<p>Yes,  <span class="math-container">$Q$</span> is convex and closed precisely because <span class="math-container">$P$</span> is and Q is its image under a continuous function. Therefore, <span class="math-container">$Q$</span> is compact.</p>

<p>The function you are maximizing is continuous as well, so the maximum must exist. </p>

CC BY-SA 4.0
2019-10-20T17:18:30.253Z
2019-10-20T17:18:30.253Z
Regio
<p>I would like your help to better understand the notion of <strong>Bayes Correlated Equilibrium (BCE)</strong> in a <strong>single-agent decision problem with uncertainty</strong> The notion of BCE is provided in <a href="http://www.princeton.edu/~smorris/pdfs/bce.pdf" rel="nofollow noreferrer">this</a> paper for a generic <span class="math-container">$N$</span>-player game. </p>

<hr>

<p>Consider the following single-agent choice problem under uncertainty. </p>

<p>Let <span class="math-container">$V$</span> be the state of the world with support <span class="math-container">$\mathcal{V}$</span> and probability distribution <span class="math-container">$P_V\in \Delta(\mathcal{V})$</span>. First, let nature draw a realisation <span class="math-container">$v$</span> of <span class="math-container">$V$</span> from <span class="math-container">$P_V$</span>. 
Then, let the decision maker choose an action <span class="math-container">$y\in \mathcal{Y}$</span>, with <span class="math-container">$\mathcal{Y}$</span> finite, without observing <span class="math-container">$v$</span>. Upon the decision has been made, the decision maker gets a payoff <span class="math-container">$u(y,v)$</span>.</p>

<p>Before choosing an action, the decision maker can receive some signal to refine her prior about the state of the world.</p>

<hr>

<p>Now, I want to use Theorem 1 in Bergemann and Morris (2016) to characterise the set of optimal strategies under minimal assumptions on the amount of information that is processed by the decision maker. To do that, I introduce the notion of one-player BCE.</p>

<p>A one-player BCE of the game described is a probability distribution <span class="math-container">$P_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$</span> such that:</p>

<p>1) <span class="math-container">$\forall v \in \mathcal{V}$</span>
<span class="math-container">$$\sum_{y\in \mathcal{Y}}P_{Y,V}(y,v)=P_V(v)$$</span></p>

<p>2) <span class="math-container">$\forall y\in \mathcal{Y}$</span> and <span class="math-container">$\forall \tilde{y}\neq y$</span>
<span class="math-container">$$
\sum_{v\in \mathcal{V}} [u(y,v)-u(\tilde{y},v)] \times P_{Y,V}(y,v)\geq 0
$$</span></p>

<hr>

<p>Let <span class="math-container">$\mathcal{P}$</span> be the set of BCE. 
Let 
<span class="math-container">$$
\mathcal{Q}\equiv \{P_Y\in  \Delta(\mathcal{Y}): P_Y(y)\equiv \sum_{v\in \mathcal{V}}P_{Y,V}(y,v) \text{ }\forall y \in \mathcal{Y}, P_{Y,V}\in \mathcal{P}\}
$$</span></p>

<hr>

<p><strong>Questions:</strong></p>

<p>1) Is <span class="math-container">$\mathcal{Q}$</span> convex? I guess it is because <span class="math-container">$\mathcal{P}$</span> is convex. Could you confirm?</p>

<p>2) Let <span class="math-container">$ \mathcal{B}\equiv \{b\in \mathbb{R}^{|\mathcal{Y}|}: b^Tb\leq 1\}$</span> be the unit ball in <span class="math-container">$\mathbb{R}^{|\mathcal{Y}|}$</span>. Let us stack in a <span class="math-container">$|\mathcal{Y}|\times 1$</span> vector the image set of every probability distribution <span class="math-container">$P_{Y}\in \mathcal{Q}$</span>. With some abuse of notation, let us still denote any such a vector by <span class="math-container">$P_{Y}\in \mathcal{Q}$</span>. Does 
<span class="math-container">$$
\max_{P_Y\in \mathcal{Q}} b^T P_Y
$$</span>
exists <span class="math-container">$\forall b\in \mathcal{B}$</span>? I guess the answer to this question depends on the features of <span class="math-container">$\mathcal{Q}$</span>. For example, if <span class="math-container">$\mathcal{Q}$</span> is compact, then the maximum above exists. Could you help?</p>

CC BY-SA 4.0
2019-08-11T02:32:35.800Z
Some notations are undefined---$v_y$, $v_{\tilde{y}}$, $\mathcal{P}$, etc. For 1), if $\mathcal{P}$ is convex, than $\mathcal{Q}$ is also, since it's the image of a convex set under a linear map. (The map of taking marginal distribution). For 2), if $\mathcal{Q}$ is convex, then it's a linear program with convex constraint, which always have a solution.
Michael
CC BY-SA 4.0
2019-08-11T11:34:54.027Z
Thanks. Notation on $v_y, v_{\tilde{y}}$ fixed. $\mathcal{P}$ is the set of BCE.
Star
CC BY-SA 4.0
2019-08-12T03:19:46.417Z
From your definition of $\mathcal{P}$, you can see that indeed it is convex. It is the intersection of two convex sets, one from each condition.
Michael
CC BY-SA 4.0
2019-08-12T08:39:25.663Z
@Michael. I dont think that your claim about 2) is correct. If convexity was sufficient than $\underset{x \in B_1(0)}{max} 1 \times x$ would have a solution as the unit open ball around zero($B_1(0)$) is convex. You need closedness at some level.
Grada Gukovic
<game-theory><bayesian-game>
Features of the set of Bayes Correlated Equilibrium in a single-agent decision problem with uncertainty
Star
# Features of the set of Bayes Correlated Equilibrium in a single-agent decision problem with uncertainty

(Asked by: username_0 on 2019-08-06)

I would like your help to better understand the notion of **Bayes Correlated Equilibrium (BCE)** in a **single-agent decision problem with uncertainty** The notion of BCE is provided in [this](http://www.princeton.edu/~smorris/pdfs/bce.pdf) paper for a generic $N$-player game.

---

Consider the following single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P\_V\in \Delta(\mathcal{V})$. First, let nature draw a realisation $v$ of $V$ from $P\_V$.
Then, let the decision maker choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the decision maker gets a payoff $u(y,v)$.

Before choosing an action, the decision maker can receive some signal to refine her prior about the state of the world.

---

Now, I want to use Theorem 1 in Bergemann and Morris (2016) to characterise the set of optimal strategies under minimal assumptions on the amount of information that is processed by the decision maker. To do that, I introduce the notion of one-player BCE.

A one-player BCE of the game described is a probability distribution $P\_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ such that:

1) $\forall v \in \mathcal{V}$
$$\sum\_{y\in \mathcal{Y}}P\_{Y,V}(y,v)=P\_V(v)$$

2) $\forall y\in \mathcal{Y}$ and $\forall \tilde{y}\neq y$
$$
\sum\_{v\in \mathcal{V}} [u(y,v)-u(\tilde{y},v)] \times P\_{Y,V}(y,v)\geq 0
$$

---

Let $\mathcal{P}$ be the set of BCE.
Let
$$
\mathcal{Q}\equiv \{P\_Y\in \Delta(\mathcal{Y}): P\_Y(y)\equiv \sum\_{v\in \mathcal{V}}P\_{Y,V}(y,v) \text{ }\forall y \in \mathcal{Y}, P\_{Y,V}\in \mathcal{P}\}
$$

---

**Questions:**

1) Is $\mathcal{Q}$ convex? I guess it is because $\mathcal{P}$ is convex. Could you confirm?

2) Let $ \mathcal{B}\equiv \{b\in \mathbb{R}^{|\mathcal{Y}|}: b^Tb\leq 1\}$ be the unit ball in $\mathbb{R}^{|\mathcal{Y}|}$. Let us stack in a $|\mathcal{Y}|\times 1$ vector the image set of every probability distribution $P\_{Y}\in \mathcal{Q}$. With some abuse of notation, let us still denote any such a vector by $P\_{Y}\in \mathcal{Q}$. Does
$$
\max\_{P\_Y\in \mathcal{Q}} b^T P\_Y
$$
exists $\forall b\in \mathcal{B}$? I guess the answer to this question depends on the features of $\mathcal{Q}$. For example, if $\mathcal{Q}$ is compact, then the maximum above exists. Could you help?

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

> **Comments:**
> **username_1**: Some notations are undefined---$v_y$, $v_{\tilde{y}}$, $\mathcal{P}$, etc. For 1), if $\mathcal{P}$ is convex, than $\mathcal{Q}$ is also, since it's the image of a convex set under a linear map. (The map of taking marginal distribution). For 2), if $\mathcal{Q}$ is convex, then it's a linear program with convex constraint, which always have a solution.  
> **username_0**: Thanks. Notation on $v_y, v_{\tilde{y}}$ fixed. $\mathcal{P}$ is the set of BCE.  
> **username_1**: From your definition of $\mathcal{P}$, you can see that indeed it is convex. It is the intersection of two convex sets, one from each condition.  
> **username_2**: @username_1. I dont think that your claim about 2) is correct. If convexity was sufficient than $\underset{x \in B_1(0)}{max} 1 \times x$ would have a solution as the unit open ball around zero($B_1(0)$) is convex. You need closedness at some level.  

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## 1 answers

### Answer by username_3 on 2019-10-20. Score: 3 | Accepted answer

Yes, $Q$ is convex and closed precisely because $P$ is and Q is its image under a continuous function. Therefore, $Q$ is compact.

The function you are maximizing is continuous as well, so the maximum must exist.


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