<p>You have to be careful what you can say is that if <span class="math-container">$\underline{S}$</span> be given by <span class="math-container">$(T^1,\pi^1)$</span> and <span class="math-container">$S$</span> is any information structure, say given by <span class="math-container">$(T^2,\pi^2)$</span>, then there exists an expansion <span class="math-container">$S^*$</span> of <span class="math-container">$\underline S$</span> given by <span class="math-container">$(T^1\times T^2, \pi^1\cdot \pi^2)$</span> where <span class="math-container">$\pi^1\cdot \pi^2$</span> is the product distribution of <span class="math-container">$\pi^1$</span> and  <span class="math-container">$\pi^2$</span>. Note that I have constructed the expansion, thus shown its existence.</p>

<p>The information structures, <span class="math-container">$S$</span> and <span class="math-container">$S^*$</span> are different since their support is in different spaces, <span class="math-container">$T^2$</span> vs <span class="math-container">$T^1\times T^2$</span>. However, they are informationally equivalent. The equivalence here is in terms of the distribution of posterior beliefs induced by <span class="math-container">$S$</span> and <span class="math-container">$S^*$</span>. Alternatively, if <span class="math-container">$\succsim$</span> is the Blackwell partial order, then <span class="math-container">$S\succsim S^*$</span> and <span class="math-container">$S^*\succsim S$</span>. Your intuition, however, is correct, any information structure will be "essentially" an expansion of <span class="math-container">$\underline S$</span>.</p>

CC BY-SA 4.0
2019-05-21T12:53:31.463Z
I have posted another question where I'm trying to elaborate on what you wrote. Sorry if I'm being redundant, but I'm trying to understand as much as I can. https://economics.stackexchange.com/questions/29456/implications-of-the-concepts-of-combination-expansion-of-an-information-structur
Star
CC BY-SA 4.0
2019-05-09T21:00:10.990Z
2019-05-09T21:00:10.990Z
Regio
<p>I would like your help to understand the concept of <strong>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>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>My question:</strong>  </p>

<ul>
<li><p>The game, as it is described by the authors, seems to assume that, before receiving the signal <span class="math-container">$T_i$</span>, each player <span class="math-container">$i$</span> knows <strong>nothing</strong> about what will be the realisation of the state. I call this as the <strong>baseline level of information</strong> assumed. </p>

<p>(For example, in other contexts, one may assume that the state is a vector of size <span class="math-container">$N\times 1$</span> and,  before receiving the signal <span class="math-container">$T_i$</span>, each player <span class="math-container">$i$</span> knows the realisation of the <span class="math-container">$i$</span>th component of such a vector. This would correspond to another kind of baseline level of information)</p></li>
<li><p>Let <span class="math-container">$\underline{S}$</span> denote the 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">$T_i$</span>, where we write 
<span class="math-container">$T\equiv T_1\times T_2\times ... \times T_N$</span>.</p>

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

<p>Notice that there are <strong>many</strong> ways to characterise the uninformative information structure (just by varying <span class="math-container">$T$</span> and <span class="math-container">$\tilde{\pi}$</span>).</p></li>
<li><p>Let <span class="math-container">$\mathcal{S}$</span> denote the <strong>collection of all possible information structures</strong>. 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">$\mathcal{S}$</span> contains all possible ways to characterise the uninformative information structure. </p></li>
<li><p><strong>Question:</strong> Can we show that, for a given <span class="math-container">$\underline{S}$</span>, each <span class="math-container">$S\in \mathcal{S}$</span> is an expansion of <span class="math-container">$\underline{S}$</span> (including <span class="math-container">$S=\underline{S}$</span>)? </p>

<p>This seems to me to hold at the light of Theorem 1 combined with reading at p.26 of the paper "<em>Now consider the case where the original information structure is degenerate (there is only one signal
which represents the prior over the states of the world). In this case, the set of Bayes correlated equilibria
correspond to joint distributions of actions and states that could arise under rational choice by a decision
maker with <strong>any</strong> information structure</em>"</p></li>
</ul>

<game-theory><bayesian-game>
The concept of expansion of an information structure
Star
# The concept of expansion of an information structure

(Asked by: username_0 on 2019-05-09)

I would like your help to understand the concept of **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
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

---

**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 $S^2$.

---

**My question:**

* The game, as it is described by the authors, seems to assume that, before receiving the signal $T\_i$, each player $i$ knows **nothing** about what will be the realisation of the state. I call this as the **baseline level of information** assumed.

  (For example, in other contexts, one may assume that the state is a vector of size $N\times 1$ and, before receiving the signal $T\_i$, each player $i$ knows the realisation of the $i$th component of such a vector. This would correspond to another kind of baseline level of information)
* Let $\underline{S}$ denote the 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 $T\_i$, where we write
  $T\equiv T\_1\times T\_2\times ... \times T\_N$.

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

  Notice that there are **many** ways to characterise the uninformative information structure (just by varying $T$ and $\tilde{\pi}$).
* 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 $\mathcal{S}$ contains all possible ways to characterise the uninformative information structure.
* **Question:** Can we show that, for a given $\underline{S}$, each $S\in \mathcal{S}$ is an expansion of $\underline{S}$ (including $S=\underline{S}$)?

  This seems to me to hold at the light of Theorem 1 combined with reading at p.26 of the paper "*Now consider the case where the original information structure is degenerate (there is only one signal
  which represents the prior over the states of the world). In this case, the set of Bayes correlated equilibria
  correspond to joint distributions of actions and states that could arise under rational choice by a decision
  maker with **any** information structure*"

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


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## 1 answers

### Answer by username_1 on 2019-05-09. Score: 3 | Accepted answer

You have to be careful what you can say is that if $\underline{S}$ be given by $(T^1,\pi^1)$ and $S$ is any information structure, say given by $(T^2,\pi^2)$, then there exists an expansion $S^\*$ of $\underline S$ given by $(T^1\times T^2, \pi^1\cdot \pi^2)$ where $\pi^1\cdot \pi^2$ is the product distribution of $\pi^1$ and $\pi^2$. Note that I have constructed the expansion, thus shown its existence.

The information structures, $S$ and $S^\*$ are different since their support is in different spaces, $T^2$ vs $T^1\times T^2$. However, they are informationally equivalent. The equivalence here is in terms of the distribution of posterior beliefs induced by $S$ and $S^\*$. Alternatively, if $\succsim$ is the Blackwell partial order, then $S\succsim S^\*$ and $S^\*\succsim S$. Your intuition, however, is correct, any information structure will be "essentially" an expansion of $\underline S$.

> **Comments:**
> **username_0**: I have posted another question where I'm trying to elaborate on what you wrote. Sorry if I'm being redundant, but I'm trying to understand as much as I can. https://economics.stackexchange.com/questions/29456/implications-of-the-concepts-of-combination-expansion-of-an-information-structur  

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