Economic natural selection in Bertrand and Cournot settings

We study economic natural selection in classical oligopoly settings. When u nderlying pure strategies consist of a finite number of prices, convex mono tonic dynamics always converge under a weak condition of the smallest price in the support of the initial state that exceeds marginal cost. When under lying pure strategies consist of a finite number of quantities, monotonic d ynamics always converge under a specific condition to a quantity equal or s imilar classical Cournot equilibrium.