We discuss the Witsenhausen counterexample from the perspective of varying power budgets and propose a low-power estimation (LoPE) strategy. Specifically, our LoPE approach designs the first decision-maker (DM) a quantization step function of the Gaussian source state, making the target system state a piecewise linear function of the source with slope one. This approach contrasts with Witsenhausen's original two-point strategy, which instead designs the system state itself to be a binary step. While the two-point strategy can outperform the linear strategy in estimation cost, it, along with its multi-step extensions, typically requires a substantial power budget. Analogous to Binary Phase Shift Keying (BPSK) communication for Gaussian channels, we show that the binary LoPE strategy attains first-order optimality in the low-power regime, matching the performance of the linear strategy as the power budget increases from zero. Our analysis also provides an interpretation of the previously observed near-optimal sloped step function ("sawtooth") structure to the Witsenhausen counterexample: In the low-power regime, power saving is prioritized, in which case the LoPE strategy dominates, making the system state a piecewise linear function with slope close to one. Conversely, in the high-power regime, setting the system state as a step function with the slope approaching zero facilitates accurate estimation. Hence, the sawtooth solution can be seen as a combination of both strategies.

We propose a zero estimation cost (ZEC) scheme for causal-encoding noncausal-decoding vector-valued Witsenhausen counterexample based on the coordination coding result. In contrast to source coding, our goal is to communicate a controlled system state. The introduced ZEC scheme is a joint controlcommunication approach that transforms the system state into a sequence that can be efficiently communicated using block coding. The noncausal decoder receives sufficient information for reconstructing the system state perfectly, enabling the achievable estimation cost to be zero. Numerical results show that our approach significantly reduces the power budget required for achieving zero-estimation-cost state reconstruction at the decoder. In the second part, we introduce a more general non-zero estimation cost (Non-ZEC) scheme. We observe numerically that the Non-ZEC scheme operates as a time-sharing mechanism between Witsenhausen's original two-point strategy and the ZEC scheme. Overall, by leveraging block-coding gain, our proposed methods substantially improve the power-estimation trade-off for Witsenhausen counterexample.

We study the continuous vector-valued Witsen-hausen counterexample with Gaussian states 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.

We investigate the Witsenhausen counterexample in a continuous vector-valued context with a causal encoder and noncausal decoder. Our main result is the optimal single-letter condition that characterizes the set of achievable Witsenhausen power costs and estimation costs, leveraging a modified weak typicality approach. In particular, we accommodate our power analysis to the causal encoder constraint, and provide an improved distortion error analysis for the challenging estimation of the interim state. Interestingly, the idea of dual role of control is explicitly captured by the two auxiliary random variables.

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 tradeoff 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.