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

Abstract

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 nonstandard 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.P − E c P (U n 1 ) ≤ ε, S − E c S (X n 1 , U n 2 ) ≤ ε. (5)

State Leakage and Coordination With Causal State Knowledge at the Encoder

“…The need to coordinate with receiver action V is motivated by problems in which the terminals represent decision makers that choose actions (X, V ) as a function of the system state S, as in [7]. The state can also be used to represent a system to control, in which case coordination also ties to the Witsenhausen's counterexample [35], [36].…”

Continuous Random Variable Estimation is not Optimal for the Witsenhausen Counterexample

“…In [10], we have provided an overview on the individual findings and completed the missing cases using coding results from [11]. In [12] we have derived an achievability result considering non-causal encoding and causal decoding using a continuous alphabet building on proof methods from [13]. In this work, we now derive a genieaided outer bound for this case considering only decision strategies that result in continuous random variables which have a probability density functions.…”