Structural Equivalence and Learning Dynamics in Delayed MARL

We formally establish the equivalence between Observation Delay (OD) and Action Delay (AD) in cooperative partially observable multi-agent systems using observation-action histories. We show that both systems generate identical admissible joint-policy sets, and their induced state-action-observation trajectories are identical in distribution, leading to identical optimal solutions in Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs). This formally generalizes existing infinite-horizon single-agent results to any-horizon partially observable cooperative multi-agent problems with decentralized policy execution, and allows any mixed-delay configuration to be reduced to a pure OD system. We further prove that in Transition-Independent MDPs (TI-MDPs), the observation-action history reduces to a tractable minimal local augmented state. However, we show through numerical experiments that although the optimal solution spaces are structurally isomorphic, the practical learning dynamics are fundamentally different. First, using the minimal local augmented state, the equivalence no longer holds when transitions are not independent. Second, operational constraints and causal credit-assignment errors in Temporal Difference (TD) algorithms induce different learning behaviors across regimes. Finally, leveraging this structural equivalence to bypass these learning challenges, we demonstrate successful multi-agent zero-shot policy transfer from OD to AD, paving the way for unified, efficient solution methods in complex delayed systems.