The concept of information structure in incomplete information games
What you are defining is the complete information structure $\bar{S}$ with $\bar{T}_{i}=\Theta$ for all $i$ and  $$ \bar{\pi}(t|\theta) = \begin{cases} \ 1, &amp; \text{if} \;\; t_{i}= \theta \text{ for all} \; i \\ 0, &amp; \text{otherwise}, \end{cases} $$ for all $\theta \in \Theta$. 

Note that this is not the same as the case in which $\Theta$ is a singleton, but it is rather an extreme information structure. The same authors define this here (Bayes Correlated Equilibrium and the Comparison of Information Structures - Bergemann &amp; Morris), it beats me why they didn't include this detail in the paper you are looking at.