Suppose we have $P_1,\dots, P_m$, $m$ person, starting a yes-no game, let us denote yes=Y and no=N, $\mathbb P(P_i=Y)=p_i, \mathbb P(P_i=N)=1-p_i.$
If I start a Markov chain with a state when all persons say 'No', thus the chain will stay in this state with probability $\Pi_{i=1}^{m}(1-p_i)$. Now could anyone tell me the next few possible states with correct expressions of probabilities? When we have two-person, I can see the game is
(1) Start with both people says 'No' with prob $(1-p_1)(1-p_2)$ and they can remain in this state forever with this probability.
(2) Next step we can get two state:
(i) $P_1$ says 'Yes' but $P_2$ says 'No' or the other way ( i.e one of them says no another yes) with prob $p_1(1-p_2)+p_2(1-p_1)$.
At this point, I find when there is $n$ person, this becomes very complex? many cases may arise? one of them no, rest yes, two of them no rest yes, and so on.
(ii) Both says 'Yes' with prob $p_1p_2$.
Anyway, the chain will then go on and on. I need to draw an approximate picture of this scenario when the game is played by $m$ person. Thanks for a sketch with latex for the states of the Markov chain with arrows representing probabilities and any sort of help to find the possible states and probabilities.
I can't tell how this game works. Why can't the next step in (2) be that both $P_1$ and $P_2$ say “No”?
Guessing that you meant to give four possible steps, whether you can “merge” states (i) and (ii) depends on your definition of state. Is state two “One player says no,” or is it “Player 1 says “No” but Player 2 says “Yes”?
If it's either or these, it's not much of a Markov process. The next state doesn't depend on the current state. So all the rows of the transition matrix are the same. Let $P$ be the set of people.
If your state is just the number of “No”s, the $k$th entry in the transition matrix represent the probability of the there will be $k$ “No”s, or $$\sum_{\substack N\subseteq P\\|N|=k} \prod_{1≤i≤m} [[i\in N]]p_1+[[i\notin N]](1-p_i)$$
If your state is the function from $i$ to $i$’s answer, there are $2^m$ states, and the transition from one state to another is just the probability of the answers making up the second state.