<p>Based on <a href="https://onlinelibrary.wiley.com/doi/epdf/10.3982/TE1808" rel="nofollow noreferrer">Begemann and Morris (2016)</a> we have the following definition about a standard game of incomplete information</p>
<p>A standard game <span class="math-container">$\Gamma = &lt;G,S&gt;$</span> of incomplete information consists of a set of <span class="math-container">$I$</span> players, <span class="math-container">$1,2,\cdots,I$</span> and we write <span class="math-container">$i$</span> for a typical player. There is a finite set of states, <span class="math-container">$\Theta$</span>, and we write <span class="math-container">$\theta$</span> for a typical state, where</p>
<p><span class="math-container">$\textbf{Definition 1: Basic Game}$</span> A basic game <span class="math-container">$G=\left((A_i,u_i)_{i=1}^I,\psi\right)$</span> consists of</p>
<ol>
<li>a finite set of actions <span class="math-container">$A_i$</span> (where <span class="math-container">$A=A_1\times\dots\times A_I$</span>) for every player <span class="math-container">$i$</span> and a utility function
<span class="math-container">$$u_i:A\times\Theta\to\mathbb{R}$$</span></li>
<li>and a full support common prior <span class="math-container">$\psi\in\Delta(\Theta)_{++}$</span></li>
</ol>
<p><span class="math-container">$\textbf{Definition 2: Information structure}$</span> An information structure <span class="math-container">$S$</span> consists of</p>
<ol>
<li>a finite set of signals (or types) <span class="math-container">$T_i$</span> for every player <span class="math-container">$i$</span> (where <span class="math-container">$T=T_1\times\dots\times T_I$</span>)</li>
<li>and a signal probability distribution <span class="math-container">$\pi:\Theta\to\Delta(T)$</span></li>
</ol>
<p>And this completes the game definition.</p>
<p>A decision rule in the incomplete information game <span class="math-container">$\Gamma = &lt;G,S&gt;$</span> is a mapping <span class="math-container">$\sigma$</span> such that:</p>
<p><span class="math-container">\begin{equation}\tag{1}\sigma:T\times\Theta\to\Delta(A)\end{equation}</span></p>
<p>To understand the game, <span class="math-container">$\sigma$</span> is the strategy of an omniscient mediator who first observes the realization of <span class="math-container">$\theta\in\Theta$</span> chosen according to <span class="math-container">$\psi$</span> and the realization of <span class="math-container">$t\in T$</span> chosen according to <span class="math-container">$\pi(\cdot|\theta)$</span>, and then picks an action profile <span class="math-container">$a\in A$</span>. From the players perspective in order to follow the action recommendation, it would have to be the case that the recommended action <span class="math-container">$a_i$</span> was always proffered to any other alternative <span class="math-container">$a_i{'}$</span> conditional on the signal <span class="math-container">$t_i$</span> that player <span class="math-container">$i$</span> had received and his knowledge of the recommended action <span class="math-container">$a_i$</span>. This is reflected in the obedience condition</p>
<p><span class="math-container">$\textbf{Definition 3: Obedience}$</span> A decision rule <span class="math-container">$\sigma$</span> is obedient for <span class="math-container">$\Gamma$</span>, if for every <span class="math-container">$i\in I$</span>, <span class="math-container">$t_i\in T_i$</span> and <span class="math-container">$a_i\in A_i$</span> we have</p>
<p><span class="math-container">\begin{equation}
\sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\sigma((a_{i},a_{-i})|(t_i,t_{-i}),\theta)u_i((a_{i},a_{-i}),\theta) \\
\geq \sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\sigma((a_{i},a_{-i})|(t_i,t_{-i}),\theta)u_i((a_{i}^{'},a_{-i}),\theta)
\end{equation}</span></p>
<p>for all <span class="math-container">$a_{i}^{'}\in A_i$</span>.</p>
<p><span class="math-container">$\textbf{Definition 4: Bayesian Correlated Equilibrium}$</span> A decision rule <span class="math-container">$\sigma$</span> is a Bayesian correlated equilibrium (BCE) of <span class="math-container">$\Gamma$</span>, if it is obedient for <span class="math-container">$\Gamma$</span>.</p>
<p>Suppose that a game <span class="math-container">$\Gamma^{e}=(G, S^{e} )$</span>  is a new game where with respect to the original game <span class="math-container">$\Gamma$</span> where the information structure <span class="math-container">$S^{e}=(T, \pi^{e})$</span>  is an expansion of the original information structure <span class="math-container">$S$</span>. An expansion of an information structure <span class="math-container">$S$</span> means that it can be written as an combination of <span class="math-container">$S$</span> with another information structure <span class="math-container">$S^{'}$</span>, where the combined information structure is obtained by forming a product space of the signals <span class="math-container">$T_i^{e}= T_i\times T_i^{'}$</span> for every <span class="math-container">$i$</span> and the signal distribution <span class="math-container">$\pi^{e}:\Theta\to\Delta(T\times T^{'})$</span> that preserves the marginal distribution of its constituent information structures.</p>
<p><span class="math-container">$\textbf{Definition 5: Combination}$</span> The information structure <span class="math-container">$S^{e}=(T, \pi^{e})$</span> is a combination of the information structures <span class="math-container">$S=(T, \pi)$</span> and <span class="math-container">$S^{'}=(T, \pi^{'})$</span> if</p>
<p><span class="math-container">$$T_i^{e}= T_i\times T_i^{'},\quad\text{for every $i$}$$</span></p>
<p>and</p>
<p><span class="math-container">$$\sum_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)=\pi(t|\theta),\quad\text{for every $t\in T$ and $\theta\in \Theta$}$$</span></p>
<p><span class="math-container">$$\sum_{t\in T}\pi^{e}(t,t^{'}|\theta)=\pi^{'}(t^{'}|\theta),\quad\text{for every $t^{'}\in T^{'}$ and $\theta\in \Theta$}$$</span></p>
<p><span class="math-container">$\textbf{Definition 6: Expansion}$</span> An information structure <span class="math-container">$S^{e}$</span> is an expansion of an information structure <span class="math-container">$S$</span> if <span class="math-container">$S^{e}$</span> is a combination of <span class="math-container">$S$</span> and some other information structure <span class="math-container">$S^{'}$</span>.</p>
<p><span class="math-container">$\textbf{Definition 7: Bayesian Nash Equilibrium}$</span> A strategy profile <span class="math-container">$\beta$</span> is a Bayesian Nash Equilibrium of <span class="math-container">$\Gamma$</span>, if for each <span class="math-container">$i\in I$</span>, <span class="math-container">$t_i\in T_i$</span>, and <span class="math-container">$a_i\in A_i$</span> the (behavioral) strategy <span class="math-container">$\beta_i:T_i\to\Delta(A_i)$</span> with <span class="math-container">$\beta_i(a_i|t_i)&gt;0$</span> we have that</p>
<p><span class="math-container">\begin{equation}
\sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\left(\Pi_{j\neq i}\beta_{j}(a_j|t_j)\right)u_i((a_{i},a_{-i}),\theta) \\
\geq \sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\left(\Pi_{j\neq i}\beta_{j}(a_j|t_j)\right)u_i((a_{i}^{'},a_{-i}),\theta)
\end{equation}</span></p>
<p>for all <span class="math-container">$a_{i}^{'}\in A_i$</span>.</p>
<p>All these are needed to understand the model of Bergemman and Morris. My question is about theorem <span class="math-container">$1$</span> in the paper and a statistic that uses to connect BCE with the Bayesian Nash Equilibrium. The statistic is the following:</p>
<p>Suppose that the profile <span class="math-container">$\beta$</span> was played in <span class="math-container">$\Gamma^{e}$</span> where the information structure <span class="math-container">$S^{e}$</span> is the expansion of the information structure <span class="math-container">$S$</span> in the game <span class="math-container">$\Gamma$</span> and it is the combination of <span class="math-container">$S$</span> and <span class="math-container">$S^{'}$</span> Now if the analyst did not observe the signals of the combined information structure <span class="math-container">$S^{e}$</span>, but only the signals of <span class="math-container">$S$</span>, then the behaviour under the strategy profile <span class="math-container">$\beta$</span> would induce a decision rule for <span class="math-container">$\Gamma$</span> (Why?). Formally the strategy profile <span class="math-container">$\beta$</span> of <span class="math-container">$\Gamma^{e}$</span> induces the decision rule <span class="math-container">$\sigma$</span> of <span class="math-container">$\Gamma$</span> such that</p>
<p><span class="math-container">\begin{equation}\tag{2}\sigma(a|t,\theta):= \frac{\sum_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)\Pi_{j=1}\beta_{j}(a_j|t_j)}{\pi(t|\theta)}\end{equation}</span></p>
<p><span class="math-container">$\textbf{Theorem 1:}$</span>. A decision rule <span class="math-container">$\sigma$</span> is a Bayes correlated equilibrium of <span class="math-container">$\Gamma$</span> if and only if, for some expansion <span class="math-container">$S^{e}$</span> of <span class="math-container">$S$</span>, there is a Bayes Nash equilibrium of <span class="math-container">$\Gamma^{e}$</span> that induces <span class="math-container">$\sigma$</span>.</p>
<p>For the proof is needed one more definition about the BNE.</p>
<p>My questions are the following</p>
<ol>
<li>Why the strategy profile <span class="math-container">$\beta$</span> of <span class="math-container">$\Gamma^{e}$</span> would induce a decision rule <span class="math-container">$\sigma$</span> for <span class="math-container">$\Gamma$</span>?</li>
<li>Could someone give a little more intuition about equation <span class="math-container">$(2)$</span>? In the proof of theorem <span class="math-container">$1$</span> in the second part of the proof it is written that</li>
</ol>
<p><span class="math-container">$$\pi(t|\theta)\sigma(a|t,\theta)=\sum_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)\Pi_{j=1}\beta_{j}(a_j|t_j)$$</span></p>
<p>which comes from equation <span class="math-container">$(2)$</span>. Could someone explain how this equality is defined and make a simple calclulation of this formula in the case where <span class="math-container">$I=2$</span>, <span class="math-container">$T=\{t_1,t_2\}$</span> and <span class="math-container">$T=\{t_1^{'},t_2^{'}\}$</span>. It confuses me.</p>

CC BY-SA 4.0
2023-05-20T14:58:17.373Z
You can use the prior, signal structures, and strategies in the extended game to calculate the joint probability distribution on $\Theta\times T\times T'\times A$. From this, you obtain the decision rule by letting $\sigma(\theta,t)$ be the conditional distribution on $A$ if $(\theta,t)$ has positive probability, and specifying $\sigma(\theta,t)$ to be arbitrary distribution if $(\theta,t)$ has probability zero. All details are in messy calculations of conditional probabilities.
Michael Greinecker
CC BY-SA 4.0
2023-05-20T22:37:54.097Z
@MichaelGreinecker didn't understand a word...but my problem I suppose...how is the decision rule is $\sigma : T\times\Theta\to \Delta(A)$ calculated conditional on $A$? what is its meaning? Given that we know the strategy what state and signal is someone supposed to privately observe? Anyway, I thought it was something simple that I do not understand, but is omitted due to simplicity, not due to complexity...how did they pass the journal referee? :P
Oliver Queen
CC BY-SA 4.0
2023-05-21T07:04:29.923Z
It is not conditional on $A$. It is conditional on $(\theta,t)$. If the probability of $E$ is positive, the conditional probability of $F$ given $E$ is given by $P(E\cap F)/P(E)$.
Michael Greinecker
<microeconomics><mathematical-economics><game-theory><bayesian-game><information>
Two questions in Bergemann and Morris (2016) - BCE and Comparison of Information Structures in games
Oliver Queen
# Two questions in Bergemann and Morris (2016) - BCE and Comparison of Information Structures in games

(Asked by: username_0 on 2023-05-20)

Based on [Begemann and Morris (2016)](https://onlinelibrary.wiley.com/doi/epdf/10.3982/TE1808) we have the following definition about a standard game of incomplete information

A standard game $\Gamma = <G,S>$ of incomplete information consists of a set of $I$ players, $1,2,\cdots,I$ and we write $i$ for a typical player. There is a finite set of states, $\Theta$, and we write $\theta$ for a typical state, where

$\textbf{Definition 1: Basic Game}$ A basic game $G=\left((A\_i,u\_i)\_{i=1}^I,\psi\right)$ consists of

1. a finite set of actions $A\_i$ (where $A=A\_1\times\dots\times A\_I$) for every player $i$ and a utility function
   $$u\_i:A\times\Theta\to\mathbb{R}$$
2. and a full support common prior $\psi\in\Delta(\Theta)\_{++}$

$\textbf{Definition 2: Information structure}$ An information structure $S$ consists of

1. a finite set of signals (or types) $T\_i$ for every player $i$ (where $T=T\_1\times\dots\times T\_I$)
2. and a signal probability distribution $\pi:\Theta\to\Delta(T)$

And this completes the game definition.

A decision rule in the incomplete information game $\Gamma = <G,S>$ is a mapping $\sigma$ such that:

\begin{equation}\tag{1}\sigma:T\times\Theta\to\Delta(A)\end{equation}

To understand the game, $\sigma$ is the strategy of an omniscient mediator who first observes the realization of $\theta\in\Theta$ chosen according to $\psi$ and the realization of $t\in T$ chosen according to $\pi(\cdot|\theta)$, and then picks an action profile $a\in A$. From the players perspective in order to follow the action recommendation, it would have to be the case that the recommended action $a\_i$ was always proffered to any other alternative $a\_i{'}$ conditional on the signal $t\_i$ that player $i$ had received and his knowledge of the recommended action $a\_i$. This is reflected in the obedience condition

$\textbf{Definition 3: Obedience}$ A decision rule $\sigma$ is obedient for $\Gamma$, if for every $i\in I$, $t\_i\in T\_i$ and $a\_i\in A\_i$ we have

\begin{equation}
\sum\_{a\_{-i},t\_{-i},\theta}\psi(\theta)\pi((t\_i,t\_{-i})|\theta)\sigma((a\_{i},a\_{-i})|(t\_i,t\_{-i}),\theta)u\_i((a\_{i},a\_{-i}),\theta) \\
\geq \sum\_{a\_{-i},t\_{-i},\theta}\psi(\theta)\pi((t\_i,t\_{-i})|\theta)\sigma((a\_{i},a\_{-i})|(t\_i,t\_{-i}),\theta)u\_i((a\_{i}^{'},a\_{-i}),\theta)
\end{equation}

for all $a\_{i}^{'}\in A\_i$.

$\textbf{Definition 4: Bayesian Correlated Equilibrium}$ A decision rule $\sigma$ is a Bayesian correlated equilibrium (BCE) of $\Gamma$, if it is obedient for $\Gamma$.

Suppose that a game $\Gamma^{e}=(G, S^{e} )$ is a new game where with respect to the original game $\Gamma$ where the information structure $S^{e}=(T, \pi^{e})$ is an expansion of the original information structure $S$. An expansion of an information structure $S$ means that it can be written as an combination of $S$ with another information structure $S^{'}$, where the combined information structure is obtained by forming a product space of the signals $T\_i^{e}= T\_i\times T\_i^{'}$ for every $i$ and the signal distribution $\pi^{e}:\Theta\to\Delta(T\times T^{'})$ that preserves the marginal distribution of its constituent information structures.

$\textbf{Definition 5: Combination}$ The information structure $S^{e}=(T, \pi^{e})$ is a combination of the information structures $S=(T, \pi)$ and $S^{'}=(T, \pi^{'})$ if

$$T\_i^{e}= T\_i\times T\_i^{'},\quad\text{for every $i$}$$

and

$$\sum\_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)=\pi(t|\theta),\quad\text{for every $t\in T$ and $\theta\in \Theta$}$$

$$\sum\_{t\in T}\pi^{e}(t,t^{'}|\theta)=\pi^{'}(t^{'}|\theta),\quad\text{for every $t^{'}\in T^{'}$ and $\theta\in \Theta$}$$

$\textbf{Definition 6: Expansion}$ An information structure $S^{e}$ is an expansion of an information structure $S$ if $S^{e}$ is a combination of $S$ and some other information structure $S^{'}$.

$\textbf{Definition 7: Bayesian Nash Equilibrium}$ A strategy profile $\beta$ is a Bayesian Nash Equilibrium of $\Gamma$, if for each $i\in I$, $t\_i\in T\_i$, and $a\_i\in A\_i$ the (behavioral) strategy $\beta\_i:T\_i\to\Delta(A\_i)$ with $\beta\_i(a\_i|t\_i)>0$ we have that

\begin{equation}
\sum\_{a\_{-i},t\_{-i},\theta}\psi(\theta)\pi((t\_i,t\_{-i})|\theta)\left(\Pi\_{j\neq i}\beta\_{j}(a\_j|t\_j)\right)u\_i((a\_{i},a\_{-i}),\theta) \\
\geq \sum\_{a\_{-i},t\_{-i},\theta}\psi(\theta)\pi((t\_i,t\_{-i})|\theta)\left(\Pi\_{j\neq i}\beta\_{j}(a\_j|t\_j)\right)u\_i((a\_{i}^{'},a\_{-i}),\theta)
\end{equation}

for all $a\_{i}^{'}\in A\_i$.

All these are needed to understand the model of Bergemman and Morris. My question is about theorem $1$ in the paper and a statistic that uses to connect BCE with the Bayesian Nash Equilibrium. The statistic is the following:

Suppose that the profile $\beta$ was played in $\Gamma^{e}$ where the information structure $S^{e}$ is the expansion of the information structure $S$ in the game $\Gamma$ and it is the combination of $S$ and $S^{'}$ Now if the analyst did not observe the signals of the combined information structure $S^{e}$, but only the signals of $S$, then the behaviour under the strategy profile $\beta$ would induce a decision rule for $\Gamma$ (Why?). Formally the strategy profile $\beta$ of $\Gamma^{e}$ induces the decision rule $\sigma$ of $\Gamma$ such that

\begin{equation}\tag{2}\sigma(a|t,\theta):= \frac{\sum\_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)\Pi\_{j=1}\beta\_{j}(a\_j|t\_j)}{\pi(t|\theta)}\end{equation}

$\textbf{Theorem 1:}$. A decision rule $\sigma$ is a Bayes correlated equilibrium of $\Gamma$ if and only if, for some expansion $S^{e}$ of $S$, there is a Bayes Nash equilibrium of $\Gamma^{e}$ that induces $\sigma$.

For the proof is needed one more definition about the BNE.

My questions are the following

1. Why the strategy profile $\beta$ of $\Gamma^{e}$ would induce a decision rule $\sigma$ for $\Gamma$?
2. Could someone give a little more intuition about equation $(2)$? In the proof of theorem $1$ in the second part of the proof it is written that

$$\pi(t|\theta)\sigma(a|t,\theta)=\sum\_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)\Pi\_{j=1}\beta\_{j}(a\_j|t\_j)$$

which comes from equation $(2)$. Could someone explain how this equality is defined and make a simple calclulation of this formula in the case where $I=2$, $T=\{t\_1,t\_2\}$ and $T=\{t\_1^{'},t\_2^{'}\}$. It confuses me.

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game, information

> **Comments:**
> **username_1**: You can use the prior, signal structures, and strategies in the extended game to calculate the joint probability distribution on $\Theta\times T\times T'\times A$. From this, you obtain the decision rule by letting $\sigma(\theta,t)$ be the conditional distribution on $A$ if $(\theta,t)$ has positive probability, and specifying $\sigma(\theta,t)$ to be arbitrary distribution if $(\theta,t)$ has probability zero. All details are in messy calculations of conditional probabilities.  
> **username_0**: @username_1 didn't understand a word...but my problem I suppose...how is the decision rule is $\sigma : T\times\Theta\to \Delta(A)$ calculated conditional on $A$? what is its meaning? Given that we know the strategy what state and signal is someone supposed to privately observe? Anyway, I thought it was something simple that I do not understand, but is omitted due to simplicity, not due to complexity...how did they pass the journal referee? :P  
> **username_1**: It is not conditional on $A$. It is conditional on $(\theta,t)$. If the probability of $E$ is positive, the conditional probability of $F$ given $E$ is given by $P(E\cap F)/P(E)$.  

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## 0 answers

