  We study a decision-maker's problem of finding optimal monetary incentive
schemes for retention when faced with agents whose participation decisions
(stochastically) depend on the incentive they receive. Our focus is on policies
constrained to fulfill two fairness properties that preclude outcomes wherein
different groups of agents experience different treatment on average. We
formulate the problem as a high-dimensional stochastic optimization problem,
and study it through the use of a closely related deterministic variant. We
show that the optimal static solution to this deterministic variant is
asymptotically optimal for the dynamic problem under fairness constraints.
Though solving for the optimal static solution gives rise to a non-convex
optimization problem, we uncover a structural property that allows us to design
a tractable, fast-converging heuristic policy. Traditional schemes for
retention ignore fairness constraints; indeed, the goal in these is to use
differentiation to incentivize repeated engagement with the system. Our work
(i) shows that even in the absence of explicit discrimination, dynamic policies
may unintentionally discriminate between agents of different types by varying
the type composition of the system, and (ii) presents an asymptotically optimal
policy to avoid such discriminatory outcomes.
