Remote Empirical Coordination

Abstract

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 $\hat{X}$ 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 $L<\infty$, we prove that it suffices $R_i\ge I\left(\hat{X};\hat{Y}\right)$ for at least one agent whereas for $L⟶\infty$, we show that it suffices $R_i\ge I\left(\hat{X};\hat{Y}|X\right)$ for all agents where $\hat{Y}$ is a random variable such that $X-\hat{X}-\hat{Y}$ and $\Vert p_{X,\hat{Y}}\left(x,y\right)-p_{X,Y}\left(x,y\right){\Vert }_{TV}\le \Delta$ ( $\Delta$ is the pre-specified fidelity).