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 rst provide some general results for the oligopoly, highlighting features of a duopolistic mixed strategy equilibrium that generalize to oligopoly. Unlike in the duopoly, however, there can be in nitely many equilibria when the capacity of a subset of rms is so large that no strategic interaction among smaller rms exists. Then we focus on the triopoly, providing a complete characterization of the mixed strategy equilibrium of the Bertrand-Edgeworth game. The mixed-strategy region of the capacity space is partitioned according to key equilibrium features. We also prove the possibility of a disconnected support of an equilibrium strategy and show how gaps are then determined. Computing the mixed strategy equilibrium then becomes quite a simple task