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Implicit coordination in two-agent team problems with continuous action sets. Application to the Witsenhausen cost function

The main contribution of this paper is the characterization of the limiting average performance of a system involving two agents who coordinate their actions which belong to a continuous set. One agent has complete and noncausal knowledge of the sequence of i.i.d. realizations of a random state X0, which is called the system state and affects the common team payoff. For the other agent, two scenarios in terms of observation assumptions are considered: in the first scenario, the other agent has a strictly causal knowledge of the system state while in the second scenario it has no direct knowledge of the state at all. However, in both scenarios, the less informed agent can always observe a noisy and strictly causal version of the actions taken by the (most) informed agent. There exists no dedicated communication channel between the two agents, and thus, the informed agent can only communicate via its actions which in turn affect the common payoff, hence the term implicit communication. Thus, there is a tradeoff to be found for the informed agent between communicating information about the incoming realizations of the system state and maximizing the payoff at the current stage. We use this general framework and apply it to a specific cost function, namely the Witsenhausen cost function. Although the problem tackled differs from the famous Witsenhausen's counterexample, the authors believe interesting new connections which help to understand the corresponding open problem might be established over time. A numerical analysis is conducted to assess the Witsenhausen's cost for two sub-optimal classes of strategies.