## Worst-case equilibria (1999)

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Venue: | IN PROCEEDINGS OF THE 16TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE |

Citations: | 677 - 17 self |

### BibTeX

@INPROCEEDINGS{Koutsoupias99worst-caseequilibria,

author = {Elias Koutsoupias and Christos Papadimitriou},

title = {Worst-case equilibria},

booktitle = {IN PROCEEDINGS OF THE 16TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE},

year = {1999},

pages = {404--413},

publisher = {}

}

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### Abstract

In a system in which noncooperative agents share a common resource, we propose the ratio between the worst possible Nash equilibrium and the social optimum as a measure of the effectiveness of the system. Deriving upper and lower bounds for this ratio in a model in which several agents share a very simple network leads to some interesting mathematics, results, and open problems.

### Citations

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Citation Context ...refer the reader to the book [14] which devotes a significant fraction to these issues. The popularity of the concept of the price of anarchy owes much to the follow-up work of Roughgarden and Tardos =-=[18]-=- which opened the way to studying the price of anarchy in atomic and nonatomic congestion games. The conference version of this work together with the work of Nisan and Ronen [13, 12] on algorithmic m... |

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Citation Context ...du 1 The conference version of this work [9] appeared a decade ago and it has been followed by a large amount of work on the concept of the price of anarchy (as witnessed by the extensive coverage in =-=[14]-=-). In this journal version we tried to keep as much as possible the text of the original paper. There are, though, important changes because results that were substantially improved in the meantime ar... |

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Citation Context ...rm “price of anarchy” instead of the original term “coordination ratio”. The use of the latter term faded away in the literature, replaced by the term “price of anarchy” which was first introduced in =-=[16]-=-. 1which are more resilient to video-like traffic. Their point of view is also that of the mechanism design aspect of game theory, in that they try to design games (strategy spaces and reward tables)... |

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Citation Context ..., with the Prisoner's Dilemma [8, 10] being the best-known example. Conditions under which Nash equilibria can achieve or approximate the overall optimum have been studied extensively ([10]� see also =-=[5, 7, 11]-=- for studies on networks). However, this line of previous work compares the overall optimum with the best Nash equilibrium, not the worst, as be ts our line of reasoning. To put it otherwise, this pre... |

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Citation Context ...links, log m the price of anarchy is Ω( log log m ) (Theorem 1). In the conference version of the paper we had a nonmatching upper bound. It was later shown that this lower bound is essentially tight =-=[3, 8]-=-. 2 All Nash equilibria We consider the case of n agents sharing m identical links. Before describing all Nash equilibria, we need a few definitions. We usually use subscripts for agents and superscri... |

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Citation Context ...link and P wi its load [7]. More important is the study of realistic Internet metrics, that result from the employed protocols such as the one related to TCP and the square root of the drop frequency =-=[3]-=-. Finally, it would be extremely interesting, once the relative quality of the Nash equilibria in such situations is better understood, to employ suchunderstanding in the design of improved protocols ... |

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Citation Context ...e sum of all delays over all agents, say. The question we want to ask is, how far from the optimum total delay can be the total delay achieved by a Nash equilibrium? Numerical experiments reported in =-=[6]-=- imply that there are Nash equilibria which can be more than 20% o the overall optimum. In this paper we address a very simple special case of this problem, in which the network is just a set of m par... |

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Citation Context ... with the Prisoner 's Dilemma [8, 10] being the best-known example. Conditions under which Nash equilibria can achieve or approximate the overall optimum have been studied extensively ([10]; see also =-=[5, 7, 11]-=- for studies on networks). However, this line of previous work compares the overall optimum with the best Nash equilibrium, not the worst, as befits our line of reasoning. To put it otherwise, this pr... |

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Citation Context ...orithmic and combinatorial questions and results. It is nontrivial to arrive at a compelling mathematical formulation of this question. Independent, non-cooperative agents obviously evoke game theory =-=[8]-=-, and its main concept of rational behavior, the Nash equilibrium: In an environment inwhicheachagentisaware of the situation facing all other agents, a Nash equilibrium is a combination of choices (d... |

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Citation Context ...links, log m the price of anarchy is Ω( log log m ) (Theorem 1). In the conference version of the paper we had a nonmatching upper bound. It was later shown that this lower bound is essentially tight =-=[3, 8]-=-. 2 All Nash equilibria We consider the case of n agents sharing m identical links. Before describing all Nash equilibria, we need a few definitions. We usually use subscripts for agents and superscri... |

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Citation Context ...istic or randomized), one for each agent, from whichnoagent has an incentive to unilaterally moveaway.Nash equilibria are known not to always optimize overall performance, with the Prisoner's Dilemma =-=[8, 10]-=- being the best-known example. Conditions under which Nash equilibria can achieve or approximate the overall optimum have been studied extensively ([10]� see also [5, 7, 11] for studies on networks). ... |

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Citation Context ...robabilities p 1 i of the stochastic agents (p2 i =1; p1 i ). p 1 s2 i = ; s1 + s2 (s2 ; s1) P i wi +(s2d1 ; s1d2 ) (k ; 1)(s1 + s2)wi It is not hard to verify that these probabilities indeed satisfy =-=(9)-=-. To prove the theorem, we consider the case of no initial loads and two agents with jobs w1 = s2 and w2 = s1. The probabilities are p1 1 = s 2 1 s2(s1+s2) and p12 =1; s 2 2 .We can s1(s1+s2) then com... |

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Citation Context ... easy to see that M j = L j + X i p j i wi: (1)sFrom the point of view of agent i, its nish time when its own tra c wi is assigned to link j is c j i = wi + L j + X i6=t p j twt = M j +(1; p j i )wi: =-=(2)-=- Probabilities p j de ne a Nash equilibrium if there is no incentive for agent i to i change its strategy. Thus, agent i will assign nonzero probabilities only to links j that minimize c j i .We will ... |

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Citation Context ..., with the Prisoner's Dilemma [8, 10] being the best-known example. Conditions under which Nash equilibria can achieve or approximate the overall optimum have been studied extensively ([10]� see also =-=[5, 7, 11]-=- for studies on networks). However, this line of previous work compares the overall optimum with the best Nash equilibrium, not the worst, as be ts our line of reasoning. To put it otherwise, this pre... |

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On the existence of equilibria in noncooperative optimal ow control
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Citation Context ..., with the Prisoner's Dilemma [8, 10] being the best-known example. Conditions under which Nash equilibria can achieve or approximate the overall optimum have been studied extensively ([10]� see also =-=[5, 7, 11]-=- for studies on networks). However, this line of previous work compares the overall optimum with the best Nash equilibrium, not the worst, as be ts our line of reasoning. To put it otherwise, this pre... |

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