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In this paper, we propose a novel policy for steering multiple vehicles between assigned start and goal configurations, ensuring collision avoidance. The policy rests on the assumption that all agents are cooperating by implementing the same traffic rules. However, the policy is completely decentralized, as each agent decides its own motion by applying those rules only on the locally available information, and scalable, in the sense that the amount of information processed by each agent and the computational complexity of the algorithms do not increase with the number of agents in the scenario. The proposed policy applies to systems in which new vehicles may enter the scene and start interacting with existing ones at any time, while others may leave. Under mild conditions on the initial configurations, the policy is shown to be safe, i.e., it guarantees collision avoidance throughout the system evolution. In the paper, conditions are discussed on the desired configurations of agents, under which the ultimate convergence of all vehicles to their goals can also be guaranteed. To show that such conditions are actually necessary and sufficient, which turns out to be a challenging liveness-verification problem for a complex hybrid automaton, we employ a probabilistic verification method. The paper finally presents and discusses simulations for systems of several tens of vehicles, and reports on some experimental implementation showing the practicality of the approach.