Game-Theoretic Safe Multi-Agent Motion Planning with Reachability Analysis for Dynamic and Uncertain Environments (Extended Version)

Ensuring safe, robust, and scalable motion planning for multiagent systems in dynamic and uncertain environments is a persistent challenge, driven by complex interagent interactions, stochastic disturbances, and model uncertainties. To overcome these challenges, particularly the computational complexity of coupled decision-making and the need for proactive safety guarantees, we propose a reachability-enhanced dynamic potential game (RE-DPG) framework, which integrates game-theoretic coordination into reachability analysis. This approach formulates multiagent coordination as a dynamic potential game, where the Nash equilibrium (NE) defines optimal control strategies across agents. To enable scalability and decentralized execution, we develop a neighborhood-dominated iterative best response scheme, built upon an iterated $\varepsilon$-BR process that guarantees finite-step convergence to an $\varepsilon$-NE. This allows agents to compute strategies based on local interactions while ensuring theoretical convergence guarantees. Furthermore, to ensure safety under uncertainty, we integrate a multiagent forward reachable set mechanism into the cost function, explicitly modeling uncertainty propagation and enforcing collision avoidance constraints. Through both simulations and real-world experiments in 2-D and 3-D environments, we validate the effectiveness of RE-DPG across diverse operational scenarios.