Dynamics of Cournot and Bertrand Firms: Exploring Imitation and Replicator Processes

This study delves into the evolution of Cournot and Bertrand firms within the framework of imitation and replicator dynamics, encompassing diverse scenarios. In each time period, firms are randomly paired with either Bertrand or Cournot counterparts for a duopoly game. Subsequently, the populations of these firms evolve in accordance with either imitation or replicator dynamics. Within this context, the potential outcomes include the establishment of globally asymptotically stable limits for replicator dynamics, such as the coexistence of both-type firms, as well as the dominance of either all-Cournot or all-Bertrand firms. However, imitation dynamics tend to yield only the latter two equilibria, excluding the possibility of both-type coexistence. In the specific case of linear demand and cost, the stable limits of replicator dynamics hinge on factors like the uniformity of product differentiation levels among Cournot and Bertrand firms, as well as the nature of substitute or complement goods. Notably, these considerations do not exert the same influence on the long-term equilibria of imitation dynamics. Moreover, the outcomes reveal that while all evolutionary stable strategies of duopoly games featuring differentiated goods fail to serve as stable limits within our replicator dynamics, some of these strategies do attain stability when homogeneous goods are produced. Lastly, the application of our models to the analysis of firms producing goods of varying quality sheds light on intriguing disparities compared to outcomes obtained from static models.