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In this paper, we propose an efficient energy-management framework in wireless sensor networks (WSNs) to address the fundamental research challenge imposed by both the maintenance of the energy supply and the support of the quality-of-information (QoI) requirements. By quantifying the QoI benefit the tasks receive in relation to the level of QoI they request as the QoI satisfaction index), we propose a QoI-aware energy-management scheme to distributedly decide the participating state of each sensor. Specifically, by using the mathematical framework of the Gur Game, we propose a novel pay-off structure taking into account the QoI and the energy consumption. We finally evaluate the proposed scheme under an event occurrence detection scenario, where the proposed scheme successfully guarantees less than 7% QoI outage, saves 80% of the energy reserve if compared with the lower bound solution, and achieves the suboptimum with only 4% gap if compared with optimal solution.