  Automated decision-making tools increasingly assess individuals to determine
if they qualify for high-stakes opportunities. A recent line of research
investigates how strategic agents may respond to such scoring tools to receive
favorable assessments. While prior work has focused on the short-term strategic
interactions between a decision-making institution (modeled as a principal) and
individual decision-subjects (modeled as agents), we investigate interactions
spanning multiple time-steps. In particular, we consider settings in which the
agent's effort investment today can accumulate over time in the form of an
internal state - impacting both his future rewards and that of the principal.
We characterize the Stackelberg equilibrium of the resulting game and provide
novel algorithms for computing it. Our analysis reveals several intriguing
insights about the role of multiple interactions in shaping the game's outcome:
First, we establish that in our stateful setting, the class of all linear
assessment policies remains as powerful as the larger class of all monotonic
assessment policies. While recovering the principal's optimal policy requires
solving a non-convex optimization problem, we provide polynomial-time
algorithms for recovering both the principal and agent's optimal policies under
common assumptions about the process by which effort investments convert to
observable features. Most importantly, we show that with multiple rounds of
interaction at her disposal, the principal is more effective at incentivizing
the agent to accumulate effort in her desired direction. Our work addresses
several critical gaps in the growing literature on the societal impacts of
automated decision-making - by focusing on longer time horizons and accounting
for the compounding nature of decisions individuals receive over time.
