186 CHAPTER 7 Energy-Efficient Power Control Games In Section 7.7 the discrete set assumption is used to apply the results of Fudenberg and Yamamoto (2009) and H orner et al. (2009) without introducing additional tech- ¨ nicalities. Finally, the receiver is always assumed to implement single-user decoding; i.e., each flow of symbols is decoded independently of the others, which are consid- ered as noise. For more advanced receivers such as those implementing successive interference cancellation, the reader is referred to Bonneau et al. (2008) and Lasaulce et al. (2009b). 7.3 THE ONE-SHOT POWER CONTROL GAME Here we study the scenario, introduced by Goodman and Mandayam (2000), where each transmitter updates his power level once per block in order to maximize his instantaneous utility. In order to define the corresponding static power control game some notation is in order. We denote by R i the information transmission rate (in bit/s) for user i, and by f an efficiency function representing the block success rate, which is assumed to be sigmoidal and identical for all the users; recall that a sigmoidal or S-shape function is convex up to a point and then concave from this point. For a given