This paper analyzes the evolutionary dynamics of a modified version of Rosenthal's "centipede," whereby the players' strategies are represented by finite-state automata. In the framework considered, the automata required to implement different strategies may differ among them both with respect to their complexity, and with with respect to their cost. This is due to the possibility that the players have to identify similar situations, and similar actions, across different information sets. This modification of the original game introduces an unstable equilibrium at which all strategies are represented in the population of the players. With the unperturbed replicator dynamics, the system eventually converges to some point belonging to a continuum of rest-points that the present model shares with the evolutionary version of Rosenthal's original game. The transient phase - characterized by fluctuations in the fractions of the two populations adopting different automata - can however be relatively long. Furthermore, numerical results point to the fact that small perturbations can turn the fluctuations characterizing the convergent paths of the unperturbed system into persistent phenomena. In this case, the players' payoffs can be significantly higher than those achievable at the Nash-equilibria.