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Chapter 7. Energy-Efficient Power Contro... > 7.5 The Hierarchical Power Control G... - Pg. 191

7.5 The Hierarchical Power Control Game 191 quasi-concavity property is not preserved by addition of a linear function, which is what happens here for u i , i K. Therefore, the Debreu­Fan­Glicksberg existence theorem cannot be used anymore. But, as noticed in Saraydar et al. (2002), by mak- ing an additional change, the resulting static game becomes super-modular, which allows one to apply an existence theorem for super-modular games (Theorem 59 in Chapter 2). Indeed, the trick proposed in Saraydar et al. (2002) is to assume that each transmitter operates in the interval P = [P c , P max ], where P c is the power level i i i which transmitter i requires to operate at an SINR equal to 0 , 0 being the unique inflection point of f . It is simple to check that if i K, p -i p -i , then the quan- tity u i (p) - u i p i , p -i is non-decreasing in p i on P i . The interpretation is that if the other transmitters, -i, increase their power (generating more interference), then transmitter i has to increase his. The game: G = (K, { P i } iK , {u i } iK ) (7.10) is therefore super-modular, which ensures the existence of at least one pure Nash equilibrium, following Theorem 59 in Chapter 2. The uniqueness problem is not trivial, and so far only simulations (Saraydar et al., 2002) have been used to analyze this issue. Another disadvantage of pricing-