Selfish Routing and the Price of Anarchy

Description

An 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 Selfish Routing and the Price of Anarchy, Tim Roughgarden studies the loss of social welfare caused by selfish, uncoordinated behavior in networks. He quantifies the price of anarchy—the worst-possible loss of social welfare from selfish routing—and also discusses several methods for improving the price of anarchy with centralized control.

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.Preface vii
I INTRODUCTION AND PRELIMINARIES 1 Introduction 3
1.1 Selfish Routing 3
1.2 Two Motivating Examples 3
1.3 Applications and Caveats 6
1.4 How to Read this Book 11
1.5 Notes 12
2 Preliminaries 17
2.1 The Model 17
2.2 Flows at Nash Equilibrium 20
2.3 The Price of Anarchy 21
2.4 A Characterization of Optimal Flows 22
2.5 Examples 28
2.6 Existence and Uniqueness of Nash Flows 34
2.7 Nash Flows in Single-Commodity Networks 37
2.8 Notes 41
II BOUNDING THE PRICE OF ANARCHY
3 How Bad Is Selfish Routing? 51
3.1 Overview 51
3.2 The Price of Anarchy with Linear Cost Functions 53
3.3 A General Upper Bound on the Price of Anarchy 58
3.4 Matching Lower Bounds in Simple Networks 62
3.5 Computing the Price of Anarchy 69
3.6 A Bicriteria Bound in General Networks 73
3.7 Notes 77
4 Extensions 85
4.1 Nonatomic Congestion Games 85
4.2 Approximate Nash Flows 88
4.3 Edge Capacities 92
4.4 Atomic Selfish Routing 99
4.5 A Quick-and-Dirty Bound on the Price of Anarchy 104
4.6 Better Bounds for Many Traffic Rates 107
4.7 Maximum Cost 110
4.8 Notes 114
III COPING WITH SELFISHNESS
5 Bounding and Detecting Braess’s Paradox 121
5.1 Overview 121
5.2 Bounding Braess’s Paradox 122
5.3 Detecting Braess’s Paradox 132
5.4 Notes 146
6 Stackelberg Routing 151
6.1 Overview 151
6.2 Stackelberg Strategies and Induced Equilibria 152
6.3 Three Stackelberg Strategies 153
6.4 Upper Bounds for Networks of Parallel Links 155
6.5 Lower Bounds in More General Networks 159
6.6 The Complexity of Computing Optimal Strategies 162
6.7 Notes 165
References 169
Index 191Tim Roughgarden is Assistant Professor of Computer Science at Stanford University and recipient of the ACM’s Grace Hopper Award.US