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Asymmetries in Competitive Location Models on the Line

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Spatial Interaction Models

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 118))

Abstract

This paper first presents a standard competitive duopoly location model on a linear market and derives an equilibrium solution as well as a solution for the sequential von Stackelberg game. The heart of the contribution then investigates scenarios, in which the duopolists face or follow asymmetric situations or strategies. In particular, we examine situations, in which the duopolists have different objectives, models, in which firms apply different pricing policies, and instances, in which the competitors have different capabilities.

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Acknowledgements

This work was in part supported by the Institute Complex Engineering Systems, through grants ICM-MIDEPLAN P-05-004-F and CONICYT FB0816.

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Authors and Affiliations

  1. Faculty of Business Administration, University of New Brunswick, Fredericton, NB, Canada

    H. A. Eiselt

  2. Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile

    Vladimir Marianov

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  1. H. A. Eiselt
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  2. Vladimir Marianov
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Correspondence to H. A. Eiselt .

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Editors and Affiliations

  1. Department of Mathematics and Applications, University of Naples Federico II, Napoli, Italy

    Lina Mallozzi

  2. Department of Industrial and Information Engineering, Second University of Naples, Aversa, Italy

    Egidio D'Amato

  3. Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

Appendix

Appendix

Given that firm A locates at the center of the market, i.e., \(a = \frac{1} {2}\ell\), firm B’s market area is symmetric about \(\frac{1} {2}\ell\). Suppose that firm B’s market area is d units near both ends of the market. (We deviate from some of the notation in the paper in order to simplify matters.) As usual, B is located b units from the right end of the market. This situation is shown in Fig. 7.

Fig. 7
figure 7

Transportation cost of a firm that uses delivered prices and whose market share extends d from both ends of the market, the form locates outside its market area

Full size image

Firm B’s transportation costs are then the two trapezoids (D, E) and (F, G). Elementary algebra indicates that the areas of D, E, F, and G are \(\frac{1} {2}td^{2}\), (ℓ − b − d)dt, \(\frac{1} {2}td^{2}\), and(b − d)dt, respectively, so that the total cost (the total area) is (ℓ − d)td. It is apparent that these costs are independent of b, the location of firm B. The scenario does not change as long as b ∈ [d, ℓ − d].

Consider now the case, in which locates at a point b ≤ d (or, alternatively, b ≥ ℓ − d). This situation is shown in Fig. 8.

Fig. 8
figure 8

Transportation cost of a firm that uses delivered prices and whose market share extends d from both ends of the market, the form locates inside its market area

Full size image

Similar to the above analysis, we have the four areas H, I, J, and K, the sum of whose areas determine the transportation cost incurred by firm B. The areas are \(\frac{1} {2}d^{2}t\), (ℓ − b − d)td, \(\frac{1} {2}(d - b)^{2}t\), and \(\frac{1} {2}tb^{2}\), respectively, so that the area is − 2btd + ℓ t d + tb 2, which is dependent on b. The minimum is found at b = d, indicating that firm B best locates somewhere in its opponent’s market area.

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Eiselt, H.A., Marianov, V. (2017). Asymmetries in Competitive Location Models on the Line. In: Mallozzi, L., D'Amato, E., Pardalos, P. (eds) Spatial Interaction Models . Springer Optimization and Its Applications, vol 118. Springer, Cham. https://doi.org/10.1007/978-3-319-52654-6_6

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