Policy Synthesis for Finite Populations of MDP Agents under Aggregate Reach-Avoid Chance Constraints

Consider a finite population of agents with decoupled Markov transition dynamics and empirical-density feedback, subject to the following constraints: with probability at least $1-{\delta }_r$, at least a fraction ${\alpha }_r$ of agents must reach a target region at some time $t^{\ast }$, while, at each time up to $t^{\ast }$, the unsafe population fraction must remain below ${\beta }_u$ with probability at least $1-{\delta }_u$. However, standard mean-field methods enforce these constraints only in expectation, which fails to account for stochastic fluctuations at finite fleet size $N$. To address this control problem, we propagate the second-order moment (variance) of the empirical density alongside the mean-field trajectory via a discrete-time Lyapunov recursion, and apply the Cantelli inequality to convert chance constraints into tractable deterministic conditions on the moments of the empirical density. We then incorporate these moment-based surrogate constraints into a gradient-based sequential convex approximation procedure for density-feedback policy synthesis. We further introduce additional moment-error bounds to construct a rigorous finite-$N$ certificate. The method is evaluated on a gridworld environment and a power-system EV-charging aggregation problem and compared with a standard deterministic population-level LP baseline.