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The Continuous Hotelling Pure Location Game with Elastic Demand Revisited

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Mathematical Optimization Theory and Operations Research (MOTOR 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12095))

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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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Notes

  1. 1.

    For an overview and discussion of the literature we refer to [5] and [6].

  2. 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. 3.

    However, see concluding remark 3 in Sect. 8.

  4. 4.

    Here \(\{ L \} - B_i(L-x)\) is the Minkowski sum of the sets \( \{ L \}\) and \(- B_i(L-x)\).

  5. 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. 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.).

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Authors and Affiliations

  1. Wageningen Universiteit, Wageningen, The Netherlands

    Pierre von Mouche

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  1. Pierre von Mouche
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Correspondence to Pierre von Mouche .

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Editors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Alexander Kononov

  2. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  3. National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Valery A Kalyagin

  4. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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von Mouche, P. (2020). The Continuous Hotelling Pure Location Game with Elastic Demand Revisited. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Lecture Notes in Computer Science(), vol 12095. Springer, Cham. https://doi.org/10.1007/978-3-030-49988-4_17

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