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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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Chapter 5. Competitive Location and Pricing with Random Utilities

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consequence, this chapter also contributes to numerical approaches for com- puting equilibrium prices under multinomial logit demand and is therefore very closely related to Morrow (2008); Morrow and Skerlos (2011). Related models can be found in the field of product positioning, cf. Choi et al. (1990) and Rhim and Cooper (2005). This chapter proceeds as follows. First, a detailed problem formulation is given in Section 5.1 with results concerning the existence of price equilibria and the computational complexity in Sections 5.1.1 and 5.1.2, respectively. In Section 5.2 we show that, given the locations of all facilities, standard numerical approaches can be combined to reliably and quickly determine local price equilibria. Finally, in Section 5.3, we show that price competition may actually affect optimal locations of facilities, and we provide first insights into the performance of heuristic algorithms for finding solutions to the location problem. The chapter closes with a conclusion in Section 5.4. 5.1 Problem Formulation Consider a network N = (V,E, λ) with vertex demand only. A firm L (leader) acts as a monopolist with multiple facilities in this spatial market. L’s facilities are located at p > 0 distinct vertices Xp ⊆ V of the network. A competitor F (follower) wants to enter the market with an a priori fixed number of facilities r > 0.1 F’s potential facility sites are restricted to the vertex set of the network. Hence, F solves a discrete (r|Xp)-medianoid problem. At most two facilities, one of the leader’s and one of the follower’s...

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