Bayes Correlated equilibrium without information expansion

Question

The questions are with reference to Bergemann and Morris (2013, 2016).

I'm trying to give an alternative interpretation to BCE from the analyst/information designer perspective provided in the paper.

Take a basic game G (as in BM, 2013) without private information, and only prior on the fundamental $θ\sim N(\bar{θ},σ_{θ}^{2})$. i.e. agents condition their actions only on $\bar{θ}$.

Suppose that there exists a decision rule $\varphi:\Theta \rightarrow\Delta(A)$ in this game chosen randomly, but that could possibly be induced by a Bayes Nash Equilibrium of an augmented game G+ (this one with private information).

If I understood how BCE works; Agents in G don't know the true state of the world, but know only the decision rule (i.e. the distribution of action profiles for each possible realisation of the fundamental). Then some omniscient mediator will suggest each of them (privately) an action based on the decision rule $\varphi$. If each recommendation is obeyed by everyone then there is a Bayes Correlated Equilibrium in game G.

But obedience will require the recommendation to be optimal (i.e. induced by a BNE in G+).So, reverse engineer....

My interpretation of the "decision rule induced by BNE" is that agents extract the information from the suggestion of the mediator to compute their own optimal action. And if the optimal action is congruent with the suggestion, then they obey. That process is made in game G+.i.e. the only source of information of agents beside common prior on fundamentals is the information extracted from the recommendation.

But if that is the case, do they extract the whole information structure from just the suggestion? It seems normal to assume that a private signal could be extracted from a recommendation, but where does the common prior on private information comes from? How do agents extract this from the mediator's recommendation? It would have to be the case that the common prior on private information was there in G, but agents could not have access to it, and this access was activated only once recommendation was made or something like that?!

Again, I'm trying to understand the concept of BCE in game G when the decision rule $\varphi∈Δ(A×Θ)$ is induced by a BNE distribution $\pi∈Δ(A×S×Θ)$(where S is the space of private information) of an augmented game.

If we reinterpret BM (2016) in my context, their expansion of information in G+ is that the recommendation adds to the private information that agents already had. What I am describing instead is that neither agents nor the mediator had any private information beforehand. Private information in G+ is just what is extracted from the random decision rule in G. But then, where the heck does the common prior on private information come from?

I hope that was clear. Thanks enormously in advance for helping me.

Answer 1

An analyst wants to study the behavior of players playing a Bayes Nash equilibrium in a game of incomplete information. The analyst can observe some signal sources that the players have, the state of nature, and the actions that the players subsequently choose. The analyst is modest enough to consider it possible that the players might have additional sources of information that the analyst is not aware of.

So the analyst observes the joint distribution of states, signals the analyst is aware of, and actions. Under some regularity conditions, such a joint distribution can be described (together with the common prior) by a function from states and signal profiles to distributions over action profiles. Such a function is a decision rule. One interpretation of a decision rule is that it could be implemented by a mediator who knows the state of nature and the realization of all signals the analyst is aware of. This mediator then tells every player what actions they would play in the resulting randomly chosen action profile, and nothing else. We think of this action as a recommendation. But this is just an interpretation, the mediator is not some magical creature that comes in addition to the decision rule.

The central condition of a Bayes correlated equilibrium is that they are obedient. If a player only knows the common prior, the decision rule, that player's signal (the one the analyst is aware of), and the recommended action, then it is optimal to follow the recommendation. In terms of the information sources the analyst is aware of, the action recommendation is an additional signal to a player and tells them something they might not have known otherwise.

But it might not tell the player anything new, since the player might have already had additional information that the analyst is not aware of. Trivially, the player might already have the recommendation among the signals.

The central result of the 2016 paper of Bergemann and Morris is that the idea works: The joint distribution of states, signals the analyst is aware of, and action profiles is a Bayes correlated equilibrium if and only if there are additional information sources for the players and a corresponding Bayes Nash equilibrium such that the marginal distribution states, signals the analyst is aware of, and action profiles is given by the Bayes correlated equilibrium.

Btw: Bayes correlated equilibria are not (!) incentive-compatible in the usual sense that players would happily give their information to an uninformed mediator. If one, in addition, requires this form of incentive compatibility, one obtains the notion of a "communication equilibrium" due to Françoise Forges.