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Bertrand-Edgeworth games under triopoly: the equilibrium strategies when the payoffs of the two smallest firms are proportional to their capacities

De Francesco, Massimo A. and Salvadori, Neri (2016): Bertrand-Edgeworth games under triopoly: the equilibrium strategies when the payoffs of the two smallest firms are proportional to their capacities.

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Abstract

The paper is the second part of a trilogy in which we extend the analysis of price competition among capacity-constrained sellers beyond duopoly to triopoly. In the first part of the trilogy we provided some general results, highlighting features of a duopolistic mixed strategy equilibrium that generalize to triopoly and provided a first partition concerning the pure strategy equilibrium regions and the mixed strategies equilibrium region and then the partition of this region in a part in which the payoffs of the two smallest firm are proportional to their capacities and another in which the smallest firm obtains a payoff proportinally higher than that of the middle sized firm. In this paper we provide a complete characterization of the set of mixed strategy equilibria in the part in which the payoffs of the two smallest firms are proportional to their capacities. This part is partitioned according to equilibrium features and in each part it is determined whether equilibria are uniquely determined or not and in the latter case it is proved that the equilibria constitute a continuum. Further we determine the circustances in which supports of an equilibrium strategy may be disconnected and show how gaps are then determined. We also prove that the union of supports is indeed connected, a property which cannot be extended to the case in which the smallest firm obtains a payoff proportinally higher than that of the middle sized firm. The third part of the trilogy will be devoted to a complete characterization of the mixed strategy equilibria when the smallest firm obtains a payoff proportinally higher than that of the middle sized firm. This will allow also to determine the payoff of the smallest firm.

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