How does this reporting correspondence is redefined?

Question

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 $R_i^t:\underbrace{...}_{?}\to\underbrace{...}_{?}$

Answer 1

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 $t$ ($f_i^t\in F_i^t$).