  This paper studies algorithmic decision-making under human's strategic
behavior, where a decision maker uses an algorithm to make decisions about
human agents, and the latter with information about the algorithm may exert
effort strategically and improve to receive favorable decisions. Unlike prior
works that assume agents benefit from their efforts immediately, we consider
realistic scenarios where the impacts of these efforts are persistent and
agents benefit from efforts by making improvements gradually. We first develop
a dynamic model to characterize persistent improvements and based on this
construct a Stackelberg game to model the interplay between agents and the
decision-maker. We analytically characterize the equilibrium strategies and
identify conditions under which agents have incentives to improve. With the
dynamics, we then study how the decision-maker can design an optimal policy to
incentivize the largest improvements inside the agent population. We also
extend the model to settings where 1) agents may be dishonest and game the
algorithm into making favorable but erroneous decisions; 2) honest efforts are
forgettable and not sufficient to guarantee persistent improvements. With the
extended models, we further examine conditions under which agents prefer honest
efforts over dishonest behavior and the impacts of forgettable efforts.
