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7.5 The Hierarchical Power Control Game > 7.5.1 Introducing Hierarchy Among the... - Pg. 192

192 CHAPTER 7 Energy-Efficient Power Control Games 3. Scenario 3 (He et al., 2010b): there are K levels of hierarchy, with one transmitter per level. This is the scenario reported in this chapter. The motivations for such a framework are as follows (Lasaulce et al., 2009b): · · To obtain an equilibrium profile which is more efficient (in the sense of Pareto) than the Nash equilibrium profile of G. As this can be reached by implementing a pricing mechanism, the second moti- vation is to avoid the four main drawbacks of the pricing-based power control policy, namely the goal is that: · the additional assumptions made on the action spaces in order to have the super-modularity property in G be removed; · the system is predictable analytically; that is the uniqueness of the game outcome is ensured; · the optimal power control policy be determined explicitly; · the transmitters exploit individual CSI only (at least in the non-saturated regime). A general motivation for this approach is to design networks where intelligence is split between the receiver (e.g., the base station) and transmitters (e.g., the mobile stations) in order to find a desired trade-off between the global network performance reached at the equilibrium and the amount of signaling needed to make it work. This cost induced by the proposed formulation is not evaluated here and both this evaluation and the mentioned tradeoff are problems left to the interested reader. 7.5.1 Introducing Hierarchy Among the Transmitters In this section, we propose a Stackelberg formulation of the power control game where there are K different levels. Without loss of generality (but possibly with loss of optimality) we assume that transmitter K can observe the action played by transmitters K - 1, K - 2, . . . , 1, transmitter K - 1 can observe the action played by transmitters K - 2, K - 3, . . . , 1, . . ., and transmitter 1 cannot observe any transmit- ter. The receiver is not a player of the game here. In this respect, we always assume that single-user decoding is implemented at the receiver. The motivations for using single-user decoding are precisely that the receiver has to remain neutral in the game, and/or to limit the receiver complexity, and/or to minimize the possible signaling cost induced by a more advanced receiver. For example, in Lasaulce et al. (2009b) the authors assume successive interference cancellation, which naturally introduces hierarchy between transmitters via the decoding order used by the receiver. Under the assumptions made, the utility of transmitter 1 only depends on p 1 since transmitter 1 knows that transmitter 2 observes his action and reacts to this observation accord- ingly. So p 2 = p 2 ( p 1 ), the utility of transmitter 2 only depends on ( p 1 , p 2 ), . . ., and the utility of transmitter K depends on the whole action profile ( p 1 , . . . , p K ). Know- ing that the reaction of a player is always a scalar-valued function, the corresponding