Decentralised partially observable Markov decision processes (DecPOMDPs) provide a general framework for modelling multi-agent decision making under uncertainty. However, DecPOMDPs are known to suffer from exponential complexity in the number of agents. One way to combat this intractability in agent numbers is to look at partitions of agents that exhibit a form of symmetry among agents, allowing for a compact encoding by counting. However, a challenge arises as the policy space explodes, even though the model complexity and evaluation cost reduce to a polynomial dependence. In this paper, we redirect our focus from counting agents to counting policies, which actually enables tractability in agent numbers for so called policy-counted DecPOMDPs. Further, we present policy-counted dynamic programming using the compact representation to solve policy-counted DecPOMDPs efficiently.
We introduce trajectory-regularized stochastic optimal control (TRSOC), which augments standard stochastic optimal control (SOC) with a Kullback--Leibler (KL) divergence between controlled and reference trajectory distributions. Using Girsanov's theorem, the trajectory KL reduces to a quadratic drift mismatch penalty, yielding a modified running cost that preserves the dynamic programming (DP) structure. We derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation and characterize the optimal policy. In the linear-quadratic (LQ) setting, the formulation admits a closed-form solution with an augmented control cost. Experiments show that the regularization parameter induces a trade-off between performance-driven and reference-preserving behavior, including cases with reference dynamics learned from offline data.
We study finite-horizon MDP planning under \emph{root-based} (resolute) risk objectives that apply a rank-dependent functional to the distribution of total returns. Such objectives are non-linear in the return distribution and generally break Bellman optimality, so direct optimization by scenario-tree enumeration is intractable. We propose \textbf{ERQDP}, an enumeration-free and sampling-free method that solves a rank--quantile surrogate via exact DP (Dynamic Programming), evaluates candidate policies exactly by DP over return Probability Mass Functions (PMFs) on a discretized return grid (with an explicit rounding bound), and refines the surrogate in an anytime loop that reports an explicit upper--lower gap (certificate) for the target objective up to discretization budgets. Across tested benchmarks, ERQDP returns certified solutions or explicit residual gaps, enables fast risk-parameter sweeps with substantial runtime gains, and supports both risk-averse and risk-seeking behaviors.