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

Dongshuang Hou (dshhou@126.com), Theo Driessen (t.s.h.driessen@ewi.utwente.nl) and Aymeric Lardon
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Dongshuang Hou: Department of Applied Mathematics [Twente] - University of Twente
Theo Driessen: Department of Applied Mathematics [Twente] - University of Twente

Working Papers from HAL

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.

Keywords: Total Balancedness; Bertrand Oligopoly situation; Bertrand Oligopoly Game; Convexity; Shapley Value; Total Balancedness. (search for similar items in EconPapers)
Date: 2011
New Economics Papers: this item is included in nep-bec, nep-com, nep-gth, nep-hpe and nep-ind
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