Many systems consist of a set of agents which must acquire exclusive access to resources from a shared pool. Coordination of agents in such systems is often implemented in the form of a centralized mechanism. The intervention of this type of mechanism, however, typically introduces significant computational overhead and reduces the amount of concurrent activity. Alternatives to centralized mechanisms exist, but they generally suffer from the need for extensive interagent communication. In this paper, we develop a randomized approach to make multiagent resource-allocation decisions with the objective of maximizing expected concurrency measured by the number of the active agents. This approach does not assume a centralized mechanism and has no need for interagent communication. Compared to existing autonomous-decentralized-decision-making (ADDM)-based approaches for resource-allocation, our work emphasizes achieving the highest degree of agent autonomy and is able to handle more general resource requirements.