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Selfish behaviors of individual machines in a Grid can potentially damage the performance of the system as a whole. However, scrutinizing the Grid by taking into account the noncooperativeness of machines is a largely unexplored research problem. In this paper, we first present a new hierarchical game-theoretic model of the Grid that matches well with the physical administrative structure in real-life situations. We then focus on the impact of selfishness in intrasite job execution mechanisms. Based on our novel utility functions, we analytically derive the Nash equilibrium and optimal strategies for the general case. To study the effects of different strategies, we have also performed extensive simulations by using a well-known practical scheduling algorithm over the NAS (Numerical Aerodynamic Simulation) and the PSA (Parameter Sweep Application) workloads. We have studied the overall job execution performance of the Grid system under a wide range of parameters. Specifically, we find that the Optimal selfish strategy significantly outperforms the Nash selfish strategy. Our performance evaluation results can serve as a valuable reference for designing appropriate strategies in a practical Grid.