<p>Answer to Question 1: Yes. To see this, note that for every combination of elements of <span class="math-container">$T^1\times T^2$</span> in the support of <span class="math-container">$\pi^*$</span> you get a, possibly different, posterior. To show <span class="math-container">$\pi^*$</span> is at least as informative as <span class="math-container">$\pi^1$</span> you need these posteriors to be a mean-preserving spread the posteriors induced by <span class="math-container">$\pi^1$</span> (there are actually several characterizations of Blackwell's partial order, I'm thinking there is a more elegant proof, but this is the one I can think of right now). You can easily show this is the case if you average up the posteriors induced by <span class="math-container">$S^*$</span>, over the realizations of <span class="math-container">$T^2$</span>.</p>

<p>Answer to Question 2: Yes. For any signal <span class="math-container">$S$</span>, you can find a "trivial" expansion by combining it with <span class="math-container">$\underline \pi$</span>, so the resulting signal <span class="math-container">$S^*$</span> (which is informationally equivalent to <span class="math-container">$S$</span>) is at least as informative than <span class="math-container">$\underline S$</span>.</p>

<p>In that sense, Bayesian decision makers can never have less information than their prior. </p>

CC BY-SA 4.0
2019-05-22T15:45:37.310Z
2019-05-22T15:45:37.310Z
Regio
<p>I would like your help to understand the concept of <strong>combination/expansion of an information structure</strong> in the incomplete information game at p.6-9 <a href="http://www.princeton.edu/~smorris/pdfs/bce.pdf" rel="nofollow noreferrer">this</a> paper.</p>

<hr>

<p>Let me summarise the game as described in the paper. </p>

<p>There are <span class="math-container">$N\in \mathbb{N}$</span> players, with <span class="math-container">$i$</span> denoting a generic player.</p>

<p>There is a finite set of states <span class="math-container">$\Theta$</span>, with <span class="math-container">$\theta$</span> denoting a generic state. </p>

<p>A basic game <span class="math-container">$G$</span> consists of</p>

<ul>
<li><p>for each player <span class="math-container">$i$</span>, a finite set of actions <span class="math-container">$A_i$</span>, where we write <span class="math-container">$A\equiv A_1\times A_2\times ... \times A_N$</span>, and a utility function <span class="math-container">$u_i: A\times \Theta \rightarrow \mathbb{R}$</span>.</p></li>
<li><p>a full support prior <span class="math-container">$\psi\in \Delta(\Theta)$</span>.</p></li>
</ul>

<p>An information structure <span class="math-container">$S$</span> consists of</p>

<ul>
<li><p>for each player <span class="math-container">$i$</span>, a finite set of signals <span class="math-container">$T_i$</span>, where we write <span class="math-container">$T\equiv T_1\times T_2\times ... \times T_N$</span>.</p></li>
<li><p>a signal distribution <span class="math-container">$\pi: \Theta \rightarrow \Delta(T)$</span>.</p></li>
</ul>

<p>A decision rule of the incomplete information game <span class="math-container">$(G,S)$</span> is a mapping 
<span class="math-container">$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$</span></p>

<hr>

<p><strong>Combination/Expansion:</strong></p>

<p>Consider two information structures, <span class="math-container">$S^1\equiv (T^1, \pi^1)$</span> and <span class="math-container">$S^2\equiv (T^2, \pi^2)$</span>. We say that <span class="math-container">$S^*\equiv (T^*, \pi^*)$</span> is a combination of <span class="math-container">$S^1$</span> and <span class="math-container">$S^2$</span> if </p>

<ul>
<li><p><span class="math-container">$T_i^*=T_i^1\times T_i^2$</span> <span class="math-container">$\forall i$</span>.</p></li>
<li><p><span class="math-container">$\pi^*:\Theta \rightarrow \Delta(T^1\times T^2)$</span> has <span class="math-container">$\pi^1$</span> and <span class="math-container">$\pi^2$</span> as marginals. </p></li>
</ul>

<p>An information structure <span class="math-container">$S^*$</span> is an expansion of an information structure <span class="math-container">$S^1$</span> if there exists an information structure <span class="math-container">$S^2$</span> such that <span class="math-container">$S^*$</span> is a combination of <span class="math-container">$S^1$</span> and <span class="math-container">$S^2$</span>. </p>

<hr>

<p><strong>Question 1:</strong>  Is it correct to say that <span class="math-container">$S^*$</span> is <em>at least as informati</em>ve as <span class="math-container">$S^1$</span>?</p>

<hr>

<p><strong>Question 2:</strong> </p>

<p>The game described above assumes that, before privately receiving the signal, each player <span class="math-container">$i$</span> knows nothing about what will be the realisation of the state. I call this "baseline level of information". </p>

<p>Let <span class="math-container">$\underline{S}$</span> denote an information structure that is totally uninformative, i.e., it does not add anything to the baseline level of information assumed (also called DEGENERATE at p.26 of the linked paper). In other words, <span class="math-container">$\underline{S}$</span> consists of </p>

<p>(a) for each player <span class="math-container">$i$</span>, a finite set of signals <span class="math-container">$\underline{T}_i$</span>, where we write 
  <span class="math-container">$\underline{T}\equiv \underline{T}_1\times \underline{T}_2\times ... \times \underline{T}_N$</span>.</p>

<p>(b) a signal distribution <span class="math-container">$\underline{\pi}: \Theta \rightarrow \Delta(\underline{T})$</span> such that <span class="math-container">$\underline{\pi}(\cdot|\theta)=\tilde{\pi}$</span> <span class="math-container">$\forall \theta \in \Theta$</span> for some <span class="math-container">$\tilde{\pi}\in \Delta(\underline{T})$</span>. In other words, the conditional probability is equal to the unconditional one and players' belief about the probability distribution of the state is not updated. </p>

<p>Let <span class="math-container">$\mathcal{S}$</span> denote the collection of all possible information structures. More precisely,</p>

<p><span class="math-container">$$
\mathcal{S}\equiv \{S| T \text{ is a separable metric space}, \text{ $\pi:\Theta \rightarrow \Delta(T)$ is a probability measure on $(T,\mathcal{B}(T))$}\}
$$</span>
where <span class="math-container">$\mathcal{B}(\cdot)$</span> denotes Borel sigma algebra.</p>

<p>Note that <span class="math-container">$\underline{S}\in \mathcal{S}$</span>. </p>

<p><strong>Is it correct to say that</strong>: <span class="math-container">$\forall S \in \mathcal{S}$</span> where <span class="math-container">$S\equiv ((T_i)_{i=1}^N, \pi)$</span>, there exists a combination <span class="math-container">$S^*\in \mathcal{S}$</span> of <span class="math-container">$\underline{S}$</span> and <span class="math-container">$S$</span> such that <span class="math-container">$S^*$</span> and <span class="math-container">$S$</span> are informationally equivalent. In particular, one can set <span class="math-container">$S^*\equiv ((\underline{T}_i\times T_i)_{i=1}^N, \underline{\pi}\times \pi)$</span>. Therefore, every <span class="math-container">$S\in \mathcal{S}$</span> is at least as informative as <span class="math-container">$\underline{S}$</span>. </p>

<game-theory><bayesian-game>
Implications of the concepts of combination/expansion of an information structure
Star
# Implications of the concepts of combination/expansion of an information structure

(Asked by: username_0 on 2019-05-21)

I would like your help to understand the concept of **combination/expansion of an information structure** in the incomplete information game at p.6-9 [this](http://www.princeton.edu/~smorris/pdfs/bce.pdf) paper.

---

Let me summarise the game as described in the paper.

There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.

There is a finite set of states $\Theta$, with $\theta$ denoting a generic state.

A basic game $G$ consists of

* for each player $i$, a finite set of actions $A\_i$, where we write $A\equiv A\_1\times A\_2\times ... \times A\_N$, and a utility function $u\_i: A\times \Theta \rightarrow \mathbb{R}$.
* a full support prior $\psi\in \Delta(\Theta)$.

An information structure $S$ consists of

* for each player $i$, a finite set of signals $T\_i$, where we write $T\equiv T\_1\times T\_2\times ... \times T\_N$.
* a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.

A decision rule of the incomplete information game $(G,S)$ is a mapping
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

---

**Combination/Expansion:**

Consider two information structures, $S^1\equiv (T^1, \pi^1)$ and $S^2\equiv (T^2, \pi^2)$. We say that $S^\*\equiv (T^\*, \pi^\*)$ is a combination of $S^1$ and $S^2$ if

* $T\_i^\*=T\_i^1\times T\_i^2$ $\forall i$.
* $\pi^\*:\Theta \rightarrow \Delta(T^1\times T^2)$ has $\pi^1$ and $\pi^2$ as marginals.

An information structure $S^\*$ is an expansion of an information structure $S^1$ if there exists an information structure $S^2$ such that $S^\*$ is a combination of $S^1$ and $S^2$.

---

**Question 1:** Is it correct to say that $S^\*$ is *at least as informati*ve as $S^1$?

---

**Question 2:**

The game described above assumes that, before privately receiving the signal, each player $i$ knows nothing about what will be the realisation of the state. I call this "baseline level of information".

Let $\underline{S}$ denote an information structure that is totally uninformative, i.e., it does not add anything to the baseline level of information assumed (also called DEGENERATE at p.26 of the linked paper). In other words, $\underline{S}$ consists of

(a) for each player $i$, a finite set of signals $\underline{T}\_i$, where we write
$\underline{T}\equiv \underline{T}\_1\times \underline{T}\_2\times ... \times \underline{T}\_N$.

(b) a signal distribution $\underline{\pi}: \Theta \rightarrow \Delta(\underline{T})$ such that $\underline{\pi}(\cdot|\theta)=\tilde{\pi}$ $\forall \theta \in \Theta$ for some $\tilde{\pi}\in \Delta(\underline{T})$. In other words, the conditional probability is equal to the unconditional one and players' belief about the probability distribution of the state is not updated.

Let $\mathcal{S}$ denote the collection of all possible information structures. More precisely,

$$
\mathcal{S}\equiv \{S| T \text{ is a separable metric space}, \text{ $\pi:\Theta \rightarrow \Delta(T)$ is a probability measure on $(T,\mathcal{B}(T))$}\}
$$
where $\mathcal{B}(\cdot)$ denotes Borel sigma algebra.

Note that $\underline{S}\in \mathcal{S}$.

**Is it correct to say that**: $\forall S \in \mathcal{S}$ where $S\equiv ((T\_i)\_{i=1}^N, \pi)$, there exists a combination $S^\*\in \mathcal{S}$ of $\underline{S}$ and $S$ such that $S^\*$ and $S$ are informationally equivalent. In particular, one can set $S^\*\equiv ((\underline{T}\_i\times T\_i)\_{i=1}^N, \underline{\pi}\times \pi)$. Therefore, every $S\in \mathcal{S}$ is at least as informative as $\underline{S}$.

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


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## 1 answers

### Answer by username_1 on 2019-05-22. Score: 1 | Accepted answer

Answer to Question 1: Yes. To see this, note that for every combination of elements of $T^1\times T^2$ in the support of $\pi^\*$ you get a, possibly different, posterior. To show $\pi^\*$ is at least as informative as $\pi^1$ you need these posteriors to be a mean-preserving spread the posteriors induced by $\pi^1$ (there are actually several characterizations of Blackwell's partial order, I'm thinking there is a more elegant proof, but this is the one I can think of right now). You can easily show this is the case if you average up the posteriors induced by $S^\*$, over the realizations of $T^2$.

Answer to Question 2: Yes. For any signal $S$, you can find a "trivial" expansion by combining it with $\underline \pi$, so the resulting signal $S^\*$ (which is informationally equivalent to $S$) is at least as informative than $\underline S$.

In that sense, Bayesian decision makers can never have less information than their prior.


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