Locally Interdependent Multi-Agent MDP: Theoretical Framework for Decentralized Agents with Dynamic Dependencies

The evolution of multi-agent systems has transitioned from centralized control to decentralized autonomous decision-making, driven by applications in robotics, autonomous vehicles, and UAV swarms. Early models like Multi-Agent MDPs provided a theoretical basis but faced scalability issues. The advent of Partially Observable MDPs (POMDP) and Decentralized POMDPs (Dec-POMDP) introduced formal frameworks for partial information, yet their computational complexity—NEXP-Complete—limits practical use. Empirical approaches, especially deep reinforcement learning, have shown success but lack rigorous performance guarantees. Recent efforts focus on scalable algorithms with theoretical backing, such as mean-field RL and decentralized Q-learning, but often assume fixed or stochastic dependence structures. This paper advances the field by modeling dynamic, proximity-based local dependencies, reflecting real-world environments where agent interactions are transient and spatially constrained.

The core challenge lies in designing decentralized policies that perform near-optimally under partial observability and dynamic local dependencies. Traditional methods struggle with the combinatorial explosion of state-action spaces and the non-stationarity introduced by changing neighborhood relations. Achieving scalable, theoretically sound solutions that can handle large numbers of agents with limited communication remains an open problem. Specifically, understanding how local information and dynamic dependencies influence global performance is critical for deploying multi-agent systems in real-world scenarios such as autonomous driving, drone swarms, and robotic teams.

This work introduces a formal model—Locally Interdependent Multi-Agent MDP—that captures dynamic neighborhood relations based on proximity within a metric space. It derives three closed-form policies, each with provable performance bounds, leveraging the concept of dependence time and Bellman inequalities. The key innovation is demonstrating that the performance gap between decentralized partial observability and centralized full observability diminishes exponentially with the visibility radius. Unlike prior models, this approach explicitly accounts for time-varying local dependencies, enabling scalable algorithms with theoretical guarantees. It bridges the gap between empirical RL success and rigorous performance analysis, providing a new foundation for decentralized control.

• �� Define multi-agent state space with spatial positions and internal states, neighborhood relations based on distance thresholds. • Formulate reward functions incorporating local and interdependent components within a radius R. • Derive three policies: Amalgam (local optimal aggregation), Cutoff (neighbor-based truncation), First Step (short-horizon approximation). • Use Bellman equations and performance bounds to analyze error, introducing the concept of dependence time c. • Prove that the performance difference decreases exponentially as visibility radius V increases, via upper and lower bounds. • Validate through simulations in obstacle avoidance, navigation, and formation control, measuring reward deviations, stability, and scalability.

Simulations involve multi-agent navigation in R^2 space with varying visibility radii V. Baselines include centralized optimal policies and empirical RL methods. Metrics include cumulative discounted reward, policy storage size, and computational time. Experiments test long-term stability, robustness to noise, and dynamic neighborhood changes. Results show the proposed policies achieve near-optimal performance within 10% error, with performance improving exponentially with V. Scalability is demonstrated by reduced memory requirements and faster computation, confirming theoretical predictions. Additional tests include environments with obstacles and formation constraints, verifying adaptability and robustness.

The policies’ performance gaps are bounded by exponential functions of V, with empirical results matching theoretical bounds. Long-term reward deviations stay below 10%, and storage/computation costs are reduced by over 80%. The models maintain high robustness under environmental uncertainties and dynamic neighborhood changes. These findings confirm that local information, when properly leveraged, suffices for near-global optimality, especially as the visibility radius grows, making the approach suitable for large-scale real-world systems.

Applicable to autonomous vehicle fleets, drone swarms, robotic teams in cluttered environments, and distributed sensor networks. The policies require only local communication within a limited radius, making them suitable for environments with communication constraints. The framework enables scalable, decentralized decision-making with performance guarantees, reducing reliance on centralized control. Future integration with deep learning can further enhance adaptability, enabling real-time deployment in complex, dynamic scenarios such as disaster response, surveillance, and industrial automation.

The model assumes distance-based neighborhood relations, which may oversimplify real-world interactions involving non-spatial dependencies. The performance heavily depends on the choice of visibility radius V; too small V degrades accuracy, too large V increases complexity. Robustness under high noise, communication failures, or rapid environment changes remains to be validated. Extending the framework to non-metric dependencies and incorporating learning-based policy adaptation are important future directions.