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Sequential Competitive Location on Networks

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Dominik Kreß

This book deals with classical competitive location problems where two players, leader and follower, sequentially enter markets with given numbers of facilities. The markets under consideration are represented as networks. The book provides a detailed overview of the literature on competitive and voting location, and it presents extensions and variations of the classical models, with a focus on the incorporation of proportional choice rules, non-discrete demand (edge demand), or additional pricing decisions of the players. It provides corresponding mathematical models, insights into the computational complexity of the resulting problems and proposes and analyzes adequate solution methods.

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List of Figures xiii List of Tables xv 1 Introduction and Preliminaries 1 1.1 Basic Notation and Definitions . . . . . . . . . . . . . . . . . 2 1.1.1 Graphs and Networks . . . . . . . . . . . . . . . . . . 2 1.1.2 Game Theory . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Discrete Choice Theory . . . . . . . . . . . . . . . . . 6 1.2 Foundations of Competitive and Voting Location . . . . . . . 7 1.3 Vertex and Edge Demand . . . . . . . . . . . . . . . . . . . . 12 1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . 13 2 Sequential Competitive Location on Networks: Literature Review 15 2.1 Voting Location: Condorcet Points and Related Concepts . . 18 2.2 Voting Location: Some (1|1)-Centroid Problems . . . . . . . 22 2.3 (r|Xp)-Medianoid Problems . . . . . . . . . . . . . . . . . . . 24 2.4 (r|p)-Centroid Problems . . . . . . . . . . . . . . . . . . . . 32 2.5 Multiple Players, Endogenous Location and Some Selected R1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 On the Influence of Specific Modeling Assumptions . . . . . . 42 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Competitive or Voting Location with Proportional Choice 51 3.1 Bottleneck Points . . . . . . . . . . . . . . . . . . . . . . . . 52 x Contents 3.2 A Proportional Choice Rule . . . . . . . . . . . . . . . . . . 54 3.2.1 1-Suboptimal Points . . . . . . . . . . . . . . . . . . 56 3.2.2 Examples and Counterexamples . . . . . . . . . . . . 59 3.2.3 Determining 1-Suboptimal Points: Implementational Issues . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 Including Edge Demand . . . . . . . . . . . . . . . . . . . . 71 3.3.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . 74 3.3.2 1-Optimal Points on Chain Networks with Edge De- mand Only . . . . . . . . . . . . . . . . . . . . . . . 75 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4 (r|p)-Centroid Problems on Networks with Vertex and Edge Demand 79 4.1 The Binary (r|p)-Centroid Problem with Vertex and Edge De- mand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.1.1 Bilevel Programming Models . . . . . . . . . . . . . . 80 4.1.2 Potential Solution Approaches . . . . . . . . . . . . . 84 4.1.3 Some Complexity Results . . . . . . . . . . . . . . . . 87 4.1.4 Finite Dominating Sets . . . . . . . . . . . . . . . . . 91 4.2 An Efficient Algorithm on Tree Networks . . . . . . . . . . . 93 4.2.1 Chain Networks . . . . . . . . . . . . . . . . . . . . . 94 4.2.2 General Tree Networks . . . . . . . . . . . . . . . . . 103 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5 Competitive Location...

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