Cognitive wireless systems are a collection of wireless network entities that are able to adapt intelligently to the environment through observation, exploration and learning. This project designs methods for cognitive radio systems to compete for inhomogeneous spectrum resources and to establish, without explicit coordination, a spectrum etiquette. Each system's resource utilization strategy must be decided with only imperfect information of other systems' actions, and the resulting adaptations must discourage any one system from manipulating the agreed spectrum etiquette. The framework of dynamic games with imperfect private monitoring is adopted to design coexistence mechanisms for cognitive radio systems sharing inhomogeneous resources and to analyze the stability, robustness, complexity, and convergence of these mechanisms. This work directly impacts the development and standardization of frequency agile wireless systems that can more efficiently use the available spectrum to support increasing demand for ubiquitous wireless connectivity at high data rates. The project is a collaborative effort between researchers in the US (at Virginia Tech and the University of Houston) and Finland (at the University of Oulu). This work will extend the state of the art in cognitive system adaptations by incorporating the ability to: efficiently adapt under imperfect information; fairly share inhomogeneous resources; and discourage manipulation of resource use by selfish adversaries. Results will be disseminated in high impact journals and conferences; the designed coexistence mechanisms will also be presented to industry and regulators in the US and Europe.