Causal Coordination for Distributed Decision-Making

In decentralized network control, communication plays a critical role by transforming local observations into shared knowledge, enabling agents to coordinate their actions. This paper investigates how communication facilitates cooperation behavior and therefore improves the overall performance in the vector-valued Witsenhausen counterexample, a canonical toy example in distributed decision-making. We consider setups where the encoder, i.e., the first decision-maker (DM) acts causally and the decoder, i.e., the second DM, operates noncausally, 1) without and 2) with access to channel feedback. Using a coordination coding framework, we characterize the achievable power-estimation cost regions in single-letter expressions for both scenarios. The first result is that, when restricted to Gaussian random variables, the cost is identical across all setups featuring at least one causal DM - regardless of the presence of feedback information. Next, building on the characterization of the power-estimation cost region, we propose a hybrid scheme that combines discrete quantization with a continuous Gaussian codebook - the Zero Estimation Cost (ZEC) scheme - which achieves an arbitrarily small estimation cost. This scheme uses coding tools that allow perfect reconstruction of the target symbols, leading to an asymptotic estimation cost equal to zero, while significantly reducing the asymptotic power consumption. Furthermore, when channel feedback is available at the first DM, we propose an analogous scheme that simultaneously achieves zero power and zero estimation cost in the low-noise regime.