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Convexity and the Shapley value in Bertrand oligopoly TU-games with Shubik's demand functions

Author

Listed:
  • Dongshuang Hou

    (Department of Applied Mathematics [Twente] - University of Twente)

  • Theo Driessen

    (Department of Applied Mathematics [Twente] - University of Twente)

  • Aymeric Lardon

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

Abstract

The Bertrand Oligopoly situation with Shubik's demand functions is modelled as a cooperative TU game. For that purpose two optimization problems are solved to arrive at the description of the worth of any coalition in the so-called Bertrand Oligopoly Game. Under certain circumstances, this Bertrand oligopoly game has clear affinities with the well-known notion in statistics called variance with respect to the distinct marginal costs. This Bertrand Oligopoly Game is shown to be totally balanced, but fails to be convex unless all the firms have the same marginal costs. Under the complementary circumstances, the Bertrand Oligopoly Game is shown to be convex and in addition, its Shapley value is fully determined on the basis of linearity applied to an appealing decomposition of the Bertrand Oligopoly Game into the difference between two convex games, besides two nonessential games. One of these two essential games concerns the square of one non- essential game.

Suggested Citation

  • Dongshuang Hou & Theo Driessen & Aymeric Lardon, 2011. "Convexity and the Shapley value in Bertrand oligopoly TU-games with Shubik's demand functions," Working Papers halshs-00610838, HAL.
    • Handle: RePEc:hal:wpaper:halshs-00610838
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00610838
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    References listed on IDEAS

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    2. Zhao, J, 1996. "A B-Core Existence Result and its Application to Oligopoly Markets," ISER Discussion Paper 0418, Institute of Social and Economic Research, Osaka University.
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    6. Zhao, Jingang, 1999. "A [beta]-Core Existence Result and Its Application to Oligopoly Markets," Games and Economic Behavior, Elsevier, vol. 27(1), pages 153-168, April.
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    Cited by:

    1. Wang, Lei & Zhao, Jingang, 2024. "The core in an N-firm dynamic Cournot oligopoly," Mathematical Social Sciences, Elsevier, vol. 129(C), pages 20-26.
    2. Aymeric Lardon, 2020. "Convexity of Bertrand oligopoly TU-games with differentiated products," Annals of Operations Research, Springer, vol. 287(1), pages 285-302, April.

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    Keywords

    Total Balancedness; Bertrand Oligopoly situation; Bertrand Oligopoly Game; Convexity; Shapley Value; Total Balancedness.;
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