<p>I am not entirely sure about what you mean with "Before choosing the DM can discover <span class="math-container">$z$</span> by searching sequentially and paying cost <span class="math-container">$c_y$</span> when uncovering <span class="math-container">$z_y$</span> for each <span class="math-container">$y\in Y$</span>." </p>

<p>However, it sounds like you want your DM to choose both a stoping time, <span class="math-container">$t$</span>, and an action, <span class="math-container">$y$</span>. Since the DM is searching sequentially, the decision to continue searching or not will depend on how searching affects their beliefs and the cost of searching, which you are not fully specifying. Searching typically gives you access to some signal that will affect your beliefs, and the cost of searching is usually defined regardless of the result of the signal. You have not specified such searching signal, and it sounds like you only pay a cost for searching if you uncover state <span class="math-container">$z_y$</span>. However, let me get to your question.</p>

<p>At a high level, you cannot re-write your problem through a signal density, <span class="math-container">$P_T(\cdot|z)$</span>, because the re-write does not include the searching costs.</p>

<p>Argument for the statemet: Suppose there is a well-defined search problem and the decision-maker chooses <span class="math-container">$y\in Y$</span> only after the search phase has concluded (If these assumptions are not true, my argument is just strengthen). Given an optimal search strategy (that can only depend on the prior <span class="math-container">$F_Z$</span>, the search costs <span class="math-container">$c_z$</span>, and (perhaps) the interim beliefs during the search process), for each state of the world the DM will (probabilistically) end with some sequence of signals (one for each period they searched). Denote each possible sequence with a different element in <span class="math-container">$T$</span> and let <span class="math-container">$P_T(t|z)$</span> be the probability that the DM ends with sequence <span class="math-container">$t$</span> when the state is <span class="math-container">$z$</span>. However, each sequence of signals will, in general, have a different cost. This cost cannot be, in general, captured by the probability density.</p>

<p>If searching was free, then sure, your problem is one of choosing <span class="math-container">$y$</span> to maximize the expected utility where expectations are taking with respect to some posterior beliefs that arise after searching, and the density <span class="math-container">$P_T(\cdot|z)$</span> is the object that summarizes the searching process that is assumed to be well-defined. </p>

CC BY-SA 4.0
2020-03-20T20:00:29.447Z
Could you help for this https://economics.stackexchange.com/questions/34538/example-of-information-structure-in-a-one-player-bayes-correlated-equilibrium? Thanks
Star
CC BY-SA 4.0
2020-03-02T18:01:22.310Z
2020-03-02T18:01:22.310Z
Regio
<p>Consider a decision maker (DM) who as to choose an action from the finite set <span class="math-container">$\mathcal{Y}$</span> with cardinality <span class="math-container">$L. $</span></p>

<p>The payoff the he gets depends on the action chosen and the state of the world. The state of the world is denoted by <span class="math-container">$z\equiv (z_1,...,z_L)\in \mathcal{Z}^L\subseteq \mathbb{R}^L$</span>. The payoff when the DM chooses action <span class="math-container">$y\in \mathcal{Y}$</span> and the state of the world is <span class="math-container">$z\in \mathcal{Z}^L$</span> is denoted by <span class="math-container">$u(y,z_y)$</span>. </p>

<p>Before choosing the DM does not observe the state of the world. His prior on <span class="math-container">$z$</span> is denoted by <span class="math-container">$F_Z\in \Delta(\mathcal{Z}^L)$</span> and has marginals that are <strong>equivalent</strong> across <span class="math-container">$y$</span>. </p>

<p>Before choosing the DM can discover <span class="math-container">$z$</span> by searching sequentially and paying cost <span class="math-container">$c_y$</span>  when uncovering <span class="math-container">$z_y$</span> for each <span class="math-container">$y\in \mathcal{Y}$</span>.</p>

<p>When search stops, the DM chooses action  <span class="math-container">$y$</span> maximising his payoff among <strong>the searched actions</strong>. </p>

<hr>

<p><strong>Question</strong> </p>

<p>Can we find a conditional signal density <span class="math-container">$P_{T}(\cdot|z)\in \Delta(\mathcal{T})$</span> such that the problem above can be rewritten as 
<span class="math-container">$$
\max_{y\in \mathcal{Y}}\sum_{z\in \mathcal{Z}} u(y,z_y) \underbrace{\frac{P_T(t|z) F_Z(z)}{\sum_{z\in \mathcal{Z}}P_T(t|z) F_Z(z)}}_{\text{Posterior belief by Bayes rule}}
$$</span>
for each <span class="math-container">$t\in \mathcal{T}$</span>?</p>

CC BY-SA 4.0
2020-03-02T20:40:24.270Z
When you say that the prior has "marginals that are equivalent across $y$," do you mean that the marginal of $z_l$ is equal to the marginal of $z_k$ for all $k,l$? If so, why don't you say equal marginals? Also, from what you say, $y$ is an action (a choice variable of the DM); what does it have to do with the distribution of $z$ which is a primitive of your model? (so, what does it mean that the marginals be equal across $y$?)
pegasus
<decision-theory><bayesian-game>
Write search model as a signal density inducing posterior belief
Star
# Write search model as a signal density inducing posterior belief

(Asked by: username_0 on 2020-02-21)

Consider a decision maker (DM) who as to choose an action from the finite set $\mathcal{Y}$ with cardinality $L. $

The payoff the he gets depends on the action chosen and the state of the world. The state of the world is denoted by $z\equiv (z\_1,...,z\_L)\in \mathcal{Z}^L\subseteq \mathbb{R}^L$. The payoff when the DM chooses action $y\in \mathcal{Y}$ and the state of the world is $z\in \mathcal{Z}^L$ is denoted by $u(y,z\_y)$.

Before choosing the DM does not observe the state of the world. His prior on $z$ is denoted by $F\_Z\in \Delta(\mathcal{Z}^L)$ and has marginals that are **equivalent** across $y$.

Before choosing the DM can discover $z$ by searching sequentially and paying cost $c\_y$ when uncovering $z\_y$ for each $y\in \mathcal{Y}$.

When search stops, the DM chooses action $y$ maximising his payoff among **the searched actions**.

---

**Question**

Can we find a conditional signal density $P\_{T}(\cdot|z)\in \Delta(\mathcal{T})$ such that the problem above can be rewritten as
$$
\max\_{y\in \mathcal{Y}}\sum\_{z\in \mathcal{Z}} u(y,z\_y) \underbrace{\frac{P\_T(t|z) F\_Z(z)}{\sum\_{z\in \mathcal{Z}}P\_T(t|z) F\_Z(z)}}\_{\text{Posterior belief by Bayes rule}}
$$
for each $t\in \mathcal{T}$?

**Tags:** decision-theory, bayesian-game
> **Comments:**
> **username_1**: When you say that the prior has "marginals that are equivalent across $y$," do you mean that the marginal of $z_l$ is equal to the marginal of $z_k$ for all $k,l$? If so, why don't you say equal marginals? Also, from what you say, $y$ is an action (a choice variable of the DM); what does it have to do with the distribution of $z$ which is a primitive of your model? (so, what does it mean that the marginals be equal across $y$?)  

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## 1 answers

### Answer by username_2 on 2020-03-02. Score: 1 | Accepted answer

I am not entirely sure about what you mean with "Before choosing the DM can discover $z$ by searching sequentially and paying cost $c\_y$ when uncovering $z\_y$ for each $y\in Y$."

However, it sounds like you want your DM to choose both a stoping time, $t$, and an action, $y$. Since the DM is searching sequentially, the decision to continue searching or not will depend on how searching affects their beliefs and the cost of searching, which you are not fully specifying. Searching typically gives you access to some signal that will affect your beliefs, and the cost of searching is usually defined regardless of the result of the signal. You have not specified such searching signal, and it sounds like you only pay a cost for searching if you uncover state $z\_y$. However, let me get to your question.

At a high level, you cannot re-write your problem through a signal density, $P\_T(\cdot|z)$, because the re-write does not include the searching costs.

Argument for the statemet: Suppose there is a well-defined search problem and the decision-maker chooses $y\in Y$ only after the search phase has concluded (If these assumptions are not true, my argument is just strengthen). Given an optimal search strategy (that can only depend on the prior $F\_Z$, the search costs $c\_z$, and (perhaps) the interim beliefs during the search process), for each state of the world the DM will (probabilistically) end with some sequence of signals (one for each period they searched). Denote each possible sequence with a different element in $T$ and let $P\_T(t|z)$ be the probability that the DM ends with sequence $t$ when the state is $z$. However, each sequence of signals will, in general, have a different cost. This cost cannot be, in general, captured by the probability density.

If searching was free, then sure, your problem is one of choosing $y$ to maximize the expected utility where expectations are taking with respect to some posterior beliefs that arise after searching, and the density $P\_T(\cdot|z)$ is the object that summarizes the searching process that is assumed to be well-defined.

> **Comments:**
> **username_0**: Could you help for this https://economics.stackexchange.com/questions/34538/example-of-information-structure-in-a-one-player-bayes-correlated-equilibrium? Thanks  

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