  The dense deployment of small-cell base stations in HetSNets requires
efficient resource allocation techniques. More precisely, the problem of
associating users to SBSs must be revised and carefully studied. This problem
is NP-hard and requires solving an integer optimization problem. In order to
efficiently solve this problem, we model it using non-cooperative game theory.
First, we design two non-cooperative games to solve the problem and show the
existence of pure Nash equilibria (PNE) in both games. These equilibria are
shown to be far from the social optimum. Hence, we propose a better game design
in order to approach this optimum. This new game is proved to have no PNE in
general. However, simulations show, for Rayleigh fading channels, that a PNE
always exists for all instances of the game. In addition, we show that its
prices of anarchy and stability are close to one. We propose a best response
dynamics (BRD) algorithm that converges to a PNE when it exists. Because of the
high information exchange of BRD, a completely distributed algorithm, based on
the theory of learning, is proposed. Simulations show that this algorithm has
tight-to-optimal performance and further it converges to a PNE (when existing)
with high probability.
