# Bayesian update in the beliefs about the signals

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.

• Hi! What do you mean you write "if she will not take $s_L$"? If she does not observe the signal $s_L$? Oct 9, 2021 at 11:22
• 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.) Oct 9, 2021 at 11:24
• Further questions: do the players get the same signal, or indepent signals? (I am guessing the latter?) Oct 9, 2021 at 11:25
• What have you tried to answer Q1? Depending on your answers to the comments above, this seems to be a very straightforward calculation? Oct 9, 2021 at 11:25
• What have you tried to answer Q2? Depending on your answers to the comments above, this seems to be a very straightforward calculation? Oct 9, 2021 at 11:26