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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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1.1 Location spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Vertex and edge demand . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Condorcet points and related concepts . . . . . . . . . . . . . . . 21 2.2 Some (1|1)-centroid problems related to the Condorcet concept . 23 3.1 Bottleneck points . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Bottleneck points bpi,leftuv and bpi,rightuv . . . . . . . . . . . . . . . . 53 3.3 Tree network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Rejection functions and 1-suboptimal points . . . . . . . . . . . 59 3.5 No vertex weight equals zero . . . . . . . . . . . . . . . . . . . . 60 3.6 Rejection functions and 1-suboptimal points . . . . . . . . . . . 60 3.7 Repeat-until block (lines 4–7) . . . . . . . . . . . . . . . . . . . 65 3.8 Exchange xk+1 and yk+1 . . . . . . . . . . . . . . . . . . . . . . 66 3.9 Illustration of the missing case in Algorithm 3.2.1 . . . . . . . . 68 3.10 No 1-optimal point . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.11 Chain network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Serving of edge customers . . . . . . . . . . . . . . . . . . . . . 82 4.2 Average comp. time for solving BBNP with fixed sets of leader’s locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Diamond structure . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Chain network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Function f(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.6 Example - tree network . . . . . . . . . . . . . . . . . . . . . . . 105 xiv List of Figures 4.7 Example - subtrees . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.8 Collapsing stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.9 Functions fμi(yμi), i = 1, ..., 4, for the example in Fig. 4.8b . . . 113 5.1 Stages of the game . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2 Nash equilibrium with ∂ΠL/∂pL > 0 . . . . . . . . . . . . . . . . 133 5.3 Diverging fixed-point iteration . . . . . . . . . . . . . . . . . . . 135 5.4 Comparison of profit and markup functions . . . . . . . . . . . . 138 5.5 Combined algorithm . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6 Starting price vectors . . . . . . . . . . . . . . . . . . . . . . . . 140 5.7 Computational results - CA convergence . . . . . . . . . . . . . 144 5.8 Computational results - CA running time . . . . . . . . . . . . . 145 5.9 Varying α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.10 Varying β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.11 Existence of equilibria and influence of network size . . . . . . . 148 5.12 Example network . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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