I am working on an exercise from previous years in my university that states that :
In a village there are living N people and they decide with an interesting way about some actions. Specifically, if someone proposes an action, then all the N villagers vote for it with YES or NO. The next day each villager re-adjusts her/his opinion independently from the other villagers, and votes again with probability equal to the probability of the total (maximum) supporters of the previous day.This voting process continues until all N agree on the same opinion.
1)
If $Y_{n}$ is the total number of YES voters the day $n$ , describe the transition probabilities of the chain $\{Y_{n} :n \in \mathbb{N}_{0}\}$
2)
Prove that the percentage of $\Pi_{n}$ supporters of YES the day $n$ defines a martingale.
**My though: **
I think that is like a polya-urn like problem that has $i$ YES voters and $j$ NO voters.
So the transition probabilities are:
$p(x,y)=P[Y_{n+1}=Y|X_{n}=x] = \begin{cases} \frac{j+k}{i+j+k} &\text{with probability} : \frac{j}{N} \\ \frac{j}{i+j+k} &\text{with probability} : \frac{i}{N} \end{cases}$
The second question I don't know it.If someone can help me with it, I will appreciate it.
I think your notation for the transition probabilities is a little off. You have an $X_n$ process that isn't defined anywhere, and have $i$ and $j$ running around that don't appear in $p(x,y)$ anywhere.
If I interpret the question statement correctly, then given $Y_n = i$, $Y_{n+1}$ is binomially distributed with parameter $\frac yN$. This implies $$\mathbb{P}(Y_{n+1} = j | Y_n = i) = \binom{j}{N} \left(\frac iN\right)^j \left(\frac{N-i}{N}\right)^{N-j}.$$
To show $\Pi_n$ is a martingale, we can just use that given $Y_n$, $Y_{n+1}$ is binomially distributed with parameter $\frac{Y_n}{N}$ to conclude $\mathbb{E}[Y_{n+1}|Y_n] = Y_n$. Hence $$\mathbb{E}[\Pi_{n+1}|\Pi_n] = \mathbb{E}\left[\left.\frac{Y_{n+1}}{N}\right|\Pi_n\right] = \frac{Y_n}{N} = \Pi_n.$$