Comparison of information structures

We introduce two ways of comparing information structures, say ${\cal I}$ and ${\cal J}$. First we say that ${\cal I}$ is richer than ${\cal J}$ when for every compact game $G$, all correlated equilibrium distributions of $G$ induced by ${\cal J}$ are also induced by ${\cal I}$. Second, we say that ${\cal J}$ is faithfully reproducable from ${\cal I}$ when all the players can compute from their information in ${\cal I}$ ``new information'' that they could have received from ${\cal J}$. We prove that ${\cal I}$ is richer than ${\cal J}$ if and only if ${\cal J}$ is faithfully reproducable from ${\cal I}$.