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

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Author Info

  • 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-Etienne - Groupe d'analyse et de théorie économique - CNRS : UMR5824 - Université Lumière - Lyon II - École Normale Supérieure de Lyon)

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.

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Bibliographic Info

Paper provided by HAL in its series Working Papers with number halshs-00610838.

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Date of creation: 2011
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Handle: RePEc:hal:wpaper:halshs-00610838

Note: View the original document on HAL open archive server: http://halshs.archives-ouvertes.fr/halshs-00610838/en/
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Keywords: Bertrand Oligopoly situation; Bertrand Oligopoly Game; Convexity; Shapley Value; Total Balancedness.;

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  1. Aymeric Lardon, 2010. "Convexity of Bertrand oligopoly TU-games with differentiated products," Post-Print halshs-00544056, HAL.
  2. CHANDER, Parkash & TULKENS, Henry, 1995. "The Core of an Economy with Multilateral Environmental Externalities," CORE Discussion Papers 1995050, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Aymeric Lardon, 2012. "The γ-core in Cournot oligopoly TU-games with capacity constraints," Theory and Decision, Springer, vol. 72(3), pages 387-411, March.
  4. Norde, H.W. & Pham Do, K.H. & Tijs, S.H., 2000. "Oligopoly Games With and Without Transferable Technologies," Discussion Paper 2000-66, Tilburg University, Center for Economic Research.
  5. Zhao, Jingang, 1999. "A necessary and sufficient condition for the convexity in oligopoly games," Mathematical Social Sciences, Elsevier, vol. 37(2), pages 189-204, March.
  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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