I would like your help to understand the concept of expansion of an information structure in the incomplete information game at p.6-9 [this][1]this paper.
Let $\underline{S}$ denote anthe information structure that is totally uninformative, i.e., it does not add anything to the baseline level of information assumed (also called DEGENERATE at p.26 of the linked paper). In other words, $\underline{S}$ consists of
(a) for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.
(b) a signal distribution $\pi: \Theta \rightarrow \Delta(T)$ such that $\pi(\cdot|\theta)=\tilde{\pi}$ $\forall \theta \in \Theta$ for some $\tilde{\pi}\in \Delta(T)$. In other words, the conditional probability is equal to the unconditional one and our belief on the probability distribution of the state is not updated.
Notice that there are many ways to characterise the uninformative information structuresstructure (just by varying $T$ and $\tilde{\pi}$).
Let $\mathcal{S}$ denote the collection of all possible information structures. More precisely,
Note that $\mathcal{S}$ contains also all possible ways to characterise the uninformative information structuresstructure.
- Question: Can we show that, for a given $\underline{S}$, each $S\in \mathcal{S}$ is an expansion of $\underline{S}$? Is this true also for (including $S=\underline{S}$)? [1]: http://www.princeton.edu/~smorris/pdfs/bce.pdf
This seems to me to hold at the light of Theorem 1 combined with reading at p.26 of the paper "Now consider the case where the original information structure is degenerate (there is only one signal which represents the prior over the states of the world). In this case, the set of Bayes correlated equilibria correspond to joint distributions of actions and states that could arise under rational choice by a decision maker with any information structure"