How helpful is a long memory on financial markets?

Wow should portfolio decisions depend on the past? In a simple model with b oundedly rational agents we show that there is no universal answer to this question. Both, lung and short memory, can he optimal in the appropriate en vironment. In most cases there is an equilibrium where both dispositions ar e equally successful. We characterize such equilibria for the case of two a ssets and two states. For dynamics based on average payoff, equilibria are global attractors whereas discrete choice dynamics in general do not conver ge to the equilibrium.