<p>1) Given that <span class="math-container">$\mathcal{Y}$</span> is finite, it is obvious that the BCE is not an empty set (or it exists). Remember that the set of BCE's is the set of decisions that are optimal for some belief about <span class="math-container">$\mathcal{V}$</span> that could be formed (as a posterior belief) after observing some signal and having a prior <span class="math-container">$P_V$</span>. </p>

<p>To show there is at least one distribution satisfying the definition if BCE, you can consider an uninformative signal. Given that there is only a finite set of actions, one of them must maximize the expected utility given the prior. If there is more than one, randomize among them as you please. Let the optimal action be <span class="math-container">$P_Y^*$</span> (in this notation I capture both situations where the maximizer is unique and when there is more than one maximizer) and define <span class="math-container">$P_{Y,V}=P_Y^*P_V$</span>, then such a distribution constitutes a BCE.</p>

<p>Now, you should not expect uniqueness because if there was only one distribution that constitutes a, or if BCE was a singleton (since we usually think of BCE as a set), then information is essentially irrelevant. It would mean that there is an action in <span class="math-container">$\mathcal{Y}$</span> that is the best action regardless of any extra information you might receive. </p>

<p>2) To characterize this, simply find the optimal strategy when the belief is equal to the prior, <span class="math-container">$P_V$</span>. If you want to  put it in terms of the two expressions you have, then <span class="math-container">$P_{Y,V}$</span> must  be a product distribution, i.e <span class="math-container">$P_{Y,V}(y,v)=P_Y(y)\cdot P_V(v)$</span> for some <span class="math-container">$P_Y(y)\in\Delta(\mathcal{Y})$</span> (This will capture the fact that the player has no further information beyond the prior and so it cannot correlate his actions with the state of the world). The second inequality will pin down <span class="math-container">$P_Y$</span>. In fact, using the notation from the previous bullet point, it should be equal to <span class="math-container">$P_Y^*$</span>.</p>

CC BY-SA 4.0
2019-06-26T10:28:13.870Z
Thanks. Regarding 2), is it consistent with your answer here https://economics.stackexchange.com/questions/29953/optimal-strategy-in-a-single-agent-choice-problem-under-uncertainty? Are you saying the same thing, right? In case there is only one maximizer (say $y$), then $P^*_Y(y)=1$, correct?
Star
CC BY-SA 4.0
2019-06-26T17:15:43.580Z
That's correct. That is what I meant.
Regio
CC BY-SA 4.0
2019-06-25T23:33:03.213Z
2019-06-25T23:33:03.213Z
Regio
<p>I would like your help to better understand the possibility of using the notion of <strong>Bayes Correlated Equilibrium (BCE)</strong> in a <strong>single-agent decision problem with uncertainty</strong> to make predictions on optimal strategies that are robust to the belief environment. 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>For example, suppose that <span class="math-container">$\mathcal{Y}\equiv \{1,2,3\}$</span>. <span class="math-container">$V$</span> is a <span class="math-container">$3\times 1$</span> random vector, <span class="math-container">$V\equiv (V_1,V_2,V_3)$</span>. <span class="math-container">$P_V$</span> is the 3-variate standard normal distribution. <span class="math-container">$u(y,v)\equiv v_y$</span>. </p>

<p>Before choosing an action, the decision maker can receive <strong>at least</strong> one  signal to refine her prior (<strong>minimal</strong> amount of information). I assume that this minimal signal is completely uninformative (<strong>degenerate</strong> information structure).  </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 (degenerate in this case). To do that, I introduce the notion of one-player Bayesian Correlated Equilibrium (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">$$
\sum_{v\in \mathcal{V}} (v_y-v_{\tilde{y}}) \times P_{Y,V}(y,v)\geq 0
$$</span></p>

<hr>

<p>Question: </p>

<p>1) Is it obvious that a BCE exists and is unique in my example?</p>

<p>2) Suppose now that I know <strong>for sure</strong> that the decision maker does not have any information in addition to the degenerate information structure. How would I characterise an optimal strategy in such a case (without the necessity of being robust to possibly richer belief environments)? How would that definition differ from the BCE definition given above?</p>

<game-theory><decision-theory><bayesian-game><uncertainty>
Robust predictions in single-agent decision problem with uncertainty
Star
# Robust predictions in single-agent decision problem with uncertainty

(Asked by: username_0 on 2019-06-25)

I would like your help to better understand the possibility of using the notion of **Bayes Correlated Equilibrium (BCE)** in a **single-agent decision problem with uncertainty** to make predictions on optimal strategies that are robust to the belief environment. 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 $u(y,v)$.

For example, suppose that $\mathcal{Y}\equiv \{1,2,3\}$. $V$ is a $3\times 1$ random vector, $V\equiv (V\_1,V\_2,V\_3)$. $P\_V$ is the 3-variate standard normal distribution. $u(y,v)\equiv v\_y$.

Before choosing an action, the decision maker can receive **at least** one signal to refine her prior (**minimal** amount of information). I assume that this minimal signal is completely uninformative (**degenerate** information structure).

---

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 (degenerate in this case). To do that, I introduce the notion of one-player Bayesian Correlated Equilibrium (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 $\forall \tilde{y}\neq y$
$$
\sum\_{v\in \mathcal{V}} (v\_y-v\_{\tilde{y}}) \times P\_{Y,V}(y,v)\geq 0
$$

---

Question:

1) Is it obvious that a BCE exists and is unique in my example?

2) Suppose now that I know **for sure** that the decision maker does not have any information in addition to the degenerate information structure. How would I characterise an optimal strategy in such a case (without the necessity of being robust to possibly richer belief environments)? How would that definition differ from the BCE definition given above?

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


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## 1 answers

### Answer by username_1 on 2019-06-25. Score: 1 | Accepted answer

1) Given that $\mathcal{Y}$ is finite, it is obvious that the BCE is not an empty set (or it exists). Remember that the set of BCE's is the set of decisions that are optimal for some belief about $\mathcal{V}$ that could be formed (as a posterior belief) after observing some signal and having a prior $P\_V$.

To show there is at least one distribution satisfying the definition if BCE, you can consider an uninformative signal. Given that there is only a finite set of actions, one of them must maximize the expected utility given the prior. If there is more than one, randomize among them as you please. Let the optimal action be $P\_Y^\*$ (in this notation I capture both situations where the maximizer is unique and when there is more than one maximizer) and define $P\_{Y,V}=P\_Y^\*P\_V$, then such a distribution constitutes a BCE.

Now, you should not expect uniqueness because if there was only one distribution that constitutes a, or if BCE was a singleton (since we usually think of BCE as a set), then information is essentially irrelevant. It would mean that there is an action in $\mathcal{Y}$ that is the best action regardless of any extra information you might receive.

2) To characterize this, simply find the optimal strategy when the belief is equal to the prior, $P\_V$. If you want to put it in terms of the two expressions you have, then $P\_{Y,V}$ must be a product distribution, i.e $P\_{Y,V}(y,v)=P\_Y(y)\cdot P\_V(v)$ for some $P\_Y(y)\in\Delta(\mathcal{Y})$ (This will capture the fact that the player has no further information beyond the prior and so it cannot correlate his actions with the state of the world). The second inequality will pin down $P\_Y$. In fact, using the notation from the previous bullet point, it should be equal to $P\_Y^\*$.

> **Comments:**
> **username_0**: Thanks. Regarding 2), is it consistent with your answer here https://economics.stackexchange.com/questions/29953/optimal-strategy-in-a-single-agent-choice-problem-under-uncertainty? Are you saying the same thing, right? In case there is only one maximizer (say $y$), then $P^*_Y(y)=1$, correct?  
> **username_1**: That's correct. That is what I meant.  

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