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We analyze the characteristics of overlay routing networks generated by selfish nodes playing competitive network construction games. We explore several networking scenarios - some simplistic, others more realistic - and analyze the resulting Nash equilibrium graphs with respect to topology, performance, and resilience. We find a fundamental tradeoff between performance and resilience, and show that limiting the degree of nodes is of great importance in controlling this balance. Further, by varying the cost function, the game produces widely different topologies; one parameter in particular - the relative cost between maintaining an overlay link and increasing the path length to other nodes - can generate topologies with node-degree distributions whose tails vary from exponential to power-law. We conclude that competitive games can create overlay routing networks satisfying very diverse goals.