  Snapshots of "best" (or "worst") experience are known to dominate human
memory and may thus also have a significant effect on future behaviour. We
consider here a model of repeated decision-making where, at every time step, an
agent takes one of two choices with probabilities which are functions of the
maximum utilities previously experienced. Depending on the utility
distributions and the level of noise in the decision process, it is possible
for an agent to become "trapped" in one of the choices on the basis of their
early experiences. If the utility distributions for the two choices are
different, then the agent may even become trapped in the choice which is
objectively worse in the sense of expected long-term returns; crucially we
extend earlier work to address this case. Using tools from statistical physics
and extreme-value theory, we show that for exponential utilities there is an
optimal value of noise which maximizes the expected returns in the long run. We
also briefly discuss the behaviour for other utility distributions.
