## Approximate Equilibria and Ball Fusion (2002)

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Venue: | Theory of Computing Systems |

Citations: | 62 - 26 self |

### BibTeX

@ARTICLE{Koutsoupias02approximateequilibria,

author = {Elias Koutsoupias and Marios Mavronicolas and Paul Spirakis},

title = {Approximate Equilibria and Ball Fusion},

journal = {Theory of Computing Systems},

year = {2002},

volume = {36},

pages = {2003}

}

### Years of Citing Articles

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### Abstract

We consider sel sh routing over a network consisting of m parallel links through which n sel sh users route their tra c trying to minimize their own expected latency. Westudy the class of mixed strategies in which the expected latency through each link is at most a constant multiple of the optimum maximum latency had global regulation been available. For the case of uniform links it is known that all Nash equilibria belong to this class of strategies. We areinterested in bounding the coordination ratio (or price of anarchy) of these strategies de ned as the worst-case ratio of the maximum (over all links) expected latency over the optimum maximum latency. The load balancing aspect of the problem immediately implies a lower bound; lnm ln lnm of the coordination ratio. We give a tight (uptoamultiplicative constant) upper bound. To show the upper bound, we analyze a variant ofthe classical balls and bins problem, in which balls with arbitrary weights are placed into bins according to arbitrary probability distributions. At the heart of our approach is a new probabilistic tool that we call

### Citations

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1870 | A course in game theory
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(Show Context)
Citation Context ...ity to monitor and regulate network operation according to global objectives. A natural framework in which to study such multiobjective problems with sel sh objectives is (noncooperative) Game Theory =-=[17]-=-. Several notions of equilibria for noncooperative games have been de ned and studied in the classical literature of Game Theory� the most famous of them is Nash equilibrium, originally de ned in John... |

1551 | Probability inequalities of Sums of Bounded Random Variables
- Hoeffding
- 1963
(Show Context)
Citation Context ...d for the coordination ratio of uniform links and they greatly expanded the result to the case of links with arbitrary capacities. To show the result on uniform links they employ a lemma by Hoeffding =-=[6]-=- which bounds the tail probability of the sum of independent random variables. This lemma is stronger than the Chernoff bound we use in our approach and applies directly to balls of arbitrary weights ... |

924 |
Non-cooperative games
- Nash
- 1951
(Show Context)
Citation Context ...quilibria for noncooperative games have been de ned and studied in the classical literature of Game Theory� the most famous of them is Nash equilibrium, originally de ned in John Nash's seminal paper =-=[15]-=-. Roughly speaking, the strategies chosen by the players in a game constitute a Nash equilibrium if no player can do better by unilaterally adopting some other strategy. Several variants and re nement... |

766 |
A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations
- Chernoff
- 1952
(Show Context)
Citation Context ... the social cost for the case where all balls have weights within a factor of 2, resulting at the end of the previous step. We there apply standard techniques for estimating tails and Chernoff bounds =-=[3]-=- to show that (roughly speaking), for this case, the social cost is at most O � � ln m times the maximum expected number of balls that is placed ln ln m into any link (Lemma 4.3). Putting together the... |

670 | Worst-case equilibria
- Koutsoupias, Papadimitriou
- 1999
(Show Context)
Citation Context ...c) sel sh routing problem we study. We shall see that Nash equilibria are a special class of approximate equilibria (for the speci c game we consider). In a recent paper Koutsoupias and Papadimitriou =-=[10]-=- introduced the notion of coordination ratio or price ofanarchy to measure the performance loss due to lack of coordination. We extend the de nition to approximate equilibria and de ne the approximate... |

467 |
Reexamination of the Perfectness Concept for Equilibrium Points
- Selten
- 1975
(Show Context)
Citation Context ... Stackelberg equilibrium [19], where a distinguished leader among players in the game holds a powerful position, while the rest of the players, called followers, act rationally� a perfect equilibrium =-=[14, 18]-=- is obtained as a result of a limiting procedure that converges to suitable mixed strategies of the players� nally, asaddle-point equilibrium [16] is specially tailored for twoperson, zero-sum nite ga... |

261 | Balanced allocations - Azar, Broder, et al. - 1999 |

170 | Tight bounds for worst-case equilibria
- Czumaj, Vöcking
- 2002
(Show Context)
Citation Context ... is just the property we require in the de nition of approximate equilibria: the maximum expected latency be at most a constant multiple of the optimum. Recently and independently, Czumaj and Vocking =-=[4]-=- obtained the same upper bound for the coordination ratio of uniform links and they greatly expanded the result to the case of links with arbitrary capacities. To show the result on uniform links they... |

154 |
Refinements of the Nash equilibrium concept
- Myerson
- 1978
(Show Context)
Citation Context ... Stackelberg equilibrium [19], where a distinguished leader among players in the game holds a powerful position, while the rest of the players, called followers, act rationally� a perfect equilibrium =-=[14, 18]-=- is obtained as a result of a limiting procedure that converges to suitable mixed strategies of the players� nally, asaddle-point equilibrium [16] is specially tailored for twoperson, zero-sum nite ga... |

128 | Architecting noncooperative networks - Korilis, Lazar, et al. - 1995 |

124 | The Price of Selfish Routing - Mavronicolas, Spirakis - 2001 |

102 | The power of two random choices: A survey of the techniques and results
- Mitzenmacher, Richa, et al.
- 2000
(Show Context)
Citation Context ...e classical balls and bins problem (cf. [8]), which has been studied extensively both in its original form and with algorithmic extensions� (see, for example, the recent survey by Mitzenmacher et al. =-=[12]-=-). A central result in the theory of the balls and bins problem is that if one places uniformly and independently m identical balls into m bins, the expected maximum number of balls in a bin is log m ... |

79 | Selfish Traffic Allocation for Server Farms - Czumaj, Krysta, et al. - 2002 |

39 |
Probability inequalities for sums of bounded random variables
- ding
- 1994
(Show Context)
Citation Context ...nd for the coordination ratio of uniform links and they greatly expanded the result to the case of links with arbitrary capacities. To show the result on uniform links they employ a lemma by Hoe ding =-=[6]-=- which bounds the tail probability of the sum of independent random variables. 4sThis lemma is stronger than the Cherno bound we use in our approach and applies directly to balls of arbitrary weights ... |

26 |
A measure of asymptotic e ciency for tests of a hypothesis based on the sum of observations
- Cherno€
- 1952
(Show Context)
Citation Context ...mes the social cost for the case where all balls haveweights within a factor of 2, resulting at the end of the previous step. We there apply standard techniques for estimating tails and Cherno bounds =-=[3]-=- toshowthat (roughly speaking), for this case, the social cost is at ln m most O times the maximum expected number of balls that is placed ln ln m into any link (Lemma 4.3). Putting together the two s... |

19 |
Theory of Games and Economic Behavior, second edition
- Neumann
- 1944
(Show Context)
Citation Context ... followers, act rationally� a perfect equilibrium [14, 18] is obtained as a result of a limiting procedure that converges to suitable mixed strategies of the players� nally, asaddle-point equilibrium =-=[16]-=- is specially tailored for twoperson, zero-sum nite games. In our work, we introduce and study a very general class of equilibria, whichwecallapproximate equilibria� the de nition is tailored to the s... |

18 |
Stackelberg, “The Theory of the Market
- von
- 1952
(Show Context)
Citation Context ...ilibria have been studied in the literature of Game Theory in order to best model the appropriate solution concept for speci c games (or classes of them). Examples include the Stackelberg equilibrium =-=[19]-=-, where a distinguished leader among players in the game holds a powerful position, while the rest of the players, called followers, act rationally� a perfect equilibrium [14, 18] is obtained as a res... |

3 | Allocating Weighted Jobs - Berenbrink, Heide, et al. - 1997 |

3 |
The Power of Two Random Choices: A
- Mitzenmacher, Richa, et al.
- 2000
(Show Context)
Citation Context ...classical balls and bins problem (cf. [8]), which has been studied extensively both in its original form and with algorithmic extensions; (see, for example, the recent 3ssurvey by Mitzenmacher et al. =-=[12]-=-). A central result in the theory of the balls and bins problem is that if one places uniformly and independently m identical balls � into m bins, the expected maximum number of balls in a bin is Θ � ... |

2 | Sel sh Tra c Allocation for Server Farms - Czumaj, Krysta, et al. - 2002 |

2 | The Price of Sel sh Routing - Mavronicolas, Spirakis - 2001 |

1 | Allocating WeightedJobsinParallel - Berenbrink, Heide, et al. - 1997 |

1 | A Tight Bound on Coordination Ratio - Koutsoupias, Mavronicolas, et al. - 2001 |