## Selfish Routing and the Price of AnarchyAn analysis of the loss in performance caused by selfish, uncoordinated behavior in networks.Most of us prefer to commute by the shortest route available, without taking into account the traffic congestion that we cause for others. Many networks, including computer networks, suffer from some type of this "selfish routing." In Roughgarden begins with a relatively nontechnical introduction to selfish routing, describing two important examples that motivate the problems that follow. The first, Pigou's Example, demonstrates that selfish behavior need not generate a socially optimal outcome. The second, the counterintiuitve Braess's Paradox, shows that network improvements can degrade network performance. He then develops techniques for quantifying the price of anarchy (with Pigou's Example playing a central role). Next, he analyzes Braess's Paradox and the computational complexity of detecting it algorithmically, and he describes Stackelberg routing, which improves the price of anarchy using a modest degree of central control. Finally, he defines several open problems that may inspire further research. Roughgarden's work will be of interest not only to researchers and graduate students in theoretical computer science and optimization but also to other computer scientists, as well as to economists, electrical engineers, and mathematicians. |

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### Contents

Introduction | 3 |

13 Applications and Caveats | 6 |

14 How to Read this Book | 11 |

15 Notes | 12 |

Preliminaries | 17 |

22 Flows at Nash Equilibrium | 20 |

23 The Price of Anarchy | 22 |

25 Examples | 28 |

43 Edge Capacities | 93 |

44 Atomic Selfish Routing | 99 |

45 A QuickandDirty Bound on the Price of Anarchy | 104 |

46 Better Bounds for Many Traffic Rates | 107 |

47 Maximum Cost | 110 |

48 Notes | 114 |

Bounding and Detecting Braesss Paradox | 121 |

52 Bounding Braesss Paradox | 122 |

26 Existence and Uniqueness of Nash Flows | 34 |

27 Nash Flows in SingleCommodity Networks | 37 |

28 Notes | 41 |

How Bad Is Selfish Routing? | 51 |

32 The Price of Anarchy with Linear Cost Functions | 53 |

33 A General Upper Bound on the Price of Anarchy | 58 |

34 Matching Lower Bounds in Simple Networks | 62 |

35 Computing the Price of Anarchy | 69 |

36 A Bicriteria Bound in General Networks | 73 |

37 Notes | 77 |

Extensions | 85 |

42 Approximate Nash Flows | 89 |

53 Detecting Braesss Paradox | 132 |

54 Notes | 146 |

Stackelberg Routing | 151 |

62 Stackelberg Strategies and Induced Equilibria | 152 |

63 Three Stackelberg Strategies | 153 |

64 Upper Bounds for Networks of Parallel Links | 155 |

65 Lower Bounds in More General Networks | 159 |

66 The Complexity of Computing Optimal Strategies | 162 |

67 Notes | 165 |

169 | |

191 | |