Bertrand-Edgeworth game under oligopoly. General results and comparisons with duopoly

This paper studies price competition among a given number of capacity-constrained producers of a homogeneous commodity under the efficient rationing rule and constant (and identical) marginal cost until full capacity, when demand is a continuous, non-increasing, and non-negative function defined on the set of non-negative prices and is positive, strictly decreasing, twice differentiable and (weakly) concave when positive. The focus is on general properties of equilibria in the region of the capacity space in which no pure strategy equilibria exist. We study how the properties that are known to hold for the duopoly are generalized to the oligopoly and we highlight the new properties that can arise in asymmetric oligopoly, which include the existence of an atom in the support of the equilibrium strategy of a firm smaller than the largest one, the properties that such an atom entails, the existence of gaps in the supports, and asymmetries in the equilibrium distributions of equally-sized firms smaller than the largest one. Further, we provide results about the boundaries of the supports. Although the characterization of equilibria is far from being complete, this paper provides substantial elements in this direction.