Surplus sharing in Cournot oligopoly

We characterize equilibria of oligopolistic markets where identical firms with constant marginal cost compete a' la Cournot. For given maximal willingness to pay and maximal total demand, we first identify all combinations of equilibrium consumer surplus and industry profit that can arise from arbitrary demand functions. Then, as a further restriction, we fix the average willingness to pay above marginal cost (i.e., first-best surplus) and identify all possible triples of consumer surplus, industry profit and deadweight loss.