196 CHAPTER 7 Energy-Efficient Power Control Games As discussed in Lasaulce et al. (2009b) other choices are possible for the receiver's utility and can lead to different conclusions. The equivalent virtual mul- tiple input multiple output (MIMO) network performance is one of these, but will not be considered here. 7.6 REPEATED POWER CONTROL GAME So far, we have only considered one-shot power control games: for a given vector of channel gains, each transmitter chooses his power level in order to maximize his instantaneous utility. In this section and the next, we consider a more general situa- tion; all the transmitters can update their power levels several times within a block or a duration less than the channel coherence time. The corresponding power control type is called "distributed fast power control", generalizing the more conventional distributed slow power control for which the power levels are updated only once per block. In both cases, we want to take into account the fact that the players (namely the transmitters) interact several times, either within a block or from block to block. This feature introduces new types of behaviors (cooperation, punishment, etc.) with respect to the one-shot game. Conventionally, for i.i.d. channels, power control schemes are designed such that the power levels are chosen in an independent manner from block to block. In dis- tributed networks with selfish transmitters, the point of view has to be re-considered even if the channels are i.i.d. due to the fact that long-term interaction may change the behavior of selfish transmitters. The framework considered here is therefore the one of dynamic games. More specifically, we will analyze the case of repeated games (RG). Formally, from a game-theoretic viewpoint we assume that: the games are with complete information; transmitters are rational; rationality is common knowledge; transmitters are only informed with their individual CSI and a public signal. However, the proposed power control scheme can be implemented even if all these assumptions do not hold. As it will be seen, a transmitter only needs to know his own CSI and the power of the received signal for each block. This means that the framework in terms of information assumptions is very similar to that for iterative water-filling algorithms (Scutari et al., 2009; Yu et al., 2004). These comments also indicate that game theory can be used as a way of inventing new power control schemes, even if, in practice, some assumptions such as "rationality is common knowledge" may not hold. 7.6.1 Fast Power Control Here, we assume that transmitters update their power levels several times within a block or a time window which has a duration of less than the channel coherence time. In the one-shot power control game, each transmitter observes the channel gain associated with the current block, i.e., |h i |, and updates his power level once accord- ing to (7.5) in order to maximize his instantaneous utility. In repeated games, players want to maximize their averaged utility. To define the latter quantity, we first need