<p>Not every probability distribution comes from a density. In particular, point masses have no density (with respect to Lebesgue measure). Here, being perfectly informed requires that the conditional distribution is concentrated on the true value and, therefore, not describable by a density function. If you want to make these things rigorous, you will have to invest in learning some measure theory.</p>
<p>Addendum:</p>
<p>There is a canonical way to represent information structures popularized in the literature on Bayesian persuasion and information design. You identify signals with the distribution over the posteriors they induce. Here is how this works: First, assume that all spaces under question are Polish (separable and completely metrizable) and endowed with their Borel <span class="math-container">$\sigma$</span>-algebra. If <span class="math-container">$X$</span> is a Polish space, we let <span class="math-container">$\Delta(X)$</span> be the space of Borel probability measures on it, endowed with the topology of weak convergence and the corresponding Borel <span class="math-container">$\sigma$</span>-algebra. One can show that this is the smallest <span class="math-container">$\sigma$</span>-algebra such that the function <span class="math-container">$\mu\mapsto\mu(E)$</span> for every Borel set <span class="math-container">$E\subseteq X$</span> is measurable.</p>
<p>Now, to fill in some details in the answer by user42421, a signal is a measurable function <span class="math-container">$\sigma:\mathcal{V}\to\Delta(\mathcal{T})$</span>. There is a probability measure <span class="math-container">$\rho$</span> on <span class="math-container">$\mathcal{V}\times\mathcal{T}$</span> defined by
<span class="math-container">$$\rho(E)=\int\int 1_E(v,t)~\mathrm d\sigma_v(t)~\mathrm dP_V(v)$$</span>
for every Borel set <span class="math-container">$E\subseteq\mathcal{V}\times\mathcal{T}$</span>. One can actually reverse this process of integration: If <span class="math-container">$\rho$</span> is a probability measure on <span class="math-container">$\mathcal{V}\times\mathcal{T}$</span> whose <span class="math-container">$\mathcal{V}$</span>-marginal is <span class="math-container">$P_V$</span>, that is, <span class="math-container">$\rho(F\times\mathcal{T})=P_V(F)$</span> for every Borel set <span class="math-container">$F\subseteq\mathcal{T}$</span>, one can find a measurable function <span class="math-container">$\sigma$</span> such that <span class="math-container">$\rho$</span> can be given as above. One calls then <span class="math-container">$\sigma$</span> a <em>disintegration</em> of <span class="math-container">$\rho$</span>. Such disintegrations are unique, except for a <span class="math-container">$P_V$</span>-null set. One can use disintegrations also to define conditional beliefs given signal realizations. If <span class="math-container">$\rho_\mathcal{T}$</span> is the <span class="math-container">$\mathcal{T}$</span>-marginal of <span class="math-container">$\rho$</span>, then there exists a measurable function <span class="math-container">$\beta:\mathcal{T}\to\Delta(\mathcal{V})$</span> such that
<span class="math-container">$$\rho(E)=\int 1_E(v,t)~\mathrm d\beta_t(v)~\mathrm d\rho_\mathcal{T}(t)$$</span>
for every Borel set <span class="math-container">$E\subseteq\mathcal{V}\times\mathcal{T}$</span>. Of course, the belief <span class="math-container">$\beta(t)$</span> for an individual signal realization <span class="math-container">$t$</span> is not very meaningful since <span class="math-container">$\beta$</span> is only unique up to <span class="math-container">$\rho_\mathcal{T}$</span>-null sets. Now, we also get a distribution over posteriors (the values of <span class="math-container">$\beta$</span>), namely the element <span class="math-container">$\tau$</span> of <span class="math-container">$\Delta(\Delta(\mathcal{V}))$</span> such that <span class="math-container">$\tau=\rho_\mathcal{T}\circ\beta^{-1}$</span>. A convenient property of <span class="math-container">$\tau$</span> is that for every Borel set <span class="math-container">$F\subseteq V$</span>, one has
<span class="math-container">$$\int \mu(F)~\mathrm d\tau(\mu)=P_V(F),$$</span>
the average posterior equals the prior. It turns out that this is the only restriction on a set of posteriors induced by a signal. If <span class="math-container">$\tau\in \Delta(\Delta(\mathcal{V}))$</span> satisfies the condition <span class="math-container">$$\int \mu(F)~\mathrm d\tau(\mu)=P_V(F),$$</span>
for every Borel set <span class="math-container">$F$</span>, then there exists a space <span class="math-container">$T$</span> and a signal <span class="math-container">$\sigma:\mathcal{V}\to\Delta(\mathcal{T})$</span> such that <span class="math-container">$\tau$</span> is the induced distribution over posteriors. Indeed, one can let <span class="math-container">$T=\Delta(\mathcal{V})$</span>. There exists a probability measure <span class="math-container">$\rho$</span> on <span class="math-container">$\mathcal{V}\times\mathcal{T}=\mathcal{V}\times \Delta(\mathcal{V})$</span> defined by
<span class="math-container">$$\rho(E)=\int\int 1_E(v,t)~\mathrm d\mu(v)~\mathrm d \tau(\mu)$$</span>
for every measurable set <span class="math-container">$E\subseteq\mathcal{V}\times\mathcal{T}$</span>, The condition that <span class="math-container">$\tau$</span> averages to <span class="math-container">$P_V$</span> guarantees that the <span class="math-container">$\mathcal{V}$</span>-marginal of <span class="math-container">$\rho$</span> is <span class="math-container">$P_V$</span>. One then takes a disintegration <span class="math-container">$\sigma:\mathcal{V}\to\Delta(\mathcal{V})$</span> of <span class="math-container">$\rho$</span> as the desired signal. It has the convenient property that one can take the belief induced by <span class="math-container">$\sigma(v)$</span> to be <span class="math-container">$\sigma(v)$</span>. Here is the intuition behind it. Suppose you cannot learn the value of the signal but only learn what you would believe if you would have learned the value. Your updated belief should be then this belief, so taking posteriors as signals works.</p>
<p>Now, what about perfect information? Well, then the posteriors should be point-masses, Dirac-measures. Indeed, you can take <span class="math-container">$\sigma$</span> to be given by <span class="math-container">$\sigma(v)=\delta_v$</span> and <span class="math-container">$\tau=P_V\circ \sigma^{-1}$</span>.</p>

CC BY-SA 4.0
2022-10-05T08:33:38.673Z
Yes, I would like to make things rigorous. I've studied measure theory, but I cannot see how to formalise this here. Any rigorous help appreciated.
Star
CC BY-SA 4.0
2022-10-05T18:15:54.040Z
The most helpful version of conditioning for this kind of problems is given by regular conditional probabilities: https://en.wikipedia.org/wiki/Regular_conditional_probability
Michael Greinecker
CC BY-SA 4.0
2023-01-24T18:20:50.140Z
Would it be possible to formalise an answer? I've tried to apply the Wiki article you linked but I'm not sure how to proceed.
Star
CC BY-SA 4.0
2023-01-24T22:21:59.287Z
@user3285148 I've added the basics of the theory.
Michael Greinecker
CC BY-SA 4.0
2023-01-26T18:37:51.270Z
Thanks. Just some clarifications on the last line "You can take $\sigma$ to be given by $\sigma_v=\delta_v$".  Take $t\in \mathcal{T}$. This means that $$\sigma_v(t)=\begin{cases} 1 & \text{if $t=v$} \\ 0 & \text{otherwise}\end{cases}$$ Now, I don't understand what is the meaning of $t=v$ in the above expression given that $\mathcal{V}$ is not finite. I am back to the issue of null sets.
Star
CC BY-SA 4.0
2023-01-27T02:29:05.850Z
$\sigma_v$ is a probability measure defined on the Borel subsets of $\mathcal{V}$. For every Borel set $E\subseteq\mathcal{V}$, we have $\sigma_v(E)=0$ if $v\notin E$ and $\sigma_v(E)=1$ if $v\in E$.
Michael Greinecker
CC BY-SA 4.0
2022-10-04T18:11:22.597Z
2023-01-24T22:21:23.570Z
2023-01-24T22:21:23.570Z
Michael Greinecker
<p><strong>Model</strong></p>
<p>A decision maker (DM) has to choose action <span class="math-container">$y\in \mathcal{Y}$</span> possibly without being fully aware of the state of the world. <span class="math-container">$\mathcal{Y}$</span> is a finite set.</p>
<p>The state of the world is a random variable <span class="math-container">$V$</span> with support <span class="math-container">$\mathcal{V}$</span>. <span class="math-container">$\mathcal{V}$</span> <strong>is not a finite set</strong>. For instance, <span class="math-container">$\mathcal{V}=[a,b]$</span> or <span class="math-container">$\mathcal{V}=\mathbb{R}$</span>.</p>
<p>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">$v\in \mathcal{V}$</span>, she receives the payoff <span class="math-container">$u(y,v)$</span>.</p>
<p>Let <span class="math-container">$P_V\in \Delta(\mathcal{V})$</span> be the DM's prior about the state of the world <span class="math-container">$V$</span> (probability density function).</p>
<p>The DM also processes some signal <span class="math-container">$T$</span> with support <span class="math-container">$\mathcal{T}$</span> and  distribution <span class="math-container">$P_{T|V}$</span> conditional on <span class="math-container">$V$</span>  to refine his prior and get a posterior on <span class="math-container">$V$</span>, denoted by <span class="math-container">$P_{V|T}$</span>, via the Bayes rule.</p>
<p>Let <span class="math-container">$S\equiv \{\mathcal{T}, P_{T|V}\}$</span> be called &quot;information structure&quot;.</p>
<p>A strategy for the DM is <span class="math-container">$P_{Y|T}$</span>. Such a strategy is optimal if it maximises his expected payoff, where the expectation is computed using the posterior, <span class="math-container">$P_{V|T}$</span>.</p>
<hr />
<p><strong>Question</strong></p>
<p>I would like to represent the information structure that gives complete information (&quot;complete info structure&quot;) to the DM. When <span class="math-container">$\mathcal{V}$</span> is finite, this is easy: we can set <span class="math-container">$\mathcal{T}=\mathcal{V}$</span> and <span class="math-container">$P_{T|V}(t|v)=1$</span> if <span class="math-container">$t=v$</span> and 0 otherwise. When <span class="math-container">$\mathcal{V}$</span> is not finite, however, I'm struggling to find an appropriate probability density function.</p>
<p><strong>Note</strong>
One way to represent the complete info structure is to set <span class="math-container">$|\mathcal{V}|=1$</span>. This is not the way I am looking for. I don't want to &quot;manipulate&quot; <span class="math-container">$\mathcal{V}$</span> to obtain the desired information structure. Let us assume that <span class="math-container">$\mathcal{V}$</span> is fixed (e.g., <span class="math-container">$\mathcal{V}=\mathbb{R}$</span>) and find a density <span class="math-container">$P_{T|V}$</span> giving complete info to the DM.</p>

<game-theory><decision-theory><bayesian-game>
Information structure for complete information
Star
# Information structure for complete information

(Asked by: username_0 on 2022-10-03)

**Model**

A decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. $\mathcal{Y}$ is a finite set.

The state of the world is a random variable $V$ with support $\mathcal{V}$. $\mathcal{V}$ **is not a finite set**. For instance, $\mathcal{V}=[a,b]$ or $\mathcal{V}=\mathbb{R}$.

When the DM chooses action $y\in \mathcal{Y}$ and the state of the world is $v\in \mathcal{V}$, she receives the payoff $u(y,v)$.

Let $P\_V\in \Delta(\mathcal{V})$ be the DM's prior about the state of the world $V$ (probability density function).

The DM also processes some signal $T$ with support $\mathcal{T}$ and distribution $P\_{T|V}$ conditional on $V$ to refine his prior and get a posterior on $V$, denoted by $P\_{V|T}$, via the Bayes rule.

Let $S\equiv \{\mathcal{T}, P\_{T|V}\}$ be called "information structure".

A strategy for the DM is $P\_{Y|T}$. Such a strategy is optimal if it maximises his expected payoff, where the expectation is computed using the posterior, $P\_{V|T}$.

---

**Question**

I would like to represent the information structure that gives complete information ("complete info structure") to the DM. When $\mathcal{V}$ is finite, this is easy: we can set $\mathcal{T}=\mathcal{V}$ and $P\_{T|V}(t|v)=1$ if $t=v$ and 0 otherwise. When $\mathcal{V}$ is not finite, however, I'm struggling to find an appropriate probability density function.

**Note**
One way to represent the complete info structure is to set $|\mathcal{V}|=1$. This is not the way I am looking for. I don't want to "manipulate" $\mathcal{V}$ to obtain the desired information structure. Let us assume that $\mathcal{V}$ is fixed (e.g., $\mathcal{V}=\mathbb{R}$) and find a density $P\_{T|V}$ giving complete info to the DM.

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

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## 1 answers

### Answer by username_1 on 2022-10-04. Score: 4 | Accepted answer

Not every probability distribution comes from a density. In particular, point masses have no density (with respect to Lebesgue measure). Here, being perfectly informed requires that the conditional distribution is concentrated on the true value and, therefore, not describable by a density function. If you want to make these things rigorous, you will have to invest in learning some measure theory.

Addendum:

There is a canonical way to represent information structures popularized in the literature on Bayesian persuasion and information design. You identify signals with the distribution over the posteriors they induce. Here is how this works: First, assume that all spaces under question are Polish (separable and completely metrizable) and endowed with their Borel $\sigma$-algebra. If $X$ is a Polish space, we let $\Delta(X)$ be the space of Borel probability measures on it, endowed with the topology of weak convergence and the corresponding Borel $\sigma$-algebra. One can show that this is the smallest $\sigma$-algebra such that the function $\mu\mapsto\mu(E)$ for every Borel set $E\subseteq X$ is measurable.

Now, to fill in some details in the answer by user42421, a signal is a measurable function $\sigma:\mathcal{V}\to\Delta(\mathcal{T})$. There is a probability measure $\rho$ on $\mathcal{V}\times\mathcal{T}$ defined by
$$\rho(E)=\int\int 1\_E(v,t)~\mathrm d\sigma\_v(t)~\mathrm dP\_V(v)$$
for every Borel set $E\subseteq\mathcal{V}\times\mathcal{T}$. One can actually reverse this process of integration: If $\rho$ is a probability measure on $\mathcal{V}\times\mathcal{T}$ whose $\mathcal{V}$-marginal is $P\_V$, that is, $\rho(F\times\mathcal{T})=P\_V(F)$ for every Borel set $F\subseteq\mathcal{T}$, one can find a measurable function $\sigma$ such that $\rho$ can be given as above. One calls then $\sigma$ a *disintegration* of $\rho$. Such disintegrations are unique, except for a $P\_V$-null set. One can use disintegrations also to define conditional beliefs given signal realizations. If $\rho\_\mathcal{T}$ is the $\mathcal{T}$-marginal of $\rho$, then there exists a measurable function $\beta:\mathcal{T}\to\Delta(\mathcal{V})$ such that
$$\rho(E)=\int 1\_E(v,t)~\mathrm d\beta\_t(v)~\mathrm d\rho\_\mathcal{T}(t)$$
for every Borel set $E\subseteq\mathcal{V}\times\mathcal{T}$. Of course, the belief $\beta(t)$ for an individual signal realization $t$ is not very meaningful since $\beta$ is only unique up to $\rho\_\mathcal{T}$-null sets. Now, we also get a distribution over posteriors (the values of $\beta$), namely the element $\tau$ of $\Delta(\Delta(\mathcal{V}))$ such that $\tau=\rho\_\mathcal{T}\circ\beta^{-1}$. A convenient property of $\tau$ is that for every Borel set $F\subseteq V$, one has
$$\int \mu(F)~\mathrm d\tau(\mu)=P\_V(F),$$
the average posterior equals the prior. It turns out that this is the only restriction on a set of posteriors induced by a signal. If $\tau\in \Delta(\Delta(\mathcal{V}))$ satisfies the condition $$\int \mu(F)~\mathrm d\tau(\mu)=P\_V(F),$$
for every Borel set $F$, then there exists a space $T$ and a signal $\sigma:\mathcal{V}\to\Delta(\mathcal{T})$ such that $\tau$ is the induced distribution over posteriors. Indeed, one can let $T=\Delta(\mathcal{V})$. There exists a probability measure $\rho$ on $\mathcal{V}\times\mathcal{T}=\mathcal{V}\times \Delta(\mathcal{V})$ defined by
$$\rho(E)=\int\int 1\_E(v,t)~\mathrm d\mu(v)~\mathrm d \tau(\mu)$$
for every measurable set $E\subseteq\mathcal{V}\times\mathcal{T}$, The condition that $\tau$ averages to $P\_V$ guarantees that the $\mathcal{V}$-marginal of $\rho$ is $P\_V$. One then takes a disintegration $\sigma:\mathcal{V}\to\Delta(\mathcal{V})$ of $\rho$ as the desired signal. It has the convenient property that one can take the belief induced by $\sigma(v)$ to be $\sigma(v)$. Here is the intuition behind it. Suppose you cannot learn the value of the signal but only learn what you would believe if you would have learned the value. Your updated belief should be then this belief, so taking posteriors as signals works.

Now, what about perfect information? Well, then the posteriors should be point-masses, Dirac-measures. Indeed, you can take $\sigma$ to be given by $\sigma(v)=\delta\_v$ and $\tau=P\_V\circ \sigma^{-1}$.

> **Comments:**
> **username_0**: Yes, I would like to make things rigorous. I've studied measure theory, but I cannot see how to formalise this here. Any rigorous help appreciated.  
> **username_1**: The most helpful version of conditioning for this kind of problems is given by regular conditional probabilities: https://en.wikipedia.org/wiki/Regular_conditional_probability  
> **username_0**: Would it be possible to formalise an answer? I've tried to apply the Wiki article you linked but I'm not sure how to proceed.  
> **username_1**: @user3285148 I've added the basics of the theory.  
> **username_0**: Thanks. Just some clarifications on the last line "You can take $\sigma$ to be given by $\sigma_v=\delta_v$".  Take $t\in \mathcal{T}$. This means that $$\sigma_v(t)=\begin{cases} 1 & \text{if $t=v$} \\ 0 & \text{otherwise}\end{cases}$$ Now, I don't understand what is the meaning of $t=v$ in the above expression given that $\mathcal{V}$ is not finite. I am back to the issue of null sets.  
> **username_1**: $\sigma_v$ is a probability measure defined on the Borel subsets of $\mathcal{V}$. For every Borel set $E\subseteq\mathcal{V}$, we have $\sigma_v(E)=0$ if $v\notin E$ and $\sigma_v(E)=1$ if $v\in E$.  

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