Remote Empirical Coordination

We apply the framework of imperfect empirical coordination to a two-node setup where the action $X$ of the first node is not observed directly but via $L$ agents who observe independently impaired measurements $\widehat{X}$ of the action. These $L$ agents, using a rate-limited communication that is available to all of them, help the second node to generate the action $Y$ in order to establish the desired coordinated behaviour. When $L<\mathrm{\infty }$, we prove that it suffices $R_i\ge I(\widehat{X};\widehat{Y})$ for at least one agent whereas for $L⟶\mathrm{\infty }$, we show that it suffices $R_i\geq I\left(\hat X;\hat Y|X\right)$forallagentswhere$\hat Y$isarandomvariablesuchthat$X-\hat X-\hat Y$and$|p_{X,\hat Y}\left(x,y\right)-p_{X,Y}\left(x,y\right)|_{TV}\leq \Delta$($\Delta$ is the pre-specified fidelity).