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In this paper, we investigate power allocation in cognitive sensor networks, where cognitive users (cognitive enabled sensor nodes) opportunistically share the common spectrum with primary users (licensed devices). Consider that sensor nodes are self-interested to maximize their own utilities, we formulate the energy-efficient power allocation problem as a non-cooperative coupled constraint game, by taking the interference temperature into account. An energy efficiency-oriented utility function is defined as a new metric to evaluate the performance of power allocation. Firstly, we prove that there exist Nash equilibriums in the proposed game. Then, we prove that the power allocation game is a super-modular game under some conditions. Finally, we design centralized and distributed Game-based Efficiency-oriented Power Allocation algorithms (i.e., centralized GEPA and distributed GEPA) to obtain the Nash equilibriums. Extensive simulations are conducted to demonstrate that the proposed power allocation algorithms can achieve satisfactory performance in terms of energy efficiency, convergence speed and fairness in cognitive sensor networks.