<p>Suppose that we have tow states of the world <span class="math-container">$\omega_1$</span> and <span class="math-container">$\omega_2$</span>, where <span class="math-container">$p(\omega_1)=p(\omega_1)=1/2$</span> and there are three different signals, <span class="math-container">$s_H,s_M,s_L$</span> that are equally likely to occur in everey state of the world that is <span class="math-container">$p(s_H|\omega_1)=p(s_M|\omega_1)=p(s_L|\omega_1)=1/3$</span> and <span class="math-container">$p(s_H|\omega_2)=p(s_M|\omega_2)=p(s_L|\omega_2)=1/3$</span>. Let two players exist in our world, say <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> and the following hold:</p>
<ul>
<li>In state <span class="math-container">$\omega_1$</span>, player <span class="math-container">$1$</span> distinguishes the signal <span class="math-container">$s_H$</span> but not the signals <span class="math-container">$s_M,s_L$</span> and for simplicity lets say that he knows that if he dodes not observe the signal <span class="math-container">$s_H$</span> he will know that his signal is either <span class="math-container">$s_M$</span> or <span class="math-container">$s_L$</span> with equal probability. For <span class="math-container">$P_2$</span> it holds that she distinguishes the signal <span class="math-container">$s_L$</span> but not the signals <span class="math-container">$s_H,s_M$</span> and if she will not take <span class="math-container">$s_L$</span> she will take with equal probability either <span class="math-container">$s_H$</span> or <span class="math-container">$s_M$</span>.</li>
<li>In state <span class="math-container">$\omega_2$</span>, <span class="math-container">$P_1$</span> distinguishes <span class="math-container">$s_L$</span> but not <span class="math-container">$s_H,s_M$</span> and <span class="math-container">$P_2$</span> <span class="math-container">$s_H$</span> but not the signals <span class="math-container">$s_M,s_L$</span>. If the signal is not <span class="math-container">$s_L$</span>, then <span class="math-container">$P_1$</span> knows that his signal is <span class="math-container">$s_H$</span> or <span class="math-container">$s_M$</span> with equal probability and for <span class="math-container">$P_2$</span> if his singla is not <span class="math-container">$s_H$</span>, then he knows that his signal is <span class="math-container">$s_M$</span> or <span class="math-container">$s_L$</span> with equal probability as well.</li>
</ul>
<p>My first question is:</p>
<p><span class="math-container">$Q1:$</span> If we make the Bayesian update, what is the probability that both players have taken the signal <span class="math-container">$s_H$</span>, or the the combinations <span class="math-container">$(s_H,s_M)$</span> or <span class="math-container">$(s_H,s_L)$</span> have occured in <span class="math-container">$\omega_1$</span>. Also what is the probablitiy that both players took <span class="math-container">$s_L$</span> or <span class="math-container">$(s_M,s_L)$</span> or <span class="math-container">$(s_M,s_H)$</span>?</p>
<p>If we make the Bayesian update for the one state it is simple to do this for the other.</p>
<p>My second question is:</p>
<p><span class="math-container">$Q2:$</span> If every player in every state can distinguish her signal but not the signal of the other player, namely for example in <span class="math-container">$\omega_1$</span>, <span class="math-container">$P_1$</span> will know with probability <span class="math-container">$1$</span> if she will obtain <span class="math-container">$s_H$</span>, or <span class="math-container">$s_l$</span> or <span class="math-container">$s_M$</span> and the same holds for <span class="math-container">$P_2$</span>, then how the bayesian update chenges for the posterior beliefs about the combinations of the signals <span class="math-container">$(s_H,s_H)$</span>, <span class="math-container">$(s_H,s_L)$</span>, <span class="math-container">$(s_H,s_M)$</span> or <span class="math-container">$(s_L,s_H)$</span> to occur?</p>
<p>P.S. if any more assumption is needed to be made in order to do the update in probabilities, feel free to mention it and make the adjustments that you want.</p>

CC BY-SA 4.0
2021-10-09T11:22:11.593Z
Hi! What do you mean you write "*if she will not take $s_L$*"? If she does not observe the signal $s_L$?
Giskard
CC BY-SA 4.0
2021-10-09T11:24:05.980Z
Also, what exactly do you mean by "*The rest of the narrative is preserved as above.*"? I assume it is not the same sentence, but a mirror, perhaps: In state $\omega_2$ $P_2$ distinguishes $s_H$  but not $s_L,s_M$? (Using copy paste you could easily write this.)
Giskard
CC BY-SA 4.0
2021-10-09T11:25:14.007Z
Further questions: do the players get the same signal, or indepent signals? (I am guessing the latter?)
Giskard
CC BY-SA 4.0
2021-10-09T11:25:55.520Z
What have you tried to answer Q1? Depending on your answers to the comments above, this seems to be a very straightforward calculation?
Giskard
CC BY-SA 4.0
2021-10-09T11:26:40.470Z
What have you tried to answer Q2? Depending on your answers to the comments above, this seems to be a very straightforward calculation?
Giskard
CC BY-SA 4.0
2021-10-09T11:27:03.887Z
Please elaborate on what exactly causes you difficulties, because it is hard to see where you get stuck within this exercise.
Giskard
CC BY-SA 4.0
2021-10-09T13:59:42.757Z
@Giskard ok let me make some changes
Nav89
CC BY-SA 4.0
2021-10-09T14:12:10.787Z
@Giskard what do you mean by ``Depending on your answers to the comments above, this seems to be a very straightforward calculation? " I mean... I have hust stuck a little...so I haven't tried something...
Nav89
CC BY-SA 4.0
2021-10-09T14:15:01.283Z
I know that mayebe it is a trivial problem, however It seems that I am not sure from where to start...
Nav89
<mathematical-economics><game-theory><bayesian-game><probability>
Bayesian update in the beliefs about the signals
Nav89
# Bayesian update in the beliefs about the signals

(Asked by: username_0 on 2021-10-09)

Suppose that we have tow states of the world $\omega\_1$ and $\omega\_2$, where $p(\omega\_1)=p(\omega\_1)=1/2$ and there are three different signals, $s\_H,s\_M,s\_L$ that are equally likely to occur in everey state of the world that is $p(s\_H|\omega\_1)=p(s\_M|\omega\_1)=p(s\_L|\omega\_1)=1/3$ and $p(s\_H|\omega\_2)=p(s\_M|\omega\_2)=p(s\_L|\omega\_2)=1/3$. Let two players exist in our world, say $P\_1$ and $P\_2$ and the following hold:

* In state $\omega\_1$, player $1$ distinguishes the signal $s\_H$ but not the signals $s\_M,s\_L$ and for simplicity lets say that he knows that if he dodes not observe the signal $s\_H$ he will know that his signal is either $s\_M$ or $s\_L$ with equal probability. For $P\_2$ it holds that she distinguishes the signal $s\_L$ but not the signals $s\_H,s\_M$ and if she will not take $s\_L$ she will take with equal probability either $s\_H$ or $s\_M$.
* In state $\omega\_2$, $P\_1$ distinguishes $s\_L$ but not $s\_H,s\_M$ and $P\_2$ $s\_H$ but not the signals $s\_M,s\_L$. If the signal is not $s\_L$, then $P\_1$ knows that his signal is $s\_H$ or $s\_M$ with equal probability and for $P\_2$ if his singla is not $s\_H$, then he knows that his signal is $s\_M$ or $s\_L$ with equal probability as well.

My first question is:

$Q1:$ If we make the Bayesian update, what is the probability that both players have taken the signal $s\_H$, or the the combinations $(s\_H,s\_M)$ or $(s\_H,s\_L)$ have occured in $\omega\_1$. Also what is the probablitiy that both players took $s\_L$ or $(s\_M,s\_L)$ or $(s\_M,s\_H)$?

If we make the Bayesian update for the one state it is simple to do this for the other.

My second question is:

$Q2:$ If every player in every state can distinguish her signal but not the signal of the other player, namely for example in $\omega\_1$, $P\_1$ will know with probability $1$ if she will obtain $s\_H$, or $s\_l$ or $s\_M$ and the same holds for $P\_2$, then how the bayesian update chenges for the posterior beliefs about the combinations of the signals $(s\_H,s\_H)$, $(s\_H,s\_L)$, $(s\_H,s\_M)$ or $(s\_L,s\_H)$ to occur?

P.S. if any more assumption is needed to be made in order to do the update in probabilities, feel free to mention it and make the adjustments that you want.

**Tags:** mathematical-economics, game-theory, bayesian-game, probability
> **Comments:**
> **username_1**: Hi! What do you mean you write "*if she will not take $s_L$*"? If she does not observe the signal $s_L$?  
> **username_1**: Also, what exactly do you mean by "*The rest of the narrative is preserved as above.*"? I assume it is not the same sentence, but a mirror, perhaps: In state $\omega_2$ $P_2$ distinguishes $s_H$  but not $s_L,s_M$? (Using copy paste you could easily write this.)  
> **username_1**: Further questions: do the players get the same signal, or indepent signals? (I am guessing the latter?)  
> **username_1**: What have you tried to answer Q1? Depending on your answers to the comments above, this seems to be a very straightforward calculation?  
> **username_1**: What have you tried to answer Q2? Depending on your answers to the comments above, this seems to be a very straightforward calculation?  
> **username_1**: Please elaborate on what exactly causes you difficulties, because it is hard to see where you get stuck within this exercise.  
> **username_0**: @Giskard ok let me make some changes  
> **username_0**: @Giskard what do you mean by ``Depending on your answers to the comments above, this seems to be a very straightforward calculation? " I mean... I have hust stuck a little...so I haven't tried something...  
> **username_0**: I know that mayebe it is a trivial problem, however It seems that I am not sure from where to start...  

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## 0 answers

