Most of the results on decentralized supervisory control are based on supervisors that make unconditional decisions: "enable" and "disable." In this paper, we introduce and study the properties of decentralized supervisory control architectures where supervisors are allowed to make conditional decisions in addition to unconditional decisions. The conditional decisions we consider are of the form: "enable if nobody disables" and "disable if nobody enables." We characterize the notion of conditional coobservability that appears in the necessary and sufficient conditions for the existence of supervisors in the context of such control architectures. This condition relaxes the previous notions of coobservability for unconditional architectures. The key properties of conditional coobservability are studied. We develop a polynomial-time algorithm for verifying the notion of conditional coobservability. A polynomial-time method of partitioning the controllable events between "enable by default" and "disable by default" is presented.