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.

翻译：本研究探讨了一种向量值Witsenhausen模型，其中第二位决策者在选择估计向量之前获取一个观测向量。在此模型中，第一位决策者以因果方式行动，而第二位决策者则以非因果方式进行估计。当向量长度增长时，我们通过单字母表达式刻画了第一位决策者的功率成本与第二位决策者的估计成本之间的最优权衡。本文证明，最佳线性方案可通过在两种仿射策略之间采用时分共享方法实现，这与1968年Witsenhausen解的凸包络相一致。此外，Witsenhausen的两点策略以及2010年Grover和Sahai提出的两个设备均以非因果方式运行的方案，均优于我们的最佳线性方案。因此，仅当所有决策者均以非因果方式运行时，才能通过分组编码方案获得性能增益。