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Desktop grids can achieve tremendous computing power at low cost through opportunistic sharing of resources. However, traditional client-server Grid architectures do not deal with all types of failures, and do not always cope well with very dynamic environments. This paper describes the design of a desktop grid implemented over a modified Peer-to-Peer (P2P) architecture. The underlying P2P system is decentralized and inherently adaptable, giving the Grid robustness, scalability, and the ability to cope with dynamic environments, while still efficiently mapping application instances to available resources throughout the system. We use simulation to compare three different types of matching algorithms under differing workloads. Overall, the P2P approach produces significantly lower wait times than prior approaches, while adapting efficiently to the dynamic environment.