Continuity, Existence and Topological Properties of Optimization on the Space of Information Structures

In this chapter, we will continue with our discussion of information structures (from the previous chapter) but from a different perspective: studying their regularity (such as continuity) and topological properties. In particular, we study topological and structural properties of optimal cost on spaces of information structures. We investigate values of measurement channels and the problem of channel optimization for a class of stochastic control problems. In addition to establishing existence and structural results for optimal policies, the topological properties (such as continuity on space of information structures) developed in this chapter will also be useful in assessing robustness of designs when a probabilistic description of a model is not fully accurate. We will introduce quantizers as a special class of measurement channels, and obtain several useful properties and existence results for optimal quantizers (quantizers will be studied further in the book in Part III). We will also show that all stochastic dynamic teams are nearly static reducible (with policy-independence). We will also investigate the effects of the prior measure on static and dynamic team decision problems.