A game-theoretical paradigm for collaborative and distributed power control in wireless networks

Abstract

The wireless revolution requires future wireless networks the capability of intelligently optimizing the spectrum by collaborating and using autonomy to determine not just the best use of the spectrum for its own system, but the best use of spectrum for other systems that share the same spectrum bands. How to develop the wireless paradigm of collaboration, therefore, is a crucial question. In this paper, we discuss how to model collaborative power control in a wireless interference network, where users share the same frequency band. By collaborating with other users, each user exchanges information to maximize not only its own performance but also others’ performances. A game theory framework is developed to determine the optimal power allocation. The proposed framework possesses several advantages over conventional methods, such as low complexity and fast converging algorithmic solutions, distributed implementation, and better user fairness. Simulation results state the proposed approach provides better fairness between users’ data rates, higher performance in the aggregate rate, and lower convergence time.

Appendices

Appendix A

1.1 Proof of Lemma 1

The first part of the Lemma easily follows from the SINR constraints (${\gamma _i \ge \gamma _i^{tar})}$ as

$$\begin{aligned} p_i \ge \frac{\gamma _i^{tar}}{h_{ii}} \left( \sum \limits _{j \ne i} h _{ji} p_j + \sigma _i^2\right) . \end{aligned}$$

(47)

Since $p_i \le p_i^{max}$ for all $i \in \mathcal {M}$, then

$$\begin{aligned} \frac{\gamma _i^{tar}}{h_{ii}} \left( \sum \limits _{j \ne i} h _{ji} p_j^{max} + \sigma _i^2\right) \ge \frac{\gamma _i^{tar}}{h_{ii}} \left( \sum \limits _{j \ne i} h _{ji} p_j + \sigma _i^2\right) . \end{aligned}$$

(48)

Hence, if $\forall i \in \mathcal {M}$, (19) holds, then there always exists a power $p_i \in \left[ 0,p_i^{max}\right]$ such that $\gamma _i \ge \gamma _i^{tar}$ is fulfilled.

Next, since the game is a potential game with a strictly concave potential function, it admits a unique maximizer $p_i \in \mathbb {R}^{+}$. Accounting for the SINR constraint (2) and imposing (22) eventually yields (20).

Appendix B

1.1 Proof of the Lemma 3

The utility function of player i is given by

$$\begin{aligned} \begin{aligned} \tilde{U}_i (p_i, \textbf{p}_{-i}, z_i)&= f(p_i, \textbf{p}_{-i}) + \tilde{g}_i (p_i, \textbf{p}_{-i}, z_i) \\ \end{aligned} . \end{aligned}$$

(49)

Since $\tilde{U}_i (p_i, z_i)$ is a concave function with $p_i$, there are unique maximizer point $p_{i}^{*}$ which is determined by

$$\begin{aligned} p_i^{+} \buildrel \Delta \over = {\text {*}}{arg\,max}_{{p_k} \in {^ + }} {\tilde{U}_i}\left( {{p_i},{\mathbf{{p}}_{ - i}}} \right) \end{aligned}$$

(50)

Thus, we have:

$$\begin{aligned}&{\left. {\frac{{\partial {{\hat{U}}_i}\left( {{p_i},{z_i}} \right) }}{{\partial {p_i}}}} \right| _{{p_i} = p_i^ + }} = 0 \\&\Rightarrow p_i^{+} = \frac{1}{{{c_i} - \sum \limits _{j \ne i} {{\alpha _j}{\varphi _i}\left( {{z_i}} \right) } }} - \sum \limits _{j \ne i} {\frac{{{h_{ji}}}}{{{h_{ii}}}}{p_j}} - \frac{{{\sigma _i^2}}}{{{h_{ii}}}} \end{aligned}$$

(51)

Appendix C

1.1 Proof of the Lemma 5

The utility function of player i is given by

$$\begin{aligned}&\bar{U}_i (p_i,\textbf{p}_{-i}, z_i) = \hat{f}(p_i)+ \hat{g}(z_i) + \bigtriangledown _{p_i} \hat{g}(z_i)^{T} \left( p_i-z_i\right) \nonumber \\ \end{aligned}$$

(52)

Since $\bar{U}_i (p_i, \textbf{p}_{-i}, z_i)$ is a concave function with $p_i$, there are unique maximizer point $p_{i}^{\star }$ which is determined by

$$\begin{aligned}&p_i^{\star } \buildrel \Delta \over = {\text {*}}{arg\,max}_{{p_k} \in {^ + }} {{\bar{U}}_i}\left( {{p_i},{\mathbf{{p}}_{ - i}}}, z_i \right) \\ \Rightarrow&p_i^{\star } = \frac{1}{{{c_i} + \sum \limits _{j \ne i} {{\alpha _j}\frac{{{h_{ij}}}}{{\ln 2\left( {{h_{ij}}{z_i} + \sum \limits _{k \ne i,j} {{h_{kj}}{p_k}} + {\sigma _j^2}} \right) }}} }}. \end{aligned}$$

(53)

Appendix D

Proof of the Proposition 6

We prove that the best response function meets the three requirements of a standard function. We first consider the function $p_i^{tar} \left( \textbf{p}_i\right)$ as following.

• Positively: $p_i^{tar} \left( \textbf{p}_{-i}\right) > 0$

• Monotonicity: $p_i^{tar} \left( \textbf{p}_{-i}\right)$ is increasing in all $\{p_j\}_{j \ne i}$

• Scalability: take any $\omega >1$ then it holds

  $$\begin{aligned} p_i^{tar} \left( \omega \textbf{p}_{-i}\right) = \omega \frac{\gamma _i^{tar}}{h_{ii}} \left( \sum \limits _{j \ne i} h _{ji} p_j +\frac{ \sigma _i^2}{\omega }\right) < \omega p_i^{tar} \left( \textbf{p}_{-i}\right) . \end{aligned}$$

  (54)

Thus, $p_i^{tar} \left( \textbf{p}_i\right)$ is a standard function. Next, we prove that $p_i^{\star } \left( \textbf{p}_i\right)$ is a standard function as followings:

• Positively: $p_i^{\star } \left( \textbf{p}_{-i}\right) > 0$

• Monotonicity: $p_i^{\star } \left( \textbf{p}_{-i}\right)$ is increasing in all $\{p_j\}_{j \ne i}$. If $\textbf{p}^{+}_{-i} \ge \textbf{p}^{++}_{-i}$ then

  $$\begin{aligned} p_i^{\star } \left( \textbf{p}^{+}_{-i}\right) = \frac{1}{{{c_i} + \sum \limits _{j \ne i} {{\alpha _j}\frac{{{h_{ij}}}}{{\ln 2\left( {{h_{ij}}{z_i} + \sum \limits _{k \ne i,j} {{h_{kj}}{p^{+}_k}} + {\sigma _j^2}} \right) }}} }} > p_i^{\star } \left( \textbf{p}^{++}_{-i}\right) . \end{aligned}$$

  (55)

• Scalability: take any $\varepsilon >1$ then it holds

  $$\begin{aligned} \varepsilon p_i^{\star } = \frac{1}{{\frac{{{c_i}}}{\varepsilon } + \sum \limits _{j \ne i} {{\alpha _j}\frac{{{h_{ij}}}}{{\ln 2\left( {\varepsilon {h_{ij}}{z_i} + \sum \limits _{k \ne i,j} {\varepsilon {h_{kj}}{p_k}} + \varepsilon {\sigma _j^2}} \right) }}} }} > p_i^*\left( {\varepsilon {\mathbf{{p}}_{ - i}}} \right) \end{aligned}$$

  (56)

Since $p_i^{\star } \left( \textbf{p}_i\right)$ is a standard functions and $p_i^{max}$ does not depend on $\textbf{p}_{-i}$, we conclude that the best response function $\bar{\mathcal {B}}_{-i}\left( \textbf{p}_{-i}\right)$ is also a standard function with variable $p_i$.