Optimal Gaussian Strategies for Vector-valued Witsenhausen Counterexample with Non-causal State Estimator

In this study, we investigate a vector-valued Witsenhausen model where the second decision maker (DM) acquires a vector of observations before selecting a vector of estimations. Here, the first DM acts causally whereas the second DM estimates non-causally. When the vector length grows, we characterize, via a single-letter expression, the optimal trade-off between the power cost at the first DM and the estimation cost at the second DM. In this paper, we show that the best linear scheme is achieved by using the time-sharing method between two affine strategies, which coincides with the convex envelope of the solution of Witsenhausen in 1968. Here also, Witsenhausen's two-point strategy and the scheme of Grover and Sahai in 2010 where both devices operate non-causally, outperform our best linear scheme. Therefore, gains obtained with block-coding schemes are only attainable if all DMs operate non-causally.