Distributed Strategic Learning for Wireless Engineers

For nonexpansive map, the Picard iteration may not converge, generally, or even if it converges, its limit is not a fixed point of f.

Definition 2.2.6.3. We say that $f:\mathcal{X}\to \mathcal{X}$ is pseudo-contractive if

$\Vert f\left(x\right)-f\left(y\right)\Vert \le \Vert x-y\Vert +\Vert x-y+f\left(y\right)-f\left(x\right)\Vert ,\forall \left(x,y\right)\in {\mathcal{X}}^2.$

Best response and Krasnoselskij iteration

$x_{t+1}\in \left(1-\lambda \right)x_t+\lambda BR\left(x_t\right),t\ge 1$

(2.11)

$x_0\in \mathcal{X}.$

(2.12)

$\lambda \in \left(0,1\right)$

(2.13)

We have seen above two examples where the best reply dynamics has a cycling behavior. We explain how the algorithm can be designed such that the size of the limit cycle diminishes.

First consider the logistic map example. We have seen that for s = 4 there a cycling behavior of the best reply dynamics. Is there a way to reduce the size of the limit cycle?

To answer to this question, we start ...