Taking a look in the paper of Bergemman and Morris in 2016, they refer to the desicion rule as mapping
$$\sigma:\Theta\times T\to\Delta(A)$$
The explanation to understand the notion of it is given as it follows. ``One way to mechanically understand the notion of the decision rule is to view $\sigma$ as the strategy of an omniscient mediator who first observes the realization of $\theta \in \Theta$, where $\theta$ is the state of the world, chosen according to $\psi$ and the realization of $t\in T$, $t$ is the type of the player, chosen according to $\pi(\cdot|\theta)$,and then picks an actio profile $a\in A$ and privately announces to each player $i$ the draw of $a_i$."
$\textbf{Question:}$ Accoding to the definition of $\sigma$ and the explanation that follows, does this mean that the information designer needs to know exactly the state of the world and the types of the players or that she can condition on them?
It seems to me a little weird if they make the assumption that she knows exactly the state of the world. In my point of view I understand that the information designer is able somehow to know $t\in T$ that is drawn, which means she knows if some types are not drawn and which ones are drawn, and according to them she gives a vector of recommendations to every player for any state $\theta\in\theta$. For example, she announces recommendation like the following vector of mixned strategies to every player in a two actions, two types and two states game:
- Play $(x_1(a_1),x_2(a_2))$ if you are $t_1$ and and $(x_3(a_1),x_4(a_2))$ if you are $t_2$, at the state $\theta_1$.
- Play $(x_3(a_1),x_4(a_2))$ if you are $t_1$ and and $(x_1(a_1),x_2(a_2))$ if you are $t_2$, at the state $\theta_2$
Am I right? Could someone provide some explanation?