# Two questions in Bergemann and Morris (2016) - BCE and Comparison of Information Structures in games

Based on Begemann and Morris (2016) we have the following definition about a standard game of incomplete information

A standard game $$\Gamma = $$ 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 $$\Gamma = $$ is a mapping $$\sigma$$ such that:

$$$$\tag{1}\sigma:T\times\Theta\to\Delta(A)$$$$

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

$$$$\sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\sigma((a_{i},a_{-i})|(t_i,t_{-i}),\theta)u_i((a_{i},a_{-i}),\theta) \\ \geq \sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\sigma((a_{i},a_{-i})|(t_i,t_{-i}),\theta)u_i((a_{i}^{'},a_{-i}),\theta)$$$$

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

$$$$\sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\left(\Pi_{j\neq i}\beta_{j}(a_j|t_j)\right)u_i((a_{i},a_{-i}),\theta) \\ \geq \sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi((t_i,t_{-i})|\theta)\left(\Pi_{j\neq i}\beta_{j}(a_j|t_j)\right)u_i((a_{i}^{'},a_{-i}),\theta)$$$$

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

$$$$\tag{2}\sigma(a|t,\theta):= \frac{\sum_{t^{'}\in T^{'}}\pi^{e}(t,t^{'}|\theta)\Pi_{j=1}\beta_{j}(a_j|t_j)}{\pi(t|\theta)}$$$$

$$\textbf{Theorem 1:}$$. 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.

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