<p>The following are from the model of <a href="http://publications.ut-capitole.fr/32358/1/renault2018_referenceworkentry_repeatedgameswithincompleteinf.pdf" rel="nofollow noreferrer">Jérôme Renault (Repeated Games of incomplete infomration)</a>.</p>
<p>The preliminaries of the model</p>
<p>Formally, a repeated game with incomplete information is given by the following data. There is a set of players <span class="math-container">$N$</span> and a set of states <span class="math-container">$K$</span>. Each player <span class="math-container">$i\in N$</span> has a set of actions <span class="math-container">$A^i$</span> and a set of signals <span class="math-container">$U^i$</span>, and we denote by <span class="math-container">$A=\Pi_{i\in N}A^i$</span> the set of action profiles and by <span class="math-container">$U=\Pi_{i\in N}U^i$</span> the set of signal profiles. Every player <span class="math-container">$i$</span> has a payoff function <span class="math-container">$g^i:K\times A\to\mathbb{R}$</span>. There is a signalling function <span class="math-container">$q:K\times A\to \Delta(S)$</span> and an initial probability <span class="math-container">$\pi\in\Delta(K\times U)$</span>. In what follows, we will always assume the sets of players, states, actions and signals to be non-empty and finite.</p>
<p>A repeated game with incomplete information can thus be denoted by <span class="math-container">$\Gamma =
(N,K, (A^i, U^i, g^i)_{i\in N}, q, \pi)$</span>. The progress of the game is the following.</p>
<ul>
<li>Initially, an element <span class="math-container">$(k, (u_0^i)_{i\in N})$</span> is selected according to <span class="math-container">$\pi$</span>: <span class="math-container">$k$</span> is the realized state of nature and will remain fixed, and each player <span class="math-container">$i$</span> learns <span class="math-container">$u_ 0^i$</span> (and nothing
more than <span class="math-container">$u_0^i$</span>).</li>
<li>At each integer stage <span class="math-container">$t\geq 1$</span>, simultaneously every player <span class="math-container">$i$</span> chooses an action <span class="math-container">$a_t^i$</span> in <span class="math-container">$A^i$</span>, and we denote by <span class="math-container">$a_t = (a_t^i)_{i\in N}$</span> the action profile played at stage <span class="math-container">$t$</span>. The stage payoff of a player <span class="math-container">$i$</span> is then given by <span class="math-container">$g^i(k, a^t)$</span>. A signal profile <span class="math-container">$(u_t^i)_{i\in N}$</span> is selected according to <span class="math-container">$q(k, a^t)$</span>, and each player <span class="math-container">$i$</span> learns <span class="math-container">$u_t^i$</span> (and nothing more than <span class="math-container">$u_t^i$</span>) before proceeding to the next stage.</li>
</ul>
<p>My questions are the following</p>
<p><span class="math-container">$\textbf{Question 1:}$</span> Every players knows her signal according to <span class="math-container">$\pi$</span> when the game begins, but after an action is played and the game proceeds the state dependent function <span class="math-container">$\pi$</span> is no longer used about the new signals that emerge. Specifically, <span class="math-container">$q$</span> is the signaling function from that point onward. Are these two functions <span class="math-container">$q$</span> and <span class="math-container">$\pi$</span> connected? If the player wants to update her beliefs which one of them is known to her?</p>
<p><span class="math-container">$\textbf{Question 2:}$</span> The author refers to the case of incomplete information, however the state remains fixed during the game and it is <span class="math-container">$k\in K$</span> however each player remains uncertain about <span class="math-container">$k$</span>. Should the author use another function about the common prior beliefs of the players based on the state of the world or the assumption that <span class="math-container">$k$</span> remains fixed makes him omit such a function? How should that be restated if the state of the world was not fixed and say that we have two states <span class="math-container">$G$</span> for the good and <span class="math-container">$B$</span> for the bad. In every repeated game we wouldn't be sure in which of them we are, so is there anyting that could change the assumptions of the game so far?</p>
<p><span class="math-container">$\textbf{Question 3:}$</span> Does <span class="math-container">$q$</span> refer to the chance function as it is stated in the book of <a href="https://d1wqtxts1xzle7.cloudfront.net/36566152/A_course_in_game_theory-with-cover-page-v2.pdf?Expires=1638265109&amp;Signature=DcJB9f710b94ktagGlDdpfwdP3CZHQKZsc6GPV5cTImkK8MDyzSd1di177IlrWaVidNDNeEDuNhp%7Eqxa7iybKNF-bxGMHWV9Y0b9yFDxy4jrVT2R3k%7EQOuyUt1SBQXUeDISBMrmtKEiB%7EG-6qqF2ZsKsdlq5zM4ZvMKUwc-9OgssANoDw2z7pTs7ZvGYiy9RgOF5mtWCYXceUOcgQQV3NfCmo4wE5rkzTg1HQn60j6afag6IBcaCFexx6eWewHGJHYj6b8WiX0g-LEHEoHB%7EZW1dQ174ljdtT8a3c5H9EsyByEQJBQ7%7EAwGjZuvhZcktphLIblUSS3Fu7GjTy-HdeQ__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA" rel="nofollow noreferrer">Osborne and Rubinstein (1994)</a>(see in chapeter <span class="math-container">$11$</span> page <span class="math-container">$216$</span>. An analogous function is denoted there as <span class="math-container">$f_c(a|h)$</span> and refers to the probability that <span class="math-container">$a$</span> occurs after the history <span class="math-container">$h$</span>.</p>

CC BY-SA 4.0
2021-11-30T09:42:53.523Z
One thing that I can tell for sure is that $q$ refers to chance moves in the sense that, if $k$ is the state of the world and this remains fixed until the repeated game ends, the signal function will update $u_ i^t$ for every $i$ from stage to stage. $q$ gives her the available information about which non terminal history the game has reached depending on the state $k$ and this is reflected to $u_i$. In case $k$ was not fixed and more states where introduced, then this may change after a move of chance. However in any case the signal may remain the same or change in any case according to $q$
Nav89
CC BY-SA 4.0
2021-11-30T09:45:12.987Z
By introducing more states it gives another dimension to the problem, making it a little more complex, but the player is only aware of $\pi$ about the signals. In other words, whatever update is going to occur based on $q$, she only uses $\pi$ to update her beliefs. So yes, It is the chance function.
Nav89
CC BY-SA 4.0
2021-11-30T09:49:53.663Z
Finally, by keeping $k$ fixed they do not have to use a common prior beleif function in order to find the expected payoff. You would be in the need of such a function if they where two or more states of the world realized. They only find the expected utility with respect to $\pi$ and $\sigma$ if you check the paper. In the case of more than one states you would need a third function say $\phi$ about the states of the world.
Nav89
<microeconomics><mathematical-economics><game-theory><bayesian-game><dynamic-games>
Repeated Games of Incomplete information set up and questions
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# Repeated Games of Incomplete information set up and questions

(Asked by: username_0 on 2021-11-30)

The following are from the model of [Jérôme Renault (Repeated Games of incomplete infomration)](http://publications.ut-capitole.fr/32358/1/renault2018_referenceworkentry_repeatedgameswithincompleteinf.pdf).

The preliminaries of the model

Formally, a repeated game with incomplete information is given by the following data. There is a set of players $N$ and a set of states $K$. Each player $i\in N$ has a set of actions $A^i$ and a set of signals $U^i$, and we denote by $A=\Pi\_{i\in N}A^i$ the set of action profiles and by $U=\Pi\_{i\in N}U^i$ the set of signal profiles. Every player $i$ has a payoff function $g^i:K\times A\to\mathbb{R}$. There is a signalling function $q:K\times A\to \Delta(S)$ and an initial probability $\pi\in\Delta(K\times U)$. In what follows, we will always assume the sets of players, states, actions and signals to be non-empty and finite.

A repeated game with incomplete information can thus be denoted by $\Gamma =
(N,K, (A^i, U^i, g^i)\_{i\in N}, q, \pi)$. The progress of the game is the following.

* Initially, an element $(k, (u\_0^i)\_{i\in N})$ is selected according to $\pi$: $k$ is the realized state of nature and will remain fixed, and each player $i$ learns $u\_ 0^i$ (and nothing
  more than $u\_0^i$).
* At each integer stage $t\geq 1$, simultaneously every player $i$ chooses an action $a\_t^i$ in $A^i$, and we denote by $a\_t = (a\_t^i)\_{i\in N}$ the action profile played at stage $t$. The stage payoff of a player $i$ is then given by $g^i(k, a^t)$. A signal profile $(u\_t^i)\_{i\in N}$ is selected according to $q(k, a^t)$, and each player $i$ learns $u\_t^i$ (and nothing more than $u\_t^i$) before proceeding to the next stage.

My questions are the following

$\textbf{Question 1:}$ Every players knows her signal according to $\pi$ when the game begins, but after an action is played and the game proceeds the state dependent function $\pi$ is no longer used about the new signals that emerge. Specifically, $q$ is the signaling function from that point onward. Are these two functions $q$ and $\pi$ connected? If the player wants to update her beliefs which one of them is known to her?

$\textbf{Question 2:}$ The author refers to the case of incomplete information, however the state remains fixed during the game and it is $k\in K$ however each player remains uncertain about $k$. Should the author use another function about the common prior beliefs of the players based on the state of the world or the assumption that $k$ remains fixed makes him omit such a function? How should that be restated if the state of the world was not fixed and say that we have two states $G$ for the good and $B$ for the bad. In every repeated game we wouldn't be sure in which of them we are, so is there anyting that could change the assumptions of the game so far?

$\textbf{Question 3:}$ Does $q$ refer to the chance function as it is stated in the book of [Osborne and Rubinstein (1994)](https://d1wqtxts1xzle7.cloudfront.net/36566152/A_course_in_game_theory-with-cover-page-v2.pdf?Expires=1638265109&Signature=DcJB9f710b94ktagGlDdpfwdP3CZHQKZsc6GPV5cTImkK8MDyzSd1di177IlrWaVidNDNeEDuNhp%7Eqxa7iybKNF-bxGMHWV9Y0b9yFDxy4jrVT2R3k%7EQOuyUt1SBQXUeDISBMrmtKEiB%7EG-6qqF2ZsKsdlq5zM4ZvMKUwc-9OgssANoDw2z7pTs7ZvGYiy9RgOF5mtWCYXceUOcgQQV3NfCmo4wE5rkzTg1HQn60j6afag6IBcaCFexx6eWewHGJHYj6b8WiX0g-LEHEoHB%7EZW1dQ174ljdtT8a3c5H9EsyByEQJBQ7%7EAwGjZuvhZcktphLIblUSS3Fu7GjTy-HdeQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA)(see in chapeter $11$ page $216$. An analogous function is denoted there as $f\_c(a|h)$ and refers to the probability that $a$ occurs after the history $h$.

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game, dynamic-games

> **Comments:**
> **username_1**: One thing that I can tell for sure is that $q$ refers to chance moves in the sense that, if $k$ is the state of the world and this remains fixed until the repeated game ends, the signal function will update $u_ i^t$ for every $i$ from stage to stage. $q$ gives her the available information about which non terminal history the game has reached depending on the state $k$ and this is reflected to $u_i$. In case $k$ was not fixed and more states where introduced, then this may change after a move of chance. However in any case the signal may remain the same or change in any case according to $q$  
> **username_1**: By introducing more states it gives another dimension to the problem, making it a little more complex, but the player is only aware of $\pi$ about the signals. In other words, whatever update is going to occur based on $q$, she only uses $\pi$ to update her beliefs. So yes, It is the chance function.  
> **username_1**: Finally, by keeping $k$ fixed they do not have to use a common prior beleif function in order to find the expected payoff. You would be in the need of such a function if they where two or more states of the world realized. They only find the expected utility with respect to $\pi$ and $\sigma$ if you check the paper. In the case of more than one states you would need a third function say $\phi$ about the states of the world.  

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## 0 answers

