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 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.