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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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Derivation of the Logit Choice Probabilities

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The following derivation of the logit choice probabilities is based on McFad- den (1974); Train (2009). From (1.1) and (1.2) we get prij = Prob(ik < ij + vij − vik ∀ k ∈ J, k = j). Hence, with (1.3) and (1.4) and since the stochastic utility components are independent, we have prij = +∞∫ −∞ ⎛ ⎝ ∏ k∈J,k =j e−e −(ij+vij−vik)/σ ⎞ ⎠ 1 σ e−ij/σe−e −ij/σ dij. After substituting ˆij = (1/σ)ij and denoting ˆij by z and vij/σ by vˆij for all i ∈ I and j ∈ J for the sake of brevity, prij becomes1 prij = +∞∫ −∞ ⎛ ⎝ ∏ k∈J,k =j e−e −(z+vˆij−vˆik) ⎞ ⎠ e−ze−e−z dz, and thus 1 Note that, for readability reasons, we use two identical expressions, exp and e, for the exponential function. 178 Appendix A. Derivation of the Logit Choice Probabilities prij = +∞∫ −∞ (∏ k∈J e−e −(z+vˆij−vˆik) ) e−z dz = +∞∫ −∞ exp ( −e−z ∑ k∈J e−(vˆij−vˆik) ) e−z dz. Now define t = e−z. Then prij = 0∫ +∞ − exp ( −t ∑ k∈J e−(vˆij−vˆik) ) dt = +∞∫ 0 exp ( −t ∑ k∈J e−(vˆij−vˆik) ) dt = exp (−t∑k∈J e−(vˆij−vˆik)) −∑k∈J e−(vˆij−vˆik) ∣∣∣∣∣ +∞ 0 = evij/σ∑ k∈J evik/σ as in (1.5).

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