<p>It is true that BCE is more appropriate for games. You can certainly use it for single-agent decision problems, but it is really an overkill. </p>

<p>Further, BCE is a solution concept that is agnostic about what other information people receive, or how they receive it. Therefore, I will not say that BCE nests search models. If anything, the BCE of a single-agent problem simply describes the actions that such an agent could possibly choose after receiving some information (perhaps after searching). </p>

<p>In that sense, you can study how the BCE evolves in a search model. For example, in most search models the agent starts agnostic enough so that all actions can be a best response for some posterior belief, (so the BCE will be a relatively large set). As the player searches and learns more information, her beliefs usually concentrate somewhere. For example, the agent becomes more confident about the state of the world and in most search models a cascade forms (or the agent stops searching). A cascade is simply a situation where the agent's current prior, <span class="math-container">$\underline S$</span>, is concentrated enough that the BCE is a singleton. That is, regardless of any possible extra information that the agent could get, there is only one rationalizable action. </p>

<p>I cannot see how using BCE tools to study a search model could be helpful, but this is how I would connect those two kinds of literature. </p>

CC BY-SA 4.0
2020-02-13T21:06:30.867Z
Could you help with this other question? Thanks https://economics.stackexchange.com/questions/34000/set-of-bayes-correlated-equilibria-when-complete-information-is-not-available
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CC BY-SA 4.0
2020-02-08T21:52:46.833Z
2020-02-08T21:52:46.833Z
Regio
<p>Consider a game where a decision maker (DM) has to choose action <span class="math-container">$l\in \mathcal{Y}$</span> possibly without being fully aware of the state of the world <span class="math-container">$V$</span>. </p>

<p>The choice set <span class="math-container">$\mathcal{Y}$</span> has cardinality <span class="math-container">$L$</span>. The state of the world is an <span class="math-container">$L\times 1 $</span> vector and we denote its <span class="math-container">$l$</span>-th element by <span class="math-container">$V_l$</span>. </p>

<p>The state of the world has support <span class="math-container">$\mathcal{V}$</span>. </p>

<p>When DM chooses action <span class="math-container">$l\in \mathcal{Y}$</span>, she receives the payoff <span class="math-container">$V_l$</span>. That is, she receives a payoff equal to the <span class="math-container">$l$</span>-th element of the vector <span class="math-container">$V$</span>.  </p>

<p>Let <span class="math-container">$P_V\in \Delta(\mathcal{V})$</span> be the DM's prior. </p>

<p>The DM may also process some signal (formalised by the concept of information structure) to refine his prior and get a posterior.</p>

<p>Let us define the concept of 1 player Bayesian Correlated Equilibrium provided in Bergemann and Morris (2013,2016,etc.).</p>

<p><span class="math-container">$P_{Y,V}\in  \Delta(\mathcal{Y}\times \mathcal{V})$</span> is a 1 player Bayesian Correlated Equilibrium if</p>

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

<p>2) <span class="math-container">$\sum_{v\in \mathcal{V}}V_l P_{Y,V}(l,v)\geq \sum_{v\in \mathcal{V}}V_k P_{Y,V}(l,v)$</span> for each <span class="math-container">$l$</span> and <span class="math-container">$k\neq y$</span>.</p>

<p>Bergemann and Morris show  that the set of Bayesian Correlated Equilibrium is equal to set of optimal optimal behaviours under a range of information structures. </p>

<p><strong>QUESTION:</strong> Consider a search model, in which we design a protocol according to which the DM discovers information about the state of the world. </p>

<p>For example, suppose that the DM engages in a sequential search where he discovers the <span class="math-container">$l$</span>-th element of <span class="math-container">$V$</span> if and only if the maximum utility secured up to that moment is lower than a reservation value. </p>

<p>Can this model always be written as prior/information structure/posterior? In other words, does the framework of Bergemann and Morris nest search models?</p>

<game-theory><bayesian-game>
Information and search
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# Information and search

(Asked by: username_0 on 2020-02-06)

Consider a game where a decision maker (DM) has to choose action $l\in \mathcal{Y}$ possibly without being fully aware of the state of the world $V$.

The choice set $\mathcal{Y}$ has cardinality $L$. The state of the world is an $L\times 1 $ vector and we denote its $l$-th element by $V\_l$.

The state of the world has support $\mathcal{V}$.

When DM chooses action $l\in \mathcal{Y}$, she receives the payoff $V\_l$. That is, she receives a payoff equal to the $l$-th element of the vector $V$.

Let $P\_V\in \Delta(\mathcal{V})$ be the DM's prior.

The DM may also process some signal (formalised by the concept of information structure) to refine his prior and get a posterior.

Let us define the concept of 1 player Bayesian Correlated Equilibrium provided in Bergemann and Morris (2013,2016,etc.).

$P\_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ is a 1 player Bayesian Correlated Equilibrium if

1) $\sum\_{l\in \mathcal{Y}}P\_{Y,V}(l,v)=P\_V(v)$ for each $v\in \mathcal{V}$

2) $\sum\_{v\in \mathcal{V}}V\_l P\_{Y,V}(l,v)\geq \sum\_{v\in \mathcal{V}}V\_k P\_{Y,V}(l,v)$ for each $l$ and $k\neq y$.

Bergemann and Morris show that the set of Bayesian Correlated Equilibrium is equal to set of optimal optimal behaviours under a range of information structures.

**QUESTION:** Consider a search model, in which we design a protocol according to which the DM discovers information about the state of the world.

For example, suppose that the DM engages in a sequential search where he discovers the $l$-th element of $V$ if and only if the maximum utility secured up to that moment is lower than a reservation value.

Can this model always be written as prior/information structure/posterior? In other words, does the framework of Bergemann and Morris nest search models?

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

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## 1 answers

### Answer by username_1 on 2020-02-08. Score: 1

It is true that BCE is more appropriate for games. You can certainly use it for single-agent decision problems, but it is really an overkill.

Further, BCE is a solution concept that is agnostic about what other information people receive, or how they receive it. Therefore, I will not say that BCE nests search models. If anything, the BCE of a single-agent problem simply describes the actions that such an agent could possibly choose after receiving some information (perhaps after searching).

In that sense, you can study how the BCE evolves in a search model. For example, in most search models the agent starts agnostic enough so that all actions can be a best response for some posterior belief, (so the BCE will be a relatively large set). As the player searches and learns more information, her beliefs usually concentrate somewhere. For example, the agent becomes more confident about the state of the world and in most search models a cascade forms (or the agent stops searching). A cascade is simply a situation where the agent's current prior, $\underline S$, is concentrated enough that the BCE is a singleton. That is, regardless of any possible extra information that the agent could get, there is only one rationalizable action.

I cannot see how using BCE tools to study a search model could be helpful, but this is how I would connect those two kinds of literature.

> **Comments:**
> **username_0**: Could you help with this other question? Thanks https://economics.stackexchange.com/questions/34000/set-of-bayes-correlated-equilibria-when-complete-information-is-not-available  

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