Distributed Strategic Learning for Wireless Engineers

as t goes to infinity and,

$\begin{array}{c}{\dot{x}}_{1,t}\left(s_1\right)=x_{1,t}\left(s_1\right)[\mathbb{E}{R}_1\left(\text{w},{\text{e}}_{s_1},{\beta }_{2,ϵ}\left({\mathrm{x}}_{1,t}\right)\right)\\ -{\displaystyle \sum _{{s^{\prime }}_1\in {\mathcal{A}}_1}{\mathrm{x}}_{1,t}\left({s^{\prime }}_1\right)\mathbb{E}{R}_1\left(\text{w},{\text{e}}_{s_1},{\beta }_{2,ϵ}\left({\mathrm{x}}_{1,t}\right)\right)}]\end{array}$

(4.20)

is the asymptotic pseudo-trajectory of ${\left\{{\mathrm{x}}_{1t}\right\}}_{t\ge 0}$.

•  Assume that Player 1 is a slow learner of (M-IBG) and Player 2 is a fast learner of (IBG). Then almost surely,

$\Vert {\mathrm{x}}_{2t}-{\sigma }_{2,ϵ}\left({\mathrm{x}}_1\right)\Vert \to 0,$

as t goes to infinity and,

${\dot{x}}_{1,t}\left(s_1\right)=x_{1,t}\left(s_1\right)[\mathbb{E}{R}_1\left(\text{w},{\text{e}}_{s_1},{\sigma }_{2,ϵ}\left({\mathrm{x}}_{1,t}\right)\right)$

(4.21)

$-{\displaystyle \sum _{{s^{\prime }}_1\in {\mathcal{A}}_1}{x}_{1,t}\left({s^{\prime }}_1\right)\mathbb{E}{R}_1\left(\text{w},{\text{e}}_{s_1},{\sigma }_{2,ϵ}\left({\mathrm{x}}_{1,t}\right)\right)}],$

(4.22)

is the asymptotic pseudo-trajectory of ${\left\{{\mathrm{x}}_{1t}\right\}}_{t\ge 0}$.

4.4.3    Aggregative Robust Games in Wireless Networks

The focus of our analysis in this subsection is on aggregative games. In an aggregative game, the payoff of each player is a function of the player’s own action and of the sum of the actions (or, the weighted sum action) ...