On the stability of best reply and gradient systems with applications to imperfectly competitive models

We present a general result on the convergence to an equilibrium of a class of dynamic adjustment procedures — which includes gradient systems and best reply dynamics as special cases — when there are two players and strategy sets are one dimensional. We also show that there are no restrictions on the form of the gradient or best reply dynamics, even under strong restrictions on the functional form of both demand and costs. This implies that we can construct examples with three players where the above dynamical procedures yield chaotic behavior.