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Based on Begemann and Morris (2016) 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.

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  • $\begingroup$ 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. $\endgroup$ May 20, 2023 at 14:58
  • $\begingroup$ @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 $\endgroup$ May 20, 2023 at 22:37
  • $\begingroup$ 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 $P(E\cap F)/P(E)$. $\endgroup$ May 21, 2023 at 7:04

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