  Under non-exponential discounting, we develop a dynamic theory for stopping
problems in continuous time. Our framework covers discount functions that
induce decreasing impatience. Due to the inherent time inconsistency, we look
for equilibrium stopping policies, formulated as fixed points of an operator.
Under appropriate conditions, fixed-point iterations converge to equilibrium
stopping policies. This iterative approach corresponds to the hierarchy of
strategic reasoning in Game Theory, and provides "agent-specific" results: it
assigns one specific equilibrium stopping policy to each agent according to her
initial behavior. In particular, it leads to a precise mathematical connection
between the naive behavior and the sophisticated one. Our theory is illustrated
in a real options model.
