Stochastic nestedness and the belief sharing information pattern in decentralized control

In a dynamic decentralized control problem, a common information state supplied to each of the decision makers leads to a tractable dynamic programming recursion. However, communication requirements for such conditions require exchange of very large data noiselessly, hence these assumptions are generally impractical. We present a weaker notion of nestedness, which we term as stochastic nestedness, which is characterized by a sequence of Markov chain conditions. It is shown that if the information structure is stochastically nested, then an optimization problem is tractable, and in particular for LQG problems, the team optimal solution is linear, despite the lack of deterministic nestedness or partial nestedness. One other contribution of this paper is that, by regarding the multiple decision makers as a single decision maker and using Witsenhausen´s equivalent model for discrete-stochastic control, it is shown that the common state required need not consist of observations and it suffices to share beliefs on the state and control actions; a pattern we refer to as k-stage belief sharing pattern. We evaluate a precise expression for the minimum amount of information required to achieve such an information pattern for k = 1. The information exchange needed is generally strictly less than the information exchange needed for deterministic nestedness and is zero whenever stochastic nestedness applies.