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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?

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Bayes Correlated Equilibrium characterizes (by Theorem 1 in the paper) what can happen in a Bayes Nash Equilibrium in which the players might have more information than is specified in the Bayesian game.

One way to think of it is that some "omniscient mediator" figures out what the agents would do with the additional information and simply tells them what action they would play with which probability. It turns out to be optimal to follow the mediator, even if the players do not have the additional information at hand.

But the solution concept is defined for the initially given Bayesian game that might not include all relevant information. In particular, players only observe their types, and a recommendation to do something in a certain state of nature is useless because they do not observe states of nature directly.

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  • $\begingroup$ You say ``In particular, players only observe their types and a recommendation to do something in a certain state of nature is useless because they do not observe states of nature directly.'' you mean that the omniscient mediator tells to every player to choose a single mixed action, no matter the state, right? How can this recommendation be optimal for any state? When the player will solve her problem she will check if this maximizes her expected payoff, because she will never knwo the state of the world $\endgroup$
    – Hunger Learn
    Dec 3, 2021 at 14:20
  • $\begingroup$ @HungerLearn The mediator actually recommends a single pure action to each player. Effectively, the mediator says "If you would know even more than you know now, you would play $a$." Of course, you want to do what the mediator says. The recommendation the mediator makes to you will, in general, depend on both the state and the types of all agents. $\endgroup$
    – Michael Greinecker
    Dec 3, 2021 at 14:47
  • $\begingroup$ If you check this survery, cowles.yale.edu/sites/default/files/files/pub/d20/d2075.pdf, in pages 8-10, the decision rule is a pair $(p_G,p_B)$. Namely it says invest with probability $p_G$ in the good state and $p_B$ in the bad state. Does this mean that it gives a recommendation for every state or that it gives a mixed action? And why do they differ? They use the same model $\endgroup$
    – Hunger Learn
    Dec 3, 2021 at 15:21
  • $\begingroup$ @HungerLearn You seem to refer to the section that deals with a single player who has no prior information. The decision rule recommends in each state a single action with a certain probability. It is actually quite important in this example that the firm does not completely learn the state when being told to invest or not. The rule conditions only on states and not types, since the player is uninformed and has no different types to condition on. $\endgroup$
    – Michael Greinecker
    Dec 3, 2021 at 16:21
  • $\begingroup$ So, when the uncertainty has two sources, where the one comes from the types and the other from the states of the world, then the case is that, a mediator tells to every player, a single recommendtaion, which is a single action ( or mixed action). Always, under the assumption that that the mediator is able to condition with respect to the state and the signals. $\endgroup$
    – Hunger Learn
    Dec 3, 2021 at 16:42

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