Bertrand-Edgeworth competition in an almost symmetric oligopoly

We analyze a Bertrand-Edgeworth game in homogeneous product industry, under effcient rationing, constant marginal cost until full capacity utilization, and identical technology across firms. We solve for the equilibrium and establish its uniqueness for capacity configurations in the mixed strategy region of the capacity space such that the capacities of the largest and smallest firm are sufficiently close.Bertrand-Edgeworth competition; mixed strategy equilibrium; almost symmetric oligopoly; Mixed strategy equilibrium

Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly

The paper extends the analysis of price competition among capacity-constrained sellers beyond the cases of duopoly and symmetric oligopoly. We first provide some general results for the oligopoly and then focus on the triopoly, providing a complete characterization of the mixed strategy equilibrium of the price game. The region of the capacity space where the equilibrium is mixed is partitioned according to the features of the mixed strategy equilibrium arising in each subregion. Then computing the mixed strategy equilibrium becomes a quite simple task. The analysis reveals features of the mixed strategy equilibrium which do not arise in the duopoly (some of them have also been discovered by Hirata (2008)).Bertrand-Edgeworth; Price game; Oligopoly; Triopoly; Mixed strategy equilibrium

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Bertrand-Edgeworth oligopoly: Characterization of mixed strategy equilibria when some firms are large and the others are small

This paper studies Bertrand–Edgeworth competition among firms producing a homogeneous commodity under efficient rationing and constant (and identical across firms) marginal cost until full capacity utilization isreached. Ourfocusis on a subset of the no pure-strategy equilibrium region of the capacity space in which, in a well-defined sense, some firms are large and the others are small. We characterize equilibria for such subset. For eachfirm,the payoffs are the same at any equilibriumand, for each type of firm, they are proportional to capacity. While there is a single profile of equilibrium distributions for the large firms, there is a continuum of equilibrium distributions for the small firms: what is uniquely determined, forthe latter, isthe capacity-weighted sum of their equilibrium distributions and hence the union of the supports of their equilibrium strategies

Bertrand-Edgeworth games under triopoly: the payoffs

The paper extends the analysis of price competition among capacity constrained sellers beyond duopoly and symmetric oligopoly. The main focus is on the equilibrium payoffs under triopoly. The paper also includes insightful examples highlighting features of equilibrium which can arise in a triopoly but not in a duopoly. Most notably, the supports of the equilibrium strategies need not be connected, nor need be connected the union of the supports; further, an atom may exist for a firm different from the largest one

Bertrand-Edgeworth competition in an almost symmetric oligopoly

We analyze a Bertrand-Edgeworth game in homogeneous product industry, under effcient rationing, constant marginal cost until full capacity utilization, and identical technology across firms. We solve for the equilibrium and establish its uniqueness for capacity configurations in the mixed strategy region of the capacity space such that the capacities of the largest and smallest firm are sufficiently close

Bertrand-Edgeworth games under triopoly: the payoffs

The paper extends the analysis of price competition among capacity constrained sellers beyond duopoly and symmetric oligopoly. The main focus is on the equilibrium payoffs under triopoly. The paper also includes insightful examples highlighting features of equilibrium which can arise in a triopoly but not in a duopoly. Most notably, the supports of the equilibrium strategies need not be connected, nor need be connected the union of the supports; further, an atom may exist for a firm different from the largest one

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

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

Bertrand-Edgeworth oligopoly: Characterization of mixed strategy equilibria when some firms are large and the others are small

This paper studies Bertrand-Edgeworth competition among firms producing a homogeneous commodity under efficient rationing and constant (andidentical across firms) marginal cost until full capacity utilization is reached. Our focus is on a subset of the no pure-strategy equilibrium region of the capacity space in which, in a well-defined sense, some firms are large and the others are small. We characterize equilibria for such subset. For each firm, the payoffs are the same at any equilibrium and, for each type of firm, they are proportional to capacity. While there is a single profile of equilibrium distributions for the large firms, there is a continuum of equilibrium distributions for the small firms: what is uniquely determined, for the latter, is the capacity-weighted sum of their equilibrium distributions and hence the union of the supports of their equilibrium strategies