The premise of the Mafia game goes like this: There are a set of players (all residents) and they are divided into 2 groups, citizens and mafias. Each night mafias kill a member of the citizen and the following morning the residence lynch off one person. We assume the probability of lynching a mafia member and citizen are the same. So by this after each round there are 2 residents gone. What I am trying to do is calculate the expected number of mafia members on any given day. Is this possible if not can someone suggest an equally difficult problem in which I can do my research paper on.
Let $f(m,t,d)$ be the expected number of mafia starting with $m$ mafia, $t$ town, and after $d$ days. Then, we have the following recursion $$f(m,t,d) = \frac{t-1}{t+m - 1} f(m, t-2, d-1) + \frac{m}{t+m-1} f(m-1, t-1, d-1).$$
Our initial conditions are $f(m,t,0) = m, f(m, t, 1) = \frac{m(m+t-2)}{t+m-1}.$
I claim that $$ f(m,t,d) = m \prod_{k = 1}^d \frac{m+t - 2k}{m+t-2k+1}. $$ We can check that this satisfies the initial conditions. Proving this formula is just induction.
This formula doesn't seem to work for edges cases though, like when the mafia kills off all of the town before day d.