Table of Contents
Acknowledgments v
Table of contents vii
1 Introduction 1
1.1 Introduction to game theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The basics of game theory . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Introduction to the applications studied in this thesis . . . . . . . . . . . . . 4
1.2.1 Optimal toll design . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Electricity market liberalization . . . . . . . . . . . . . . . . . . . 6
1.2.3 Theory of incentives . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Road map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Results from Classical Game Theory 11
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Stackelberg equilibria and terminology . . . . . . . . . . . . . . . . . . . . 12
2.4 Open loop versus closed loop . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Tools for one-person optimization . . . . . . . . . . . . . . . . . . . . . . 13
2.5.1 Dynamic programming for continuous-time systems . . . . . . . . 14
2.5.2 The minimum principle . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.3 Affine quadratic optimal control problems . . . . . . . . . . . . . . 16
3 Inverse Stackelberg Games 19
3.1 Static inverse Stackelberg games and equilibria . . . . . . . . . . . . . . . 19
3.1.1 One leader – one follower games . . . . . . . . . . . . . . . . . . 19
3.1.2 One leader – multiple followers games . . . . . . . . . . . . . . . 22
3.2 Dynamic inverse Stackelberg games and equilibria . . . . . . . . . . . . . 25
3.2.1 One leader – one follower games . . . . . . . . . . . . . . . . . . . 26
3.2.2 One leader – multiple followers games . . . . . . . . . . . . . . . . 33
3.3 Extension: Two leaders – one follower . . . . . . . . . . . . . . . . . . . . 35
3.4 Conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . 38
vii
viii Contents
4 Static Optimal Toll Design 39
4.1 Introduction and literature overview . . . . . . . . . . . . . . . . . . . . . 39
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 Game-theoretic interpretation of the optimal toll design problem . . 43
4.3 Drivers’ behavior – static traffic assignment . . . . . . . . . . . . . . . . . 43
4.3.1 Deterministic user (Wardrop) equilibrium . . . . . . . . . . . . . . 44
4.3.2 Probabilistuc (stochastic) user equilibrium . . . . . . . . . . . . . . 45
4.4 The problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 General problem properties . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6 Solution of problem (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6.1 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6.2 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6.3 Supervised learning . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6.4 Solving the optimal toll design problem . . . . . . . . . . . . . . . 52
4.6.5 Application of FAUN 1.1 simulator . . . . . . . . . . . . . . . . . 54
4.7 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7.1 One origin–destination pair with multiple parallel links . . . . . . . 55
4.7.2 Beltway network . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.8 Conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . 69
5 Dynamic Optimal Toll Design 71
5.1 Introduction and literature overview . . . . . . . . . . . . . . . . . . . . . 71
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Game-theoretic interpretation of the optimal toll design problem . . 76
5.3 Drivers’ behavior – dynamic traffic assignment . . . . . . . . . . . . . . . 76
5.3.1 Dynamic traffic equilibrium conditions . . . . . . . . . . . . . . . 77
5.3.2 The dynamic network loading model . . . . . . . . . . . . . . . . 78
5.4 The problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 General problem properties . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7.1 Three-links network . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.7.2 Chen network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.8 Conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . 91
6 Electricity Market Problem 93
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Games of the European electricity market . . . . . . . . . . . . . . . . . . 95
6.2.1 Game formulations . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.2 Model specifications . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.1 Games with one country . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.2 Games with two countries . . . . . . . . . . . . . . . . . . . . . . 105
6.3.3 Games with eight countries . . . . . . . . . . . . . . . . . . . . . . 106
6.4 Extension: Dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5 Conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . 110
Contents ix
7 Theory of Incentives 113
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Complete-information principal-agent model . . . . . . . . . . . . . . . . 115
7.4 Adverse-selection principal-agent model . . . . . . . . . . . . . . . . . . . 116
7.5 Conclusions and future research . . . . . . . . . . . . . . . . . . . . . . . 120
8 Conclusions and Future Research 121
8.1 Contributions to the state-of-the-art . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Bibliography 127
NGInfra PhD Thesis Series on Infrastructures 135
Samenvatting 137
Summary 139
Curriculum vitae 141
Abstract
This thesis studies the so-called inverse Stackelberg games, that are new in the world of game theory, their properties, and their applications in the optimal toll design problem, the energy markets liberalization problem, and in the theory of incentives.
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