Wireless sensor nodes are usually densely deployed to completely cover (monitor) a set of targets. Consequently, redundant sensor nodes that are not currently needed in the covering task can be powered off to conserve energy. These sensors can take over the covering task later to prolong network lifetime. The coverage problem, concerns picking up a set of working sensors that collectively meet the coverage requirements. The problem is complicated by the possibility that targets may have different coverage requirements while sensor nodes may have different amounts of energy. This article proposes a game-theoretic approach to the coverage problem, where each sensor autonomously decides its state with a simple rule based on local information. We give rigorous proofs to show stability, correctness, and efficiency of the proposed game. Implementation variants of the game consider specific issues, such as game convergence time and different amounts of sensor energy. Simulation results show significant improvement in network lifetime by the proposed approach when compared with representative alternatives.