<p>Two points.</p>

<ol>
<li><p>Common knowledge is defined by Aumann in terms of partitions, not <span class="math-container">$\sigma$</span>-algebras. There is generally no natural correspondence between these.</p></li>
<li><p>The grand state space is trivially common knowledge. So however you conceive of the relevant states, something that holds everywhere, such as the structure of the model, is common knowledge even if agents have only the trivial knowledge partition with one element. This is actually useful even when modeling more explicitly epistemic phenomena. For example, one way to guarantee common knowledge of rationality is to assume agents are rational at each state of the world. This is what Aumann did in his <a href="http://www.ma.huji.ac.il/~raumann/pdf/Correlated%20Equilibrium%20as%20an%20Expression.pdf" rel="nofollow noreferrer">1987 paper</a> on correlated equilibria.</p></li>
</ol>

CC BY-SA 4.0
2020-06-08T23:44:20.797Z
2. answers the question; the structure of the model is a probability 1 event. Regarding 1, I would say that Aumann is really making statements about the sigma-algebra generated by the partition. Operations like meet and join are standard ones for sigma-algebras and his definition translates verbatim.  Only reason to restrict to "information sets"/partitions is expository. The sigma-algebra formulation is implicitly invoked when assuming common knowledge in, e.g. the Grossman-Stiglitz model, where the information sigma-algebra is not a finite partition.
Michael
CC BY-SA 4.0
2020-06-09T06:42:13.420Z
For a countable state space, there isn't really any difference since there we get an isomorphism between partitions and $\sigma$-algebras that does preserves order. But in general, these concepts are very different. Take a look at [this paper](https://link.springer.com/article/10.1007%2Fs00199-012-0723-1), for example.
Michael Greinecker
CC BY-SA 4.0
2020-06-09T16:36:41.687Z
That is very interesting. But that is an issue only if one agrees with the definition/insistence that that information is the partition given by preimages of singletons (or equivalently, the discrete sigma algebra). That would not be an issue if one defines information to be the smallest sigma-algebra such that the signal is, say, Borel measurable, correct? In the Billingley example, signal is the identity map and information would the Borel sigma-field itself, which one can interpret as full information.
Michael
CC BY-SA 4.0
2020-06-09T16:48:43.483Z
Seems to me the latter approach, defining *information* to smallest sigma-algebra such that the signal is Borel measurable, is still pretty standard, both in probability where Billingsley comes from (e.g. the interpretation of filtration as representing information flow) and most of economics (implicitly, as most seem unaware). Has the economic theory community come to a consensus on this issue (maybe it's basic at this point, I don't know)? If having partitions as a primitive is awkward due to countability issues, why not simply bypass it for sigma-algebras?
Michael
CC BY-SA 4.0
2020-06-09T18:19:43.277Z
In practice, one usually works with type spaces and argues with certainty instead of knowledge. I personally think that can be made to work for most practical problems.
Michael Greinecker
CC BY-SA 4.0
2020-06-08T22:23:49.063Z
2020-06-08T22:23:49.063Z
Michael Greinecker
<p>Economics models usually assume that the structure of the economy is <strong>common knowledge</strong> among agents. </p>

<p>Mathematically, an event is common knowledge if it lies in the <em>meet</em> of all agents' information sets. (The meet of a family of $\sigma$-algebras is the finest common coarsification of all $\sigma$-algebras in the family. See Aumann 1976.)</p>

<p>However, in the literature when a paper makes the common knowledge assumption, one almost never sees the meet of any information sets anywhere. Let me give an example (Grossman and Stiglitz 1980).</p>

<p><strong>Grossman-Stiglitz model</strong></p>

<p>This is a model where two traders with (essentially) mean-variance utility are asymmetrically informed about the mean. By assumption, structure of the economy is common knowledge and traders are rational. Equilibrium price must therefore, first, clear the market and, second, be consistent with the uninformed trader's expectation. Details given as follows.</p>

<p>Let $I$ and $U$ denote the informed and uninformed traders respectively. The traders have same CARA utility, $u_I(W) = u_U(W) = - e^{-\gamma W}$. The risk-free rate is $r$. The price of risky asset tomorrow is $P_2 \sim \mathcal{N}(\bar{P}, \sigma^2) | \bar{P}$. The informed trader $I$ knows $\bar{P}$; the uninformed trader $U$ only knows the prior distribution $\bar{P} \sim \mathcal{N}(P_0, \sigma_0^2)$.</p>

<p>Let both trader have wealth endowments $E$. (CARA utility has no wealth effect, so $E$ plays no role in portfolio choice below.) </p>

<p>Given today's price $P_1$ for the risky asset, trader's wealth tomorrow is
$W(\phi) = rW + \phi (P_2 - rP_1)$ if he choose to hold $\phi$ units of risky asset.</p>

<p>Maximizing expected utility, a traders' demand is</p>

<p>$$
\frac{E[P_2|\mathcal{F}] - rP_1}{\gamma Var(P_2|\mathcal{F})}
$$</p>

<p>conditional on his information set $\mathcal{F}$. For $I$, $\mathcal{F}_I = \{ \bar{P}, P_1\} $. For $U$, $\mathcal{F}_U = \{ P_1 \}$.</p>

<p>The supply $x$ of risky asset is distributed $\mathcal{N}(\bar{x}, \sigma_x^2)$. </p>

<p>Equilibrium is a price function $P_1(\bar{P}, x)$ such that</p>

<p>$$
\frac{\bar{P} - r P_1(\bar{P}, x)}{\gamma \sigma^2}+
\frac{E[P_2|P_1(\bar{P}, x)] - r P_1(\bar{P}, x)}{\gamma Var(P_2|P_1(\bar{P}, x))} = x.
$$</p>

<p>In other words, market clears and the uninformed trader computes his demand using the correct pricing function in equilibrium.</p>

<p>Now in this CARA-normal setting, things are linear and one can solve for equilibrium by guessing a pricing function</p>

<p>$$
P_1(\bar{P}, x) = A\bar{P} + Bx + C
$$</p>

<p>and find $A$, $B$ and $C$ by matching coefficients. For example, compute</p>

<p>$$
E[P_2|P_1] = P_0 + \frac{A \sigma_0^2}{A^2 \sigma_0^2 + B^2 \sigma_x^2} [A(\bar{P} - P_0)  + Bx]
$$</p>

<p>and</p>

<p>$$
Var(P_2|P_1) = \frac{B^2 \sigma_x^2}{A^2 \sigma_0^2 + B^2 \sigma_x^2}\sigma_0^2 + \sigma^2 
$$</p>

<p>Substituting into the market clearing equation and some tedious algebra gives endogenous constants $A$, $B$, and $C$.</p>

<p>Instead, Grossman and Stiglitz do something more elegant.
They point out that uninformed trader $U$ can condition on the <em>residual demand</em> for any realization of $(\bar{P},x)$</p>

<p>$$
D_{resid} = x -  \frac{\bar{P} - r P_1 }{\gamma \sigma^2}.
$$</p>

<p>What's even more clever, they note that, since $P_1$ is observed in 
in equilibrium, they first conjecture that $P_1$ and</p>

<p>$$
\tilde{D}_{resid}
= x -  \frac{\bar{P} }{\gamma \sigma^2}.
$$</p>

<p>are informationally equivalent. Then equilibrium condition becomes</p>

<p>$$
\frac{ r P_1(\bar{P}, x)}{\gamma \sigma^2}+
\frac{E[P_2|\tilde{D}_{resid}] - r P_1(\bar{P}, x)}{\gamma Var(P_2|\tilde{D}_{resid})} = \tilde{D}_{resid}.
$$</p>

<p>Now things are much less tedious and there are no endogenous constants $A$ and $B$ to solve for. $E[P_2|\tilde{D}_{resid}]$ is an affine function of
$\tilde{D}_{resid}$ and $Var(P_2|\tilde{D}_{resid})$ is an exogenous constant. So that $P_1(\bar{P}, x)$ can be backed out immediately. Since
$P_1$ is an affine function of $\tilde{D}_{resid}$, it is verified ex-post 
that they are informationally equivalent.</p>

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

<p>To have the uninformed trader $U$ condition on the residual demand, the common knowledge assumption is used. Or, at least, $U$ knows that $I$ knows so that $U$ can put herself in $I$'s shoes and compute $I$'s demand. </p>

<p>However, the meet of the information sets does not appear anywhere---as it should. Result of $U$'s calculations should be measurable with respect to the meet. If one is to formulate this mathematically, where would that appear?</p>

CC BY-SA 3.0
2017-08-08T22:01:52.237Z
It looks like what is being used is common knowledge at the level of the structure of the economy---utility of agents, def of equil, etc. So that it is $U$'s action in computing residual demand  that is measurable w.r.t. the meet. The residual demand itself as an r.v. is the wrong level to pose the question.
Michael
CC BY-SA 4.0
2019-04-26T17:54:13.603Z
Isn't the common knowledge assumption showing up, precisely in that the uninformed player has certainty that the informed player knows price $\bar P$, so for any possible realization of it, she can compute the residual demand. As for the informed player, his information about the uninformed player is irrelevant. His own information is sufficient to choose his optimal demand. It is often tedious to be strictly formal about the common knowledge assumption, but it can always be seen in what are each player is conditioning on.
Regio
<mathematical-economics><asymmetric-information><information>
Common knowledge in model formulation and solution
Michael
# Common knowledge in model formulation and solution

(Asked by: username_0 on 2017-08-08)

Economics models usually assume that the structure of the economy is **common knowledge** among agents.

Mathematically, an event is common knowledge if it lies in the *meet* of all agents' information sets. (The meet of a family of $\sigma$-algebras is the finest common coarsification of all $\sigma$-algebras in the family. See Aumann 1976.)

However, in the literature when a paper makes the common knowledge assumption, one almost never sees the meet of any information sets anywhere. Let me give an example (Grossman and Stiglitz 1980).

**Grossman-Stiglitz model**

This is a model where two traders with (essentially) mean-variance utility are asymmetrically informed about the mean. By assumption, structure of the economy is common knowledge and traders are rational. Equilibrium price must therefore, first, clear the market and, second, be consistent with the uninformed trader's expectation. Details given as follows.

Let $I$ and $U$ denote the informed and uninformed traders respectively. The traders have same CARA utility, $u\_I(W) = u\_U(W) = - e^{-\gamma W}$. The risk-free rate is $r$. The price of risky asset tomorrow is $P\_2 \sim \mathcal{N}(\bar{P}, \sigma^2) | \bar{P}$. The informed trader $I$ knows $\bar{P}$; the uninformed trader $U$ only knows the prior distribution $\bar{P} \sim \mathcal{N}(P\_0, \sigma\_0^2)$.

Let both trader have wealth endowments $E$. (CARA utility has no wealth effect, so $E$ plays no role in portfolio choice below.)

Given today's price $P\_1$ for the risky asset, trader's wealth tomorrow is
$W(\phi) = rW + \phi (P\_2 - rP\_1)$ if he choose to hold $\phi$ units of risky asset.

Maximizing expected utility, a traders' demand is

$$
\frac{E[P\_2|\mathcal{F}] - rP\_1}{\gamma Var(P\_2|\mathcal{F})}
$$

conditional on his information set $\mathcal{F}$. For $I$, $\mathcal{F}\_I = \{ \bar{P}, P\_1\} $. For $U$, $\mathcal{F}\_U = \{ P\_1 \}$.

The supply $x$ of risky asset is distributed $\mathcal{N}(\bar{x}, \sigma\_x^2)$.

Equilibrium is a price function $P\_1(\bar{P}, x)$ such that

$$
\frac{\bar{P} - r P\_1(\bar{P}, x)}{\gamma \sigma^2}+
\frac{E[P\_2|P\_1(\bar{P}, x)] - r P\_1(\bar{P}, x)}{\gamma Var(P\_2|P\_1(\bar{P}, x))} = x.
$$

In other words, market clears and the uninformed trader computes his demand using the correct pricing function in equilibrium.

Now in this CARA-normal setting, things are linear and one can solve for equilibrium by guessing a pricing function

$$
P\_1(\bar{P}, x) = A\bar{P} + Bx + C
$$

and find $A$, $B$ and $C$ by matching coefficients. For example, compute

$$
E[P\_2|P\_1] = P\_0 + \frac{A \sigma\_0^2}{A^2 \sigma\_0^2 + B^2 \sigma\_x^2} [A(\bar{P} - P\_0) + Bx]
$$

and

$$
Var(P\_2|P\_1) = \frac{B^2 \sigma\_x^2}{A^2 \sigma\_0^2 + B^2 \sigma\_x^2}\sigma\_0^2 + \sigma^2
$$

Substituting into the market clearing equation and some tedious algebra gives endogenous constants $A$, $B$, and $C$.

Instead, Grossman and Stiglitz do something more elegant.
They point out that uninformed trader $U$ can condition on the *residual demand* for any realization of $(\bar{P},x)$

$$
D\_{resid} = x - \frac{\bar{P} - r P\_1 }{\gamma \sigma^2}.
$$

What's even more clever, they note that, since $P\_1$ is observed in
in equilibrium, they first conjecture that $P\_1$ and

$$
\tilde{D}\_{resid}
= x - \frac{\bar{P} }{\gamma \sigma^2}.
$$

are informationally equivalent. Then equilibrium condition becomes

$$
\frac{ r P\_1(\bar{P}, x)}{\gamma \sigma^2}+
\frac{E[P\_2|\tilde{D}\_{resid}] - r P\_1(\bar{P}, x)}{\gamma Var(P\_2|\tilde{D}\_{resid})} = \tilde{D}\_{resid}.
$$

Now things are much less tedious and there are no endogenous constants $A$ and $B$ to solve for. $E[P\_2|\tilde{D}\_{resid}]$ is an affine function of
$\tilde{D}\_{resid}$ and $Var(P\_2|\tilde{D}\_{resid})$ is an exogenous constant. So that $P\_1(\bar{P}, x)$ can be backed out immediately. Since
$P\_1$ is an affine function of $\tilde{D}\_{resid}$, it is verified ex-post
that they are informationally equivalent.

**Question**

To have the uninformed trader $U$ condition on the residual demand, the common knowledge assumption is used. Or, at least, $U$ knows that $I$ knows so that $U$ can put herself in $I$'s shoes and compute $I$'s demand.

However, the meet of the information sets does not appear anywhere---as it should. Result of $U$'s calculations should be measurable with respect to the meet. If one is to formulate this mathematically, where would that appear?

**Tags:** mathematical-economics, asymmetric-information, information
> **Comments:**
> **username_0**: It looks like what is being used is common knowledge at the level of the structure of the economy---utility of agents, def of equil, etc. So that it is $U$'s action in computing residual demand  that is measurable w.r.t. the meet. The residual demand itself as an r.v. is the wrong level to pose the question.  
> **username_1**: Isn't the common knowledge assumption showing up, precisely in that the uninformed player has certainty that the informed player knows price $\bar P$, so for any possible realization of it, she can compute the residual demand. As for the informed player, his information about the uninformed player is irrelevant. His own information is sufficient to choose his optimal demand. It is often tedious to be strictly formal about the common knowledge assumption, but it can always be seen in what are each player is conditioning on.  

---

## 1 answers

### Answer by username_2 on 2020-06-08. Score: 2 | Accepted answer

Two points.

1. Common knowledge is defined by Aumann in terms of partitions, not $\sigma$-algebras. There is generally no natural correspondence between these.
2. The grand state space is trivially common knowledge. So however you conceive of the relevant states, something that holds everywhere, such as the structure of the model, is common knowledge even if agents have only the trivial knowledge partition with one element. This is actually useful even when modeling more explicitly epistemic phenomena. For example, one way to guarantee common knowledge of rationality is to assume agents are rational at each state of the world. This is what Aumann did in his [1987 paper](http://www.ma.huji.ac.il/~raumann/pdf/Correlated%20Equilibrium%20as%20an%20Expression.pdf) on correlated equilibria.

> **Comments:**
> **username_0**: 2. answers the question; the structure of the model is a probability 1 event. Regarding 1, I would say that Aumann is really making statements about the sigma-algebra generated by the partition. Operations like meet and join are standard ones for sigma-algebras and his definition translates verbatim.  Only reason to restrict to "information sets"/partitions is expository. The sigma-algebra formulation is implicitly invoked when assuming common knowledge in, e.g. the Grossman-Stiglitz model, where the information sigma-algebra is not a finite partition.  
> **username_2**: For a countable state space, there isn't really any difference since there we get an isomorphism between partitions and $\sigma$-algebras that does preserves order. But in general, these concepts are very different. Take a look at [this paper](https://link.springer.com/article/10.1007%2Fs00199-012-0723-1), for example.  
> **username_0**: That is very interesting. But that is an issue only if one agrees with the definition/insistence that that information is the partition given by preimages of singletons (or equivalently, the discrete sigma algebra). That would not be an issue if one defines information to be the smallest sigma-algebra such that the signal is, say, Borel measurable, correct? In the Billingley example, signal is the identity map and information would the Borel sigma-field itself, which one can interpret as full information.  
> **username_0**: Seems to me the latter approach, defining *information* to smallest sigma-algebra such that the signal is Borel measurable, is still pretty standard, both in probability where Billingsley comes from (e.g. the interpretation of filtration as representing information flow) and most of economics (implicitly, as most seem unaware). Has the economic theory community come to a consensus on this issue (maybe it's basic at this point, I don't know)? If having partitions as a primitive is awkward due to countability issues, why not simply bypass it for sigma-algebras?  
> **username_2**: In practice, one usually works with type spaces and argues with certainty instead of knowledge. I personally think that can be made to work for most practical problems.  

---

