Braess's Paradox is ... |

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Citation Context ...hibited in two-terminal electrical networks [7] (where our results give analogous bounds on the largest possible increase in conductivity obtainable by removing conducting links) and queuing networks =-=[8]-=-. Organization In Section 2 we give a formal definition of our network model and of flows at Nash equilibrium, and state several lemmas needed for our main results. In Section 3 we prove our main bicr... |

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Citation Context ...tination have equal (and smallest possible) latency—if a flow does not have this property, some agent can improve its travel time by switching from a longer flow path to a shorter one. Beckman et al=-=. [3]-=- showed that if the latency of each network link is a continuous nondecreasing function of the flow on the link, then a flow corresponding to a Nash equilibrium always exists and moreover all such flo... |

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Citation Context ...died (see for example [1, 11, 16, 21, 32, 33, 34, 35, 41, 43, 47, 49]). In the past several decades, much of the work on this traffic model has been inspired by a “paradox” first discovered by Braess =-=[6]-=- and later reported by Murchland [31] (see also [2] for a non-technical account). The essence of Braess’s Paradox is captured by the example shown in Figure 1, where the edges are labeled with their l... |

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Citation Context ...ermos and Sparrow [13] were perhaps the first authors interested in computing the equilibrium efficiently, and many subsequent papers gave increasingly efficient methods for computing equilibria (see =-=[16, 17]-=- for a survey). Since these early works, many properties 2 This approach is thus in the spirit of the analyses of online algorithms via resource augmentation given by Kalyanasundaram and Pruhs [23] an... |

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Citation Context ..., 32, 33, 34, 35, 41, 43, 47, 49]). In the past several decades, much of the work on this traffic model has been inspired by a “paradox” first discovered by Braess [6] and later reported by Murchland =-=[31]-=- (see also [2] for a non-technical account). The essence of Braess’s Paradox is captured by the example shown in Figure 1, where the edges are labeled with their latency functions (each a function of ... |

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Citation Context ...s optimal for a convex program of the form (NLP) if and only if for every i ∈{1,...,k} and P1,P2 ∈Pi with fP1 > 0, c ′ P1 (f) ≤ c′ P2 (f). 4 For a formal derivation via the Karush-Kuhn-Tucker Theorem =-=[38]-=-, see [3, 13]. 8sRemark: To see that this characterization still makes sense under the weaker assumption of continuous (and not necessarily differentiable) latency functions, define c− e (x),c+ e (x) ... |

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Citation Context ...w roads in congested cities [25, 31], researchers attempted to classify networks in which the addition of a single link could degrade network performance [18, 51], discovered new types of “paradoxes” =-=[12, 15, 20, 48, 50]-=-, and proved that detecting Braess’s Paradox (even in its worst-possible manifestation) is algorithmically difficult [42]. In a related model with finitely many agents, each controlling a strictly pos... |

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Citation Context ...nterest because functions of the form ℓ(x) =1/(u−x) arise as the delay functions of M/M/1 queues with capacity or service rate u (see, e.g., [4]) and are therefore common in the networking literature =-=[4, 26, 27, 29, 35]-=-. However, in networks where the largest possible change in edge latency resulting from a single agent rerouting its flow is not too large, we can apply the results of Subsection 5.1 to derive the fol... |

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Citation Context ...a flow minimizing the total latency)? Our Results In networks in which the latency of each edge is a linear function of the edge congestion (a model that has been the focus of several previous papers =-=[7, 18, 51]-=-), we show that a flow at Nash equilibrium has total latency at most 4 times that of the optimal flow. We give 3 examples showing that this result is tight. We also consider the model in which link la... |

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Citation Context ...ermos and Sparrow [13] were perhaps the first authors interested in computing the equilibrium efficiently, and many subsequent papers gave increasingly efficient methods for computing equilibria (see =-=[16, 17]-=- for a survey). Since these early works, many properties 2 This approach is thus in the spirit of the analyses of online algorithms via resource augmentation given by Kalyanasundaram and Pruhs [23] an... |

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Citation Context ... of our results to these assumptions. Related Work Unregulated tra#c has been modeled as network flow with all flow paths between a given source-destination pair having equal latency since the 1950's =-=[3, 52]-=- (see also Knight [24]). Beckman et al. [3], observing that such an equilibrium flow is an optimal solution to a related convex program (see also Section 2), gave existence and uniqueness results for ... |

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Citation Context ...he agents! Motivated by the discovery of Braess’s Paradox and evidence of similarly counterintuitive and counterproductive traffic behavior following the construction of new roads in congested cities =-=[25, 31]-=-, researchers attempted to classify networks in which the addition of a single link could degrade network performance [18, 51], discovered new types of “paradoxes” [12, 15, 20, 48, 50], and proved tha... |

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Citation Context ...a flow minimizing the total latency)? Our Results In networks in which the latency of each edge is a linear function of the edge congestion (a model that has been the focus of several previous papers =-=[7, 18, 51]-=-), we show that a flow at Nash equilibrium has total latency at most 4 times that of the optimal flow. We give 3 examples showing that this result is tight. We also consider the model in which link la... |

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Citation Context ...roportional to its congestion; this result is implicit in the work of Dafermos and Sparrow [13], and other properties of this special case have been investigated in the context of electrical networks =-=[5, 9]-=-. Corollary 4.2 Let G be a network in which each edge latency function ℓe is of the form ℓe(x) =aex. Then for any rate vector r, aflowfeasiblefor(G, r, ℓ) is optimal if and only if it is at Nash equil... |

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Citation Context ...w roads in congested cities [25, 31], researchers attempted to classify networks in which the addition of a single link could degrade network performance [18, 51], discovered new types of “paradoxes” =-=[12, 15, 20, 48, 50]-=-, and proved that detecting Braess’s Paradox (even in its worst-possible manifestation) is algorithmically difficult [42]. In a related model with finitely many agents, each controlling a strictly pos... |

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Citation Context ...a flow minimizing the total latency)? Our Results In networks in which the latency of each edge is a linear function of the edge congestion (a model that has been the focus of several previous papers =-=[7, 18, 51]-=-), we show that a flow at Nash equilibrium has total latency at most 4 times that of the optimal flow. We give 3 examples showing that this result is tight. We also consider the model in which link la... |

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Citation Context ...w roads in congested cities [25, 31], researchers attempted to classify networks in which the addition of a single link could degrade network performance [18, 51], discovered new types of "parado=-=xes" [12, 15, 20, 48, 50]-=-, and proved that detecting Braess's Paradox (even in its worst-possible manifestation) is algorithmically di#cult [42]. In a related model with finitely many agents, each controlling a strictly posit... |

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Citation Context ...21, 32, 33, 34, 35, 41, 43, 47, 49]). In the past several decades, much of the work on this tra#c model has been inspired by a "paradox" first discovered by Braess [6] and later reported by =-=Murchland [31]-=- (see also [2] for a non-technical account). The essence of Braess's Paradox is captured by the example shown in Figure 1, where the edges are labeled with their latency functions (each a function of ... |

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Citation Context ...etwork flow, with the amount of flow between a pair of nodes in the network equal to the rate of traffic between the two nodes. A Nash equilibrium in the aforementioned noncooperative game 1 Friedman =-=[19]-=- has pointed out that this equilibrium arises not only in settings where network users can select their own paths, but also in network protocols that determine routes for users via a shortest path com... |

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Citation Context ...roportional to its congestion; this result is implicit in the work of Dafermos and Sparrow [13], and other properties of this special case have been investigated in the context of electrical networks =-=[5, 9]-=-. Corollary 4.2 Let G be a network in which each edge latency function ℓe is of the form ℓe(x) =aex. Then for any rate vector r, aflowfeasiblefor(G, r, ℓ) is optimal if and only if it is at Nash equil... |

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- Roughgarden, Tardos
(Show Context)
Citation Context ... simplicity only), we may apply the Karush-Kuhn-Tucker Theorem (see, e.g., [25]) to a convex program of the form (NLP ) to derive the following characterization of optimal flows (see the full version =-=[28]-=- for details): Lemma 2.4 A flow f is optimal for a convex program of the form (NLP ) if and only if for every i # {1, . . . , k} and P 1 , P 2 # P i with fP1 > 0, c # P1 (f) # c # P2 (f). The striking... |

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Bounding the inefficiency of Nash equilibria
- Roughgarden, Tardos
- 2002
(Show Context)
Citation Context ... remark that Theorems 3.1 and 4.5 can be extended to a broader class of games (including games without the structure provided by a network); this direction of research is pursued in a companion paper =-=[46]-=-. 5.1 Flows at Approximate Nash Equilibrium It is unreasonable to expect agents to be able to evaluate the latency of different paths with arbitrary precision. We next investigate the sensitivity of o... |

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Characterizing Braess's paradox for tra#c networks
- Hagstrom, Abrams
- 2001
(Show Context)
Citation Context ...w roads in congested cities [25, 31], researchers attempted to classify networks in which the addition of a single link could degrade network performance [18, 51], discovered new types of "parado=-=xes" [12, 15, 20, 48, 50]-=-, and proved that detecting Braess's Paradox (even in its worst-possible manifestation) is algorithmically di#cult [42]. In a related model with finitely many agents, each controlling a strictly posit... |

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Bounding the ine#ciency of Nash equilibria
- Roughgarden, Tardos
- 2002
(Show Context)
Citation Context ... remark that Theorems 3.1 and 4.5 can be extended to a broader class of games (including games without the structure provided by a network); this direction of research is pursued in a companion paper =-=[46]-=-. 5.1 Flows at Approximate Nash Equilibrium It is unreasonable to expect agents to be able to evaluate the latency of di#erent paths with arbitrary precision. We next investigate the sensitivity of ou... |

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