We prove that for any modulation, demodulation, and coding schemes and under general assumptions on the wireless channel model, the optimal power-efficient policy for transmit power assignment is necessarily of threshold nature. Although detailed features of this policy are quite complicated, it admits a simple and practical suboptimal version. Since both the optimal and suboptimal policies may have poor location-fairness characteristics, we introduce novel performance metrics (e.g., the average downtime and uptime), which quantify the short-term behavior of these policies, and develop analytical methods for their evaluation. Using these methods, we show that, although threshold policies may be extremely power efficient, users in poor locations relative to the base station may have unacceptable short-term performance characteristics. To alleviate this deficiency, we introduce an adaptive threshold policy, in which the threshold is adjusted to the user's location, and show that it is both relatively power efficient and location fair.