*as t goes to infinity and*,

$\begin{array}{c}{\dot{x}}_{1,t}\left({s}_{1}\right)={x}_{1,t}\left({s}_{1}\right)[\mathbb{E}{R}_{1}\left(\text{w},{\text{e}}_{{s}_{1}},{\beta}_{2,\u03f5}\left({\mathrm{x}}_{1,t}\right)\right)\\ -{\displaystyle \sum _{{{s}^{\prime}}_{1}\in {\mathcal{A}}_{1}}{\mathrm{x}}_{1,t}\left({{s}^{\prime}}_{1}\right)\mathbb{E}{R}_{1}\left(\text{w},{\text{e}}_{{s}_{1}},{\beta}_{2,\u03f5}\left({\mathrm{x}}_{1,t}\right)\right)}]\end{array}$ |
(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*,

$\Vert {\mathrm{x}}_{2t}-{\sigma}_{2,\u03f5}\left({\mathrm{x}}_{1}\right)\Vert \to 0,$

*as t goes to infinity and*,

${\dot{x}}_{1,t}\left({s}_{1}\right)={x}_{1,t}\left({s}_{1}\right)[\mathbb{E}{R}_{1}\left(\text{w},{\text{e}}_{{s}_{1}},{\sigma}_{2,\u03f5}\left({\mathrm{x}}_{1,t}\right)\right)$ |
(4.21) |

$-{\displaystyle \sum _{{{s}^{\prime}}_{1}\in {\mathcal{A}}_{1}}{x}_{1,t}\left({{s}^{\prime}}_{1}\right)\mathbb{E}{R}_{1}\left(\text{w},{\text{e}}_{{s}_{1}},{\sigma}_{2,\u03f5}\left({\mathrm{x}}_{1,t}\right)\right)}],$ |
(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) ...

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