<p>Without this being a clear answer onge thought could be to define the reporting correspondence as it follows
<span class="math-container">$$R_i^t(s_i^t|h^{t-1})=\{s_i^t\in S_i^t\quad \text{where player $i$ reports truthfully her type that is $s_i$ at stage $t$}\}$$</span></p>
<p>In this way the message space resembles some kind of a report space where every player confirms her index <span class="math-container">$i$</span> and her private time <span class="math-container">$s_i$</span>. However the update that you say it's kind of fellfield prophecy and let me be clear on this. When a game becomes dynamic then at any new move the previous ones are incorporated in the <span class="math-container">$\sigma-$</span> algebra of every player that encodes her information. So <span class="math-container">$h^t$</span> contains some information but not all of it. You may possibly need something more for the information set beyond the sequence <span class="math-container">$h=(a_k)_{k=1}^t$</span> or <span class="math-container">$h^t=a^t$</span> which refers to the profile of actions until stage <span class="math-container">$t$</span>. You can assume that every player may observe the actions of all the others at the end of every stage game so she does know the whole profile of actions which is true with the way that you defined history, but you will need an extended history say <span class="math-container">$f_i^t=\{(h^t,s_i^t)\quad|\quad h^t\in H^t, s_i^t\in S_i^t\}$</span>. So every player until the stage <span class="math-container">$t$</span> will know <span class="math-container">$f_i^{t-1}=(h^{t-1},s_i^{t-1})$</span>, but she will have no extra information about <span class="math-container">$s_{-i}^{t-1}$</span> and of course <span class="math-container">$s_{-i}^{t}$</span>. So the reporting correspondence could be written as it follows</p>
<p><span class="math-container">$$R_i^t(s_i^t|f_i^{t-1})=\{s_i^t\in S_i^t\quad \text{where player $i$ reports truthfully her type that is $s_i$ at stage $t$}\}\tag{1}$$</span>, namely</p>
<p><span class="math-container">$$R_i^t:F_i^t\to I\times S_i^t$$</span> such that <span class="math-container">$R_i^t(s_i^t|f_i^t)=(i,s_i^t)$</span> where the player repots her index <span class="math-container">$i$</span> and her signal <span class="math-container">$s_i$</span> at stage <span class="math-container">$t$</span> (<span class="math-container">$f_i^t\in F_i^t$</span>).</p>

CC BY-SA 4.0
2022-01-29T13:52:25.040Z
Quite good answer to model the reporting correspondence. Nice idea. Thanks!
Hunger Learn
CC BY-SA 4.0
2022-01-29T09:38:00.007Z
2022-01-29T09:49:55.303Z
2022-01-29T09:49:55.303Z
Hunger Learn
<p>Let <span class="math-container">$\mathcal{R}_i$</span> be a non-empty, finite set and define the reporting correspondence <span class="math-container">$R_i:S→2^{\mathcal{R}_i}-\{\emptyset\}$</span> to be a mapping from player i’s type space to the collection of subsets of <span class="math-container">$\mathcal{R}_i$</span>. An element <span class="math-container">$s\in\mathcal{R}_i$</span> is called a type dependent message and <span class="math-container">$R_i(s_i)$</span> is the set of type-dependent messages available to type <span class="math-container">$s_i$</span> of player <span class="math-container">$i$</span>. Type-dependent messages certify a player’s statement about his type. For example, if <span class="math-container">$L\subset S_i$</span> is the set of types of player <span class="math-container">$i$</span> who can send the message <span class="math-container">$s\in\mathcal{R}_i$</span>, then <span class="math-container">$s$</span> certifies a statement of the type “my type is in <span class="math-container">$L$</span>”. The set <span class="math-container">$L$</span> is therefore called a certifiable event.</p>
<p>I want to define the reporting correspondence <span class="math-container">$R_i$</span> differently. Suppose that the game is repeated and at the end of every stage <span class="math-container">$t$</span> of the game and before a new one starts, every player observes the history of the game that is reached until that point. We denote the history of the game as <span class="math-container">$h$</span>. It is the sequence of the past profile of strategies played by the players, namely <span class="math-container">$h=(a_k)_{k=1}^T$</span>, where <span class="math-container">$a_k$</span> is the profile of strategies played in stage <span class="math-container">$k$</span>. At the end of every stage <span class="math-container">$t$</span> of the game and before a new one starts, every player also gets an update for his type <span class="math-container">$s_i$</span> with respect to the type of the other players <span class="math-container">$s_{-i}$</span> denoted as <span class="math-container">$s_i^t$</span>. So, given the history <span class="math-container">$h^{t-1}$</span> and the updated type <span class="math-container">$s_i^t$</span> how is the correspondence <span class="math-container">$R_i$</span> redifined?</p>
<p>I ask for something simple. The reporting correspondence is defined as  <span class="math-container">$R_i:S→2^{\mathcal{R}_i}-\{\emptyset\}$</span>. Whet the game takes the dynamic dimension due to the stages, then <span class="math-container">$R_i^t:\underbrace{...}_{?}\to\underbrace{...}_{?}$</span></p>

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2022-01-29T02:23:15.317Z
Hi! You wrote nothing about the message space in the new situation. Seems like nothing really changed about the message space, and therefore certifiability is the same? It is not clear to me what you are asking.
Giskard
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2022-01-29T07:12:08.933Z
@Giskard I ask for something simple. The reporting correspondence is defined as  $R_i:S→2^{\mathcal{R}_i}-\{\emptyset\}$. Whet the game takes the dynamic dimension due to the stages, then $R_i^t:\underbrace{...}_{?}\to\underbrace{...}_{?}$
Hunger Learn
CC BY-SA 4.0
2022-01-29T07:36:49.673Z
So there is a message sent in *each* stage $t$, and the sets $S$ only depend on the type, not the stage? And you want the reporting correspondence to describe someone's possible types, provided they have so far sent messages $s_0,s_1,\dots,s_t$?
Giskard
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2022-01-29T07:44:03.837Z
@Giskard ok! Based on what you wrote how could this $R_i$ function change?
Hunger Learn
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2022-01-29T07:44:38.373Z
Sir, you are asking the question, it is your setting...or is this a problem for a university course?
Giskard
CC BY-SA 4.0
2022-01-29T07:45:15.553Z
no it is not a problem for some course. It is my personal question
Hunger Learn
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2022-01-29T07:47:42.587Z
Then please include the necessary details in the question. What do you mean by "*update*"? Does a player's type change at all? Is what I described in the comment what you mean by $R_i$? (It seems to me that I was off, and $R_i$ simply describes the messages available to a player of player $i$'s type.) Is it true that there is a message sent in each stage $t$, and the sets $S$ only directly depend on the type, not the stage? Please [edit](https://economics.stackexchange.com/posts/50247/edit) the additional information into your question.
Giskard
CC BY-SA 4.0
2022-01-29T09:38:39.763Z
@Giskard check one possible answer
Hunger Learn
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2022-01-29T09:38:54.013Z
@KingOdysseus check one possible answer
Hunger Learn
<mathematical-economics><game-theory><bayesian-game><signaling><repeated-games>
How does this reporting correspondence is redefined?
Hunger Learn
# How does this reporting correspondence is redefined?

(Asked by: username_0 on 2022-01-29)

Let $\mathcal{R}\_i$ be a non-empty, finite set and define the reporting correspondence $R\_i:S→2^{\mathcal{R}\_i}-\{\emptyset\}$ to be a mapping from player i’s type space to the collection of subsets of $\mathcal{R}\_i$. An element $s\in\mathcal{R}\_i$ is called a type dependent message and $R\_i(s\_i)$ is the set of type-dependent messages available to type $s\_i$ of player $i$. Type-dependent messages certify a player’s statement about his type. For example, if $L\subset S\_i$ is the set of types of player $i$ who can send the message $s\in\mathcal{R}\_i$, then $s$ certifies a statement of the type “my type is in $L$”. The set $L$ is therefore called a certifiable event.

I want to define the reporting correspondence $R\_i$ differently. Suppose that the game is repeated and at the end of every stage $t$ of the game and before a new one starts, every player observes the history of the game that is reached until that point. We denote the history of the game as $h$. It is the sequence of the past profile of strategies played by the players, namely $h=(a\_k)\_{k=1}^T$, where $a\_k$ is the profile of strategies played in stage $k$. At the end of every stage $t$ of the game and before a new one starts, every player also gets an update for his type $s\_i$ with respect to the type of the other players $s\_{-i}$ denoted as $s\_i^t$. So, given the history $h^{t-1}$ and the updated type $s\_i^t$ how is the correspondence $R\_i$ redifined?

I ask for something simple. The reporting correspondence is defined as $R\_i:S→2^{\mathcal{R}\_i}-\{\emptyset\}$. Whet the game takes the dynamic dimension due to the stages, then $R\_i^t:\underbrace{...}\_{?}\to\underbrace{...}\_{?}$

**Tags:** mathematical-economics, game-theory, bayesian-game, signaling, repeated-games

> **Comments:**
> **username_1**: Hi! You wrote nothing about the message space in the new situation. Seems like nothing really changed about the message space, and therefore certifiability is the same? It is not clear to me what you are asking.  
> **username_0**: @username_1 I ask for something simple. The reporting correspondence is defined as  $R_i:S→2^{\mathcal{R}_i}-\{\emptyset\}$. Whet the game takes the dynamic dimension due to the stages, then $R_i^t:\underbrace{...}_{?}\to\underbrace{...}_{?}$  
> **username_1**: So there is a message sent in *each* stage $t$, and the sets $S$ only depend on the type, not the stage? And you want the reporting correspondence to describe someone's possible types, provided they have so far sent messages $s_0,s_1,\dots,s_t$?  
> **username_0**: @username_1 ok! Based on what you wrote how could this $R_i$ function change?  
> **username_1**: Sir, you are asking the question, it is your setting...or is this a problem for a university course?  
> **username_0**: no it is not a problem for some course. It is my personal question  
> **username_1**: Then please include the necessary details in the question. What do you mean by "*update*"? Does a player's type change at all? Is what I described in the comment what you mean by $R_i$? (It seems to me that I was off, and $R_i$ simply describes the messages available to a player of player $i$'s type.) Is it true that there is a message sent in each stage $t$, and the sets $S$ only directly depend on the type, not the stage? Please [edit](https://economics.stackexchange.com/posts/50247/edit) the additional information into your question.  
> **username_0**: @username_1 check one possible answer  
> **username_0**: @KingOdysseus check one possible answer  

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## 1 answers

### Answer by username_0 on 2022-01-29. Score: 1 | Accepted answer

Without this being a clear answer onge thought could be to define the reporting correspondence as it follows
$$R\_i^t(s\_i^t|h^{t-1})=\{s\_i^t\in S\_i^t\quad \text{where player $i$ reports truthfully her type that is $s\_i$ at stage $t$}\}$$

In this way the message space resembles some kind of a report space where every player confirms her index $i$ and her private time $s\_i$. However the update that you say it's kind of fellfield prophecy and let me be clear on this. When a game becomes dynamic then at any new move the previous ones are incorporated in the $\sigma-$ algebra of every player that encodes her information. So $h^t$ contains some information but not all of it. You may possibly need something more for the information set beyond the sequence $h=(a\_k)\_{k=1}^t$ or $h^t=a^t$ which refers to the profile of actions until stage $t$. You can assume that every player may observe the actions of all the others at the end of every stage game so she does know the whole profile of actions which is true with the way that you defined history, but you will need an extended history say $f\_i^t=\{(h^t,s\_i^t)\quad|\quad h^t\in H^t, s\_i^t\in S\_i^t\}$. So every player until the stage $t$ will know $f\_i^{t-1}=(h^{t-1},s\_i^{t-1})$, but she will have no extra information about $s\_{-i}^{t-1}$ and of course $s\_{-i}^{t}$. So the reporting correspondence could be written as it follows

$$R\_i^t(s\_i^t|f\_i^{t-1})=\{s\_i^t\in S\_i^t\quad \text{where player $i$ reports truthfully her type that is $s\_i$ at stage $t$}\}\tag{1}$$, namely

$$R\_i^t:F\_i^t\to I\times S\_i^t$$ such that $R\_i^t(s\_i^t|f\_i^t)=(i,s\_i^t)$ where the player repots her index $i$ and her signal $s\_i$ at stage $t$ ($f\_i^t\in F\_i^t$).

> **Comments:**
> **username_0**: Quite good answer to model the reporting correspondence. Nice idea. Thanks!  

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