Causal Vector-valued Witsenhausen Counterexamples with Feedback

We study the continuous vector-valued Witsenhausen counterexample through the lens of empirical coordination coding. We characterize the region of achievable pairs of costs in three scenarios: (i) causal encoding and causal decoding, (ii) causal encoding and causal decoding with channel feedback, and (iii) causal encoding and noncausal decoding with channel feedback. In these vector-valued versions of the problem, the optimal coding schemes must rely on a time-sharing strategy, since the region of achievable pairs of costs might not be convex in the scalar version of the problem. We examine the role of the channel feedback when the encoder is causal and the decoder is either causal or non-causal, and we show that feedback improves the performance, only when the decoder is non-causal.