Evolutions of Cournot and Bertrand Firms Under Imitation and Replicator Dynamics

Under general demands and costs, this paper investigates the evolutions of Cournot and Bertrand firms in imitation and replicator dynamics. All-Cournot firms, all-Bertrand firms and both-type’s co-existence can be globally asymptotically stable limits of replicator dynamics, while only the first two can be the long-run equilibria of imitation dynamics. In particular, for linear demands and costs, the stable limits of replicator dynamics depend on whether both-type firms’ product differentiation levels are identical and what kinds of goods are produced, but the long-run equilibria of imitation dynamics do not. Moreover, all evolutionary stable strategies are not stable limits of our replicator dynamics when firms produce differentiated goods, but some of them are when homogeneous goods are produced. Finally, the evolutions for firms producing high- and low-quality goods are explored, and the results are different from those obtained in static models