The concept of information structure in incomplete information games I would like your help to understand the concept of information structure in the incomplete information game at p.6-7 of this paper.



Let me summarise the game as described in the paper. 

There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.

There is a finite set of states $\Theta$, with $\theta$ denoting a generic state. 

A basic game $G$ consists of


for each player $i$, a finite set of actions $A_i$, where we write $A\equiv A_1\times A_2\times ... \times A_N$, and a utility function $u_i: A\times \Theta \rightarrow \mathbb{R}$.
a full support prior $\psi\in \Delta(\Theta)$.


An information structure $S$ consists of


for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.
a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.


A decision rule of the incomplete information game $(G,S)$ is a mapping 
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$



My question:  

I interpret $\pi(t|\theta)$ as a probability that, conditional on the realisation $\theta$ of the state, the players receive as signals $t_1,...,t_N$, respectively. According to the given information structure, signals are more or less informative.

If this interpretation is correct, then I'm confused about the first sentence at p.7 of the linked paper: "If there is complete information, i.e., if $\Theta$ is singleton, [...]".

My first guess was that complete information is characterised by specifying $S$ and not by restricting the support of the state. In other words, I thought that complete information corresponds to the information structure $\bar{S}\equiv (\bar{T},\bar{\pi})$ such that


for each player $i$, $\bar{T}_i=\Theta$.
$\bar{\pi}(\theta|\theta)=1$.


Where am I wrong? Why the author defines complete information as having $|\Theta|=1$?
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.