  A game-theoretic model for studying power control in multi-carrier CDMA
systems is proposed. Power control is modeled as a non-cooperative game in
which each user decides how much power to transmit over each carrier to
maximize its own utility. The utility function considered here measures the
number of reliable bits transmitted over all the carriers per Joule of energy
consumed and is particularly suitable for networks where energy efficiency is
important. The multi-dimensional nature of users' strategies and the
non-quasiconcavity of the utility function make the multi-carrier problem much
more challenging than the single-carrier or throughput-based-utility case. It
is shown that, for all linear receivers including the matched filter, the
decorrelator, and the minimum-mean-square-error (MMSE) detector, a user's
utility is maximized when the user transmits only on its "best" carrier. This
is the carrier that requires the least amount of power to achieve a particular
target signal-to-interference-plus-noise ratio (SINR) at the output of the
receiver. The existence and uniqueness of Nash equilibrium for the proposed
power control game are studied. In particular, conditions are given that must
be satisfied by the channel gains for a Nash equilibrium to exist, and the
distribution of the users among the carriers at equilibrium is also
characterized. In addition, an iterative and distributed algorithm for reaching
the equilibrium (when it exists) is presented. It is shown that the proposed
approach results in significant improvements in the total utility achieved at
equilibrium compared to a single-carrier system and also to a multi-carrier
system in which each user maximizes its utility over each carrier
independently.
