<ul>
<li>The terms experiment and information structure are used interchangeably by the literature, so both terms are valid regardless of <span class="math-container">$N$</span>.</li>
<li>I agree if <span class="math-container">$N=1$</span> there is really no game, and it is rather a single-agent decision problem (or decision problem with uncertainty), and yes, you don't have a player, rather a decision maker.</li>
<li>For the same token, you don't want to say the solution concept if BNE, instead you want to say that the agent maximizes his expected utility given the information she received. </li>
<li>I was not aware of the term "one-player BCE" but Bergemann and Morris coined the term BCE, so if that is how they call it, it is correct.</li>
<li>In terms of other references, <a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1941708" rel="nofollow noreferrer">this</a> is an earlier version of that same paper. However, I think that the paper you linked is the right paper for citing the concept of BCE. </li>
</ul>

<p>In <a href="https://www.aeaweb.org/articles?id=10.1257/jel.20181489" rel="nofollow noreferrer">this</a> later paper, the same two authors made the remarkable connection that the problem of <a href="https://faculty.chicagobooth.edu/emir.kamenica/documents/bayesianPersuasion.pdf" rel="nofollow noreferrer">Bayesian Persuasion</a> popularized by Kamenica and Gentzkow could be interpreted in light of the concept of BCE when <span class="math-container">$N=1$</span>. So perhaps, this later reference of BM can be useful if you want to explore more the case of <span class="math-container">$N=1$</span>.</p>

CC BY-SA 4.0
2019-10-13T18:36:10.907Z
Regarding you last comment, I'm trying to formalise it here https://economics.stackexchange.com/questions/32219/bayesian-correlated-equilibrium-in-a-one-player-case-relation-with-literature. Could you help me understand whether what I wrote is correct?
Star
CC BY-SA 4.0
2019-06-22T10:09:56.923Z
Thanks. Regarding the connection with Bayesian Persuasion: is it correct to say that such a connection exists only under the degenerate information structure (i.e., not informative signal)?
Star
CC BY-SA 4.0
2019-06-22T11:19:49.447Z
I'm saying this because in the paper here http://www.princeton.edu/~smorris/pdfs/bce.pdf, BM seem to make a comparison with Bayesian Persuasion under the degenerate information structure only. But then this remark disappears in the BM's 2019 paper that you linked.
Star
CC BY-SA 4.0
2019-06-23T23:30:05.723Z
No, the comparison between BCE and Bayesian persuasion is valid even if S is not degenerate. What the authors are saying is that if S is degenerate (which means that the receiver does not have private information) their model maps to the problem presented in the original paper by Kamenica and Gentzkow. However,  in fact, BM's 2019 precisely use this connection to show how to solve the Bayesian persuasion problem when the receiver has private information (i.e. when S is not degenerate). I hope this makes sense.
Regio
CC BY-SA 4.0
2019-06-21T20:01:30.517Z
2019-06-21T20:01:30.517Z
Regio
<p>I would like your help to use the correct terminology to define a <strong>Bayes Correlated Equilibrium (BCE)</strong> and a <strong>Bayesian Nash Equilibrium (BNE)</strong> in a <strong>"game" with one player</strong>. The notion of BCE in an <span class="math-container">$N$</span>-player game has been introduced in <a href="http://www.princeton.edu/~smorris/pdfs/bce.pdf" rel="noreferrer">this</a> paper. </p>

<hr>

<p>Let me describe the game and give the notion of BCE and BNE in an <span class="math-container">$N$</span>-player game. </p>

<p>There are <span class="math-container">$N\in \mathbb{N}$</span> players, with <span class="math-container">$i$</span> denoting a generic player.</p>

<p>There is a finite set of states <span class="math-container">$\Theta$</span>, with <span class="math-container">$\theta$</span> denoting a generic state. </p>

<p>A "basic game" <span class="math-container">$G$</span> consists of</p>

<ul>
<li><p>for each player <span class="math-container">$i$</span>, a finite set of actions <span class="math-container">$A_i$</span>, where we write <span class="math-container">$A\equiv A_1\times A_2\times ... \times A_N$</span>, and a utility function <span class="math-container">$u_i: A\times \Theta \rightarrow \mathbb{R}$</span>.</p></li>
<li><p>a full support prior <span class="math-container">$\psi\in \Delta(\Theta)$</span>.</p></li>
</ul>

<p>An "information structure" <span class="math-container">$S$</span> consists of</p>

<ul>
<li><p>for each player <span class="math-container">$i$</span>, a finite set of signals <span class="math-container">$T_i$</span>, where we write <span class="math-container">$T\equiv T_1\times T_2\times ... \times T_N$</span>.</p></li>
<li><p>a signal distribution <span class="math-container">$\pi: \Theta \rightarrow \Delta(T)$</span>.</p></li>
</ul>

<p>A decision rule of the incomplete information game <span class="math-container">$(G,S)$</span> is a mapping 
<span class="math-container">$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$</span></p>

<p>A mixed strategy of player <span class="math-container">$i$</span> of the incomplete information game <span class="math-container">$(G,S)$</span> is a mapping 
<span class="math-container">$$
\beta_i: T_i\rightarrow \Delta(A_i)
$$</span></p>

<p><strong>Definition of BCE:</strong> The decision rule <span class="math-container">$\sigma$</span> is a BCE for the game <span class="math-container">$(G,S)$</span> if, for each <span class="math-container">$i=1,...,N$</span>, <span class="math-container">$t_i\in T_i$</span>, and <span class="math-container">$a_i\in A_i$</span>, we have
<span class="math-container">$$
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \sigma(a_{-i}, a_i|t_i, t_{-i}, \theta) u_i(a_i, a_{-i},\theta)
$$</span>
<span class="math-container">$$
\geq 
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \sigma(a_{-i}, a_i|t_i, t_{-i}, \theta) u_i(\tilde{a}_i, a_{-i},\theta)
$$</span>
<span class="math-container">$\forall \tilde{a}_i\in A_i$</span>.</p>

<p><strong>Definition of BNE:</strong> The strategy profile <span class="math-container">$(\beta_1,...,\beta_N)$</span> is a BNE for the game <span class="math-container">$(G,S)$</span> if, for each <span class="math-container">$i=1,...,N$</span>, <span class="math-container">$t_i\in T_i$</span>, and <span class="math-container">$a_i\in A_i$</span> such that <span class="math-container">$\beta(a_i|t_i)&gt;0$</span>, we have
<span class="math-container">$$
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \Pi_{j\neq i} \beta_j(a_j|t_j) u_i(a_i, a_{-i},\theta)
$$</span>
<span class="math-container">$$
\geq 
\sum_{a_{-i}, t_{-i}, \theta} \psi(\theta) \pi(t_{-i},t_i| \theta) \Pi_{j\neq i} \beta_j(a_j|t_j)  u_i(\tilde{a}_i, a_{-i},\theta)
$$</span>
<span class="math-container">$\forall \tilde{a}_i\in A_i$</span>.</p>

<hr>

<p><strong>Question:</strong> <strong>when <span class="math-container">$N=1$</span></strong>, I've understood from the various versions of the paper I linked that the following terminology may be appropriate. Could you check and provide better suggestions, if any?</p>

<ul>
<li><p><span class="math-container">$S$</span> is called "experiment" in the sense studied by Blackwell (1951; 1953), rather than "information structure". </p></li>
<li><p><span class="math-container">$G$</span> is called "decision problem", rather than "basic game".</p></li>
<li><p>The player is called "decision maker". </p></li>
<li><p>The BCE concept is referred to as "one-player BCE". </p></li>
<li><p>Also, how would you refer to the BNE concept with one player?</p></li>
<li><p>Lastly, are there other crucial papers in the literature introducing the concept of BCE when <span class="math-container">$N=1$</span>? Or the paper linked is still a valid reference?</p></li>
</ul>

<game-theory><bayesian-game>
The notions of Bayes Correlated Equilibrium and Bayesian Nash Equilibrium in a game with one player
Star
# The notions of Bayes Correlated Equilibrium and Bayesian Nash Equilibrium in a game with one player

(Asked by: username_0 on 2019-06-21)

I would like your help to use the correct terminology to define a **Bayes Correlated Equilibrium (BCE)** and a **Bayesian Nash Equilibrium (BNE)** in a **"game" with one player**. The notion of BCE in an $N$-player game has been introduced in [this](http://www.princeton.edu/~smorris/pdfs/bce.pdf) paper.

---

Let me describe the game and give the notion of BCE and BNE in an $N$-player game.

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
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

A mixed strategy of player $i$ of the incomplete information game $(G,S)$ is a mapping
$$
\beta\_i: T\_i\rightarrow \Delta(A\_i)
$$

**Definition of BCE:** The decision rule $\sigma$ is a BCE for the game $(G,S)$ if, for each $i=1,...,N$, $t\_i\in T\_i$, and $a\_i\in A\_i$, we have
$$
\sum\_{a\_{-i}, t\_{-i}, \theta} \psi(\theta) \pi(t\_{-i},t\_i| \theta) \sigma(a\_{-i}, a\_i|t\_i, t\_{-i}, \theta) u\_i(a\_i, a\_{-i},\theta)
$$
$$
\geq
\sum\_{a\_{-i}, t\_{-i}, \theta} \psi(\theta) \pi(t\_{-i},t\_i| \theta) \sigma(a\_{-i}, a\_i|t\_i, t\_{-i}, \theta) u\_i(\tilde{a}\_i, a\_{-i},\theta)
$$
$\forall \tilde{a}\_i\in A\_i$.

**Definition of BNE:** The strategy profile $(\beta\_1,...,\beta\_N)$ is a BNE for the game $(G,S)$ if, for each $i=1,...,N$, $t\_i\in T\_i$, and $a\_i\in A\_i$ such that $\beta(a\_i|t\_i)>0$, we have
$$
\sum\_{a\_{-i}, t\_{-i}, \theta} \psi(\theta) \pi(t\_{-i},t\_i| \theta) \Pi\_{j\neq i} \beta\_j(a\_j|t\_j) u\_i(a\_i, a\_{-i},\theta)
$$
$$
\geq
\sum\_{a\_{-i}, t\_{-i}, \theta} \psi(\theta) \pi(t\_{-i},t\_i| \theta) \Pi\_{j\neq i} \beta\_j(a\_j|t\_j) u\_i(\tilde{a}\_i, a\_{-i},\theta)
$$
$\forall \tilde{a}\_i\in A\_i$.

---

**Question:** **when $N=1$**, I've understood from the various versions of the paper I linked that the following terminology may be appropriate. Could you check and provide better suggestions, if any?

* $S$ is called "experiment" in the sense studied by Blackwell (1951; 1953), rather than "information structure".
* $G$ is called "decision problem", rather than "basic game".
* The player is called "decision maker".
* The BCE concept is referred to as "one-player BCE".
* Also, how would you refer to the BNE concept with one player?
* Lastly, are there other crucial papers in the literature introducing the concept of BCE when $N=1$? Or the paper linked is still a valid reference?

**Tags:** game-theory, bayesian-game

---

## 1 answers

### Answer by username_1 on 2019-06-21. Score: 1 | Accepted answer

* The terms experiment and information structure are used interchangeably by the literature, so both terms are valid regardless of $N$.
* I agree if $N=1$ there is really no game, and it is rather a single-agent decision problem (or decision problem with uncertainty), and yes, you don't have a player, rather a decision maker.
* For the same token, you don't want to say the solution concept if BNE, instead you want to say that the agent maximizes his expected utility given the information she received.
* I was not aware of the term "one-player BCE" but Bergemann and Morris coined the term BCE, so if that is how they call it, it is correct.
* In terms of other references, [this](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1941708) is an earlier version of that same paper. However, I think that the paper you linked is the right paper for citing the concept of BCE.

In [this](https://www.aeaweb.org/articles?id=10.1257/jel.20181489) later paper, the same two authors made the remarkable connection that the problem of [Bayesian Persuasion](https://faculty.chicagobooth.edu/emir.kamenica/documents/bayesianPersuasion.pdf) popularized by Kamenica and Gentzkow could be interpreted in light of the concept of BCE when $N=1$. So perhaps, this later reference of BM can be useful if you want to explore more the case of $N=1$.

> **Comments:**
> **username_0**: Thanks. Regarding the connection with Bayesian Persuasion: is it correct to say that such a connection exists only under the degenerate information structure (i.e., not informative signal)?  
> **username_0**: I'm saying this because in the paper here http://www.princeton.edu/~smorris/pdfs/bce.pdf, BM seem to make a comparison with Bayesian Persuasion under the degenerate information structure only. But then this remark disappears in the BM's 2019 paper that you linked.  
> **username_1**: No, the comparison between BCE and Bayesian persuasion is valid even if S is not degenerate. What the authors are saying is that if S is degenerate (which means that the receiver does not have private information) their model maps to the problem presented in the original paper by Kamenica and Gentzkow. However,  in fact, BM's 2019 precisely use this connection to show how to solve the Bayesian persuasion problem when the receiver has private information (i.e. when S is not degenerate). I hope this makes sense.  
> **username_0**: Regarding you last comment, I'm trying to formalise it here https://economics.stackexchange.com/questions/32219/bayesian-correlated-equilibrium-in-a-one-player-case-relation-with-literature. Could you help me understand whether what I wrote is correct?  

---

