Economic natural selection in Bertrand and Cournot settings

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