  Individuals working towards a goal often exhibit time inconsistent behavior,
making plans and then failing to follow through. One well-known model of such
behavioral anomalies is present-bias discounting: individuals over-weight
present costs by a bias factor. This model explains many time-inconsistent
behaviors, but can make stark predictions in many settings: individuals either
follow the most efficient plan for reaching their goal or procrastinate
indefinitely.
  We propose a modification in which the present-bias parameter can vary over
time, drawn independently each step from a fixed distribution. Following
Kleinberg and Oren (2014), we use a weighted task graph to model task planning,
and measure the cost of procrastination as the relative expected cost of the
chosen path versus the optimal path. We use a novel connection to optimal
pricing theory to describe the structure of the worst-case task graph for any
present-bias distribution. We then leverage this structure to derive conditions
on the bias distribution under which the worst-case ratio is exponential (in
time) or constant. We also examine conditions on the task graph that lead to
improved procrastination ratios: graphs with a uniformly bounded distance to
the goal, and graphs in which the distance to the goal monotonically decreases
on any path.
