Coordination Coding with Causal Decoder for Vector-valued Witsenhausen Counterexample Setups

The vector-valued extension of the famous Witsenhausen counter-example setup is studied where the first decision maker (DM1) non-causally knows and encodes the iid state sequence and the second decision maker (DM2) causally estimates the interim state. The coding scheme is transferred from the finite alphabet coordination problem for which it is proved to be optimal. The extension to the Gaussian setup is based on a non-standard weak typicality approach and requires a careful average estimation error analysis since the interim state is estimated by the decoder. Next, we provide a choice of auxiliary random variables that outperforms any linear scheme. The optimal scheme remains unknown.