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2020 | OriginalPaper | Chapter

The Continuous Hotelling Pure Location Game with Elastic Demand Revisited

Author : Pierre von Mouche

Published in: Mathematical Optimization Theory and Operations Research

Publisher: Springer International Publishing

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Abstract

The Hotelling pure location game has been revisited. It is assumed that there are two identical players, strategy sets are one-dimensional, and demand as a function of distance is constant or strictly decreasing. Besides qualitative properties of conditional payoff functions, attention is given to the structure of the equilibrium set, best-response correspondences and the existence of potentials.

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Footnotes
1
For an overview and discussion of the literature we refer to [5] and [6].
 
2
Its standard interpretation in location theory concerns two competing vendors on a beach. The vendors simultaneously and independently select a position. Customers go to the closest vendor and split themselves evenly if the vendors choose an identical position. Each vendor wants to maximize his number of customers. One can reframe the interpretation as two candidates placing themselves along an ideological spectrum, with citizens voting for whichever one is closest (see e.g. [4]).
 
3
However, see concluding remark 3 in Sect. 8.
 
4
Here \(\{ L \} - B_i(L-x)\) is the Minkowski sum of the sets \( \{ L \}\) and \(- B_i(L-x)\).
 
5
So, if f is continuous, then by Proposition 1(5), these formulas become \( D u_1^{(x_2)}(x_1) = f(x_1) - \frac{1}{2} f( \frac{x_2-x_1}{2}) \; (x_1 < x_2)\) and \( D u_1^{(x_2)}(x_1) = - f(L-x_1) + \frac{1}{2} f( \frac{x_1-x_2}{2}) \; (x_1 > x_2)\).
 
6
For the cHg, a function \(P: S \times S \rightarrow \mathbb {R}\) is (1) a generalized ordinal potential if for every \(a_1, b_1, z \in [0,L]\) it holds that \( u_1(a_1, z)< u_1(b_1; z) \; \Rightarrow \; P(a_1,z) < P(b_1,z)\) and for every \(a_2, b_2, z \in [0,L]\) it holds that \( u_2(z,a_1)< u_2(z,b_1) \; \Rightarrow \; P(z,a_1) < P(z,b_1)\); (2) a best-response potential if \( B_1(x_2) = \mathrm {argmax}_{x_1 \in S} P(x_1,x_2) \; (x_2 \in S)\) and \(B_2(x_1) = \mathrm {argmax}_{x_2 \in S} P(x_1,x_2) \; (x_1 \in S)\); (3) a quasi potential if \( \mathrm {argmax} \, P = E\). (4) a weak quasi potential if \( \mathrm {argmax} \, P \subseteq E\). In this case, one calls the game a ‘generalized ordinal potential game’ (etc.).
 
Literature
1.
2.
Smithies, A.: Optimum location in spatial competition. J. Polit. Econ. 44, 423–439 (1941)CrossRef
3.
Boulding, K.: Economics Analysis: Microeconomics. Harper & Row, New York (1955)
4.
Davis, O., Hinich, M.J., Ordeshook, P.C.: An expository development of a mathematical model of the electoral process. Am. Polit. Sci. Rev. 44, 426–448 (1970)CrossRef
5.
Eaton, B.C., Lipsey, R.G.: The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Rev. Econ. Stud. 42(1), 27–49 (1975)CrossRef
6.
Graitson, D.: Spatial competition à la hotelling: a selective survey. J. Ind. Econ. 31(1/2), 11–25 (1982)CrossRef
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Anderson, S.P., de Palma, A., Thisse, J.-F.: Discrete Choice Theory of Product Differentiation. MIT Press, Cambridge (1992)CrossRef
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Reny, P.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056 (1999)MathSciNetCrossRef
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von Mouche, P.H.M., Quartieri, F.: Cournot equilibrium uniqueness via demi-concavity. Optimization 67(4), 41–455 (2017)MathSciNetMATH
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Iimura, T.: Private Communication. Tokyo Metropolitan University, Tokyo, Japan (2017)
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Iimura, T., von Mouche, P.H.M., Watanabe, T.: Best-response potential for hotelling pure location games. Econ. Lett. 160, 73–77 (2017)MathSciNetCrossRef
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Iimura, T., von Mouche, P.H.M.: Discrete hotelling pure location games: potentials and equilibria. Working Paper, Wageningen Universiteit (2020)
Metadata
Title
The Continuous Hotelling Pure Location Game with Elastic Demand Revisited
Author
Pierre von Mouche
Copyright Year
2020
Book
Mathematical Optimization Theory and Operations Research

Print ISBN: 978-3-030-49987-7

Electronic ISBN: 978-3-030-49988-4

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-49988-4_17

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