# Bayesian Correlated Equilibrium in a one player case: relation with literature

I am curious about the connection between one-player Bayesian Correlated Equilibrium (hereafter, BCE) introduced by Bergemann and Morris for a generic $$n$$-player setting with $$n\geq 1$$ (here) and the Bayesian persuasion problem in Kamenica and Gentzkow (here) also discussed in Bergemann and Morris more recently (here).

I would like your help to summarise such relation.

Preliminaries: DM denotes decision maker. $$G$$ is the basiline choice problem that consists of the DM's prior on the state of the world. $$S$$ is the information structure that contains the probability distribution of the signal used by the DM to update the prior. $$(G,S)$$ is the augmented choice problem.

$$\underline{S}$$ denotes the degenerate information structure, i.e., the information structure that does not convey additional information on the state of the world.

The set of one-player BCE of $$G$$ consists of the set of probability distributions over actions and states that are consistent with the prior and obedient.

Note that the set of one-player BCE of $$G$$ is equal to the set of one-player BCE of $$(G,\underline{S})$$.

This is my attempt to link the two papers:

Suppose a mediator ("sender" in the Bayesian persuasion language) could pick the information structure ("experiment" in the Bayesian persuasion language) that the DM ("receiver" in the Bayesian persuasion language) could process.

Kamenica and Gentzkow (here) characterize the set of distributions over actions and states that the sender could induce through picking the null experiment (i.e., $$\underline{S}$$) and having the DM choose optimally. This set is equal to the set of one-player BCE of $$(G,\underline{S})$$.

Bergemann and Morris more recently (here) explains that such relation holds for any information structure. In other words, the set of distributions over actions and states that the sender could induce through picking ANY experiment $$S$$ and having the DM choose optimally is equal to the set of one-player BCE of $$(G,S)$$.

Is my connection correct?

There are a few imprecisions in the way you formalize things. For example saying "𝐺 is the baseline choice problem" does not make a lot of sense, because a problem should include the DM's utility, the available actions, and the belief about the state of the world. You only included the latter.

Regardless of notation, I think that you are missing the key connection between these two papers. Let me summarize what each paper does first and then draw the connection for you.

The concept of BCE characterizes the answer to the following question: If the DM were to observe some signal (informative or not) what are all the actions that would be optimal? Of course the DM can use mixed strategies, so we talk about distributions over actions. Also, it is clear that depending on what information is contained in the signal, the actions of the receiver can vary. So the set of BCE collects all possible actions for all possible signals. Note that the concept of BCE is agnostic about where this extra information is coming from.

So yes, the distributions of actions that are in the set of BCE are obedient (i.e. optimal given some belief), and the beliefs that rationalize these actions are consistent with the prior (I.e. they are the Bayesian posteriors derived from some signal given the DM's prior).

In contrast, Kamenica and Gentzkow (KG) set up a game where a sender chooses what signal to give a receiver and then the receiver takes an action that affects both players. Kamenica and Gentskow presented the simplest case, where the sender knows the prior of the DM and there is only one sender.

Their approach has some limitations because finding the optimal signal can be complicated; especially if the state space is not binary. In their paper, it is also not clear at all how to solve the problem if the receiver has more information than $$\underline S$$, or if there is more than one receiver. So yes KG assume that the receiver has some prior information $$\underline S$$, but in equilibrium the sender usually provides more information, so your statement "Kamenica and Gentzkow (here) characterize the set of distributions over actions and states that the sender could induce through picking the null experiment" is false.

Now, let me try to connect the two papers:

Bergeman and Morris (2019) found that we can drastically simplify the analysis of the game presented by KG if instead of maximizing over signals, we maximize the sender's utility choosing the distribution of actions that can be induced using some signal. They make the connection that if a distribution of actions can be chosen by the sender, it must be because it is optimal for the receiver to choose those action after observing some signal $$S\geq\underline S$$, i.e. if the distribution over actions is a BCE.

This is nice because the set of BCE's is relatively easy to find and to work with. So the game that KG setup can be dramatically simplified into a game where the sender chooses its favorite BCE.

Once you realize this connection, you can overcome many of the limitations in KG's approach. The concept of BCE can be easily extended to multiple receivers, and even receivers with private information. The authors showcase in their 2019 paper, the power of making this connection, and show how to think formally of receivers with more information than $$\underline S$$, among other things.