How helpful is a long memory on financial markets?

How should portfolio decisions depend on the past? In a simple model with boundedly rational agents we show that there is no universal answer to this question. Both, long and short memory, can be optimal in the appropriate environment. In most cases there is an equilibrium where both dispositions are equally successful. We characterize such equilibria for the case of two assets and two states. For dynamics based on average payoff, equilibria are global attractors whereas discrete choice dynamics in general do not converge to the equilibrium.