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.