<p>What you are defining is the complete information structure <span class="math-container">$\bar{S}$</span> with <span class="math-container">$\bar{T}_{i}=\Theta$</span> for all <span class="math-container">$i$</span> and  <span class="math-container">$$ \bar{\pi}(t|\theta) = \begin{cases} \ 1, &amp; \text{if} \;\; t_{i}= \theta \text{ for all} \; i \\ 0, &amp; \text{otherwise}, \end{cases} $$</span> for all <span class="math-container">$\theta \in \Theta$</span>. </p>

<p>Note that this is not the same as the case in which <span class="math-container">$\Theta$</span> is a singleton, but it is rather an extreme information structure. The same authors define this <a href="https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/3/352/files/2015/06/bce-4feb13sm-with-graphs-22vwdvb.pdf" rel="nofollow noreferrer">here</a> (Bayes Correlated Equilibrium and the Comparison of Information Structures - Bergemann &amp; Morris), it beats me why they didn't include this detail in the paper you are looking at. </p>

CC BY-SA 4.0
2019-05-09T17:14:13.463Z
2019-05-09T17:14:13.463Z
Ali
<p>The two formulations are equivalent in the sense that if every type of player always learns the state of the world, there is no uncertainty and there is really no need to carry around the realized state as a variable. By assuming <span class="math-container">$|\Theta|=1$</span> you simplify the notation without losing generality (of course you lose some information, but this is irrelevant for the purposes of that paper). If you want, you can assume that the authors' statements are true state-by-state. </p>

CC BY-SA 4.0
2019-05-09T17:18:39.123Z
2019-05-09T17:18:39.123Z
Regio
<p>I would like your help to understand the concept of information structure in the incomplete information game at p.6-7 of <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>My question:</strong>  </p>

<p>I interpret <span class="math-container">$\pi(t|\theta)$</span> as a probability that, conditional on the realisation <span class="math-container">$\theta$</span> of the state, the players receive as signals <span class="math-container">$t_1,...,t_N$</span>, respectively. According to the given information structure, signals are more or less informative.</p>

<p>If this interpretation is correct, then I'm confused about the first sentence at p.7 of the linked paper: "If there is complete information, i.e., if <span class="math-container">$\Theta$</span> is singleton, [...]".</p>

<p>My first guess was that complete information is characterised by specifying <span class="math-container">$S$</span> and not by restricting the support of the state. In other words, I thought that complete information corresponds to the information structure <span class="math-container">$\bar{S}\equiv (\bar{T},\bar{\pi})$</span> such that</p>

<ul>
<li><p>for each player <span class="math-container">$i$</span>, <span class="math-container">$\bar{T}_i=\Theta$</span>.</p></li>
<li><p><span class="math-container">$\bar{\pi}(\theta|\theta)=1$</span>.</p></li>
</ul>

<p>Where am I wrong? Why the author defines complete information as having <span class="math-container">$|\Theta|=1$</span>?</p>

CC BY-SA 4.0
2019-05-09T16:36:32.757Z
Maybe I am missing something but if the set of states is a singleton there is no uncertainty (i.e. complete information) right? There is only one state which by definition is realized.
Ali
CC BY-SA 4.0
2019-05-09T16:42:32.310Z
That is correct. I understand that $|\Theta|=1$ implies complete information. But also $\bar{S}$ as a I define implies complete information. I'm just wondering why the authors prefer to characterise complete information using $|\Theta|=1$ rather than $\bar{S}$. $\bar{S}$ seems to me more natural. Or am I making mistakes somewhere?
Star
CC BY-SA 4.0
2019-05-09T17:15:41.123Z
I posted as an answer, maybe this will help
Ali
<game-theory><bayesian-game>
The concept of information structure in incomplete information games
Star
# The concept of information structure in incomplete information games

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

I would like your help to understand the concept of information structure in the incomplete information game at p.6-7 of [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
$$
\sigma: T\times \Theta\rightarrow \Delta(A)
$$

---

**My question:**

I interpret $\pi(t|\theta)$ as a probability that, conditional on the realisation $\theta$ of the state, the players receive as signals $t\_1,...,t\_N$, respectively. According to the given information structure, signals are more or less informative.

If this interpretation is correct, then I'm confused about the first sentence at p.7 of the linked paper: "If there is complete information, i.e., if $\Theta$ is singleton, [...]".

My first guess was that complete information is characterised by specifying $S$ and not by restricting the support of the state. In other words, I thought that complete information corresponds to the information structure $\bar{S}\equiv (\bar{T},\bar{\pi})$ such that

* for each player $i$, $\bar{T}\_i=\Theta$.
* $\bar{\pi}(\theta|\theta)=1$.

Where am I wrong? Why the author defines complete information as having $|\Theta|=1$?

**Tags:** game-theory, bayesian-game
> **Comments:**
> **username_1**: Maybe I am missing something but if the set of states is a singleton there is no uncertainty (i.e. complete information) right? There is only one state which by definition is realized.  
> **username_0**: That is correct. I understand that $|\Theta|=1$ implies complete information. But also $\bar{S}$ as a I define implies complete information. I'm just wondering why the authors prefer to characterise complete information using $|\Theta|=1$ rather than $\bar{S}$. $\bar{S}$ seems to me more natural. Or am I making mistakes somewhere?  
> **username_1**: I posted as an answer, maybe this will help  

---

## 2 answers

### Answer by username_2 on 2019-05-09. Score: 2 | Accepted answer

The two formulations are equivalent in the sense that if every type of player always learns the state of the world, there is no uncertainty and there is really no need to carry around the realized state as a variable. By assuming $|\Theta|=1$ you simplify the notation without losing generality (of course you lose some information, but this is irrelevant for the purposes of that paper). If you want, you can assume that the authors' statements are true state-by-state.


---

### Answer by username_1 on 2019-05-09. Score: 2

What you are defining is the complete information structure $\bar{S}$ with $\bar{T}\_{i}=\Theta$ for all $i$ and $$ \bar{\pi}(t|\theta) = \begin{cases} \ 1, & \text{if} \;\; t\_{i}= \theta \text{ for all} \; i \\ 0, & \text{otherwise}, \end{cases} $$ for all $\theta \in \Theta$.

Note that this is not the same as the case in which $\Theta$ is a singleton, but it is rather an extreme information structure. The same authors define this [here](https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/3/352/files/2015/06/bce-4feb13sm-with-graphs-22vwdvb.pdf) (Bayes Correlated Equilibrium and the Comparison of Information Structures - Bergemann & Morris), it beats me why they didn't include this detail in the paper you are looking at.


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