Distributed channel assignment for network MIMO: game-theoretic formulation and stochastic learning

Abstract

The cooperative frequency reuse among base stations (BSs) can improve the system spectral efficiency by reducing the intercell interference through channel assignment and precoding. This paper presents a game-theoretic study of channel assignment for realizing network multiple-input multiple-output (MIMO) operation under time-varying wireless channel. We propose a new joint precoding scheme that carries enhanced interference mitigation and capacity improvement abilities for network MIMO systems. We formulate the channel assignment problem from a game-theoretic perspective with BSs as the players, and show that our game is an exact potential game given the proposed utility function. A distributed, stochastic learning-based algorithm is proposed where each BS progressively moves toward the Nash equilibrium (NE) strategy based on its own action-reward history only. The convergence properties of the proposed learning algorithm toward an NE point are theoretically and numerically verified for different network topologies. The proposed learning algorithm also demonstrates an improved capacity and fairness performance as compared to other schemes through extensive link-level simulations.

Proof of proposition 4

The Proposition is proved by investigating the nondecreasing and upper-bounded properties of $\varPsi$ along the trajectory of the ODE in (27). First, we rewrite the ODE in (27) as follows:

$$\begin{aligned} \frac{dp_{i,s_i}(t)}{dt} = p_{i,s_i}(t)\sum \limits _{s'_i \in \mathcal {A}_i}p_{i,s'_i}(t) \left[ \psi _i(\mathbf {e}_{s_i},\mathbf {P}_{-i}) - \psi _i(\mathbf {e}_{s'_i},\mathbf {P}_{-i})\right] . \end{aligned}$$

(35)

Given that $\varPsi (\mathbf {e}_{s_i}, \mathbf {P}_{-i}) = {\partial \varPsi (\mathbf {P})}/{\partial p_{i,s_i}}$ is positively correlated with $\psi _i(\mathbf {e}_{s_i}, \mathbf {P}_{-i})$, and let $D_{i,s_i,s'_i} = \psi _i(\mathbf {e}_{s_i}, \mathbf {P}_{-i})-\psi _i(\mathbf {e}_{s'_i}, \mathbf {P}_{-i})$, $E_{i,s_i,s'_i} = \varPsi (\mathbf {e}_{s_i}, \mathbf {P}_{-i})-\varPsi (\mathbf {e}_{s'_i}, \mathbf {P}_{-i})$, we may write

$$\begin{aligned} D_{i,s_i,s'_i} >0 \Leftrightarrow E_{i,s_i,s'_i} >0. \end{aligned}$$

(36)

By applying (35) and (36), the derivation of $\varPsi (\mathbf {P})$ with respect to $t$ is given by

$$\begin{aligned} \frac{d\varPsi (\mathbf {P})}{dt}&= \sum \limits _{i \in \mathcal {N}}\sum \limits _{s_i \in \mathcal {A}_i} \frac{\partial \varPsi (\mathbf {P})}{\partial p_{i,s_i}} \frac{d p_{i,s_i}}{dt} \nonumber \\&= \sum \limits _{i \in \mathcal {N}}\sum \limits _{s_i,s'_i \in \mathcal {A}_i} p_{i,s_i}p_{i,s'_i}\varPsi (\mathbf {e}_{s_i},\mathbf {P}_{-i}) \cdot D_{i,s_i,s'_i} \nonumber \\&= \frac{1}{2}\sum \limits _{i \in \mathcal {N}}\sum \limits _{\begin{array}{c} s_i,s'_i \in \mathcal {A}_i\\ s_i < s'_i \end{array}} p_{i,s_i}p_{i,s'_i} E_{i,s_i,s'_i} \cdot D_{i,s_i,s'_i} \nonumber \\&\ge 0 \end{aligned}$$

(37)

where the last inequality holds since given the condition in (36), $D_{i,s_i,s'_i}$ and $E_{i,s_i,s'_i}$ always have the same sign.

Thus, $\varPsi$ is nondecreasing along the trajectories of the ODE, and asymptotically all the trajectories will be in the set $\{\mathbf {P} \in \mathcal {P}: \frac{d\varPsi (\mathbf {P})}{dt} = 0\}$. From (35) and (37), we know

$$\begin{aligned}&\frac{d\varPsi (\mathbf {P})}{dt} = 0 \nonumber \\&\quad \Rightarrow p_{i,s_i}p_{i,s'_i}\left[ \psi _i(\mathbf {e}_{s_i}, \mathbf {P}_{-i})-\psi _i(\mathbf {e}_{s'_i}, \mathbf {P}_{-i})\right] ^2=0, \quad \forall i, s_i, s'_i \nonumber \\&\quad \Rightarrow \frac{dp_{i,s_i}}{dt} = 0, \quad \forall i, s_i, s'_i \nonumber \\&\quad \Rightarrow \mathbf {P}^*\,\text {is\;a\;stationary\;point\;of\;the\;ODE}\, (27). \end{aligned}$$

(38)

According to Proposition 3, when starting from an interior point of the simplex of the mixed strategy space $\mathcal {P}$, the trajectory of the ODE in (35) converges to a stable stationary point, i.e., an NE. Then, by Proposition 2, the linearly interpolated process of the strategy update $p_{i,s_i}(n)$ is bounded within the neighborhood of the trajectory of (35). Thus, we complete the proof [22, Theorem 3.3].