towniness-based EV calculation

[biancospino]

Do note that,

for Vanilla Nightless games only

, the EV is actually identical calculated this way and by using random lims (it's not the same for non-Nightlesses since nk'ing at the top instead of randomly affects the likelyhood that the bottom is scum).
(In fact, it is easy to see that, if p(m,x) is the chance of mafia winning when there are m mafiosi in x players, then p(m,x)=2m/x. In fact, note that one has p(0,x)=0, p(m,2m)=1 and
p(m,x)=p(last is scum)p(m-1,x-1)+p(last is town)p(m,x-1)=(m/x)p(m-1,x-1)+(1-x/m)p(m,x-1)
when 0<x<m/2; and by double induction those uniquely specify p (for all 0<=m and 2x<=m), and are satisfied by p(m,x)=2m/x. Your example has indeed p(3,9)=6/9=2/3.)

[biancospino]

For Vanilla games with nightkills, then the mafia, if there are m mafiosi and x players, assuming that x is odd, wins iff there are (x+1)/2-m mislims; this always happens unless the whole scumteam is in the bottom ((x+1)/2-m)+m-1=(x-1)/2 players.
So, the chance of mafia to win would be
p(m,x)=1-p(all m scum in the bottom (x-1)/2)
=1-(((x-1)/2)Cm)/(xCm)
where aCb is the binomial a!/(b!(a-b)!). For a 2:9 mountaneous, for instance, one has
p(2,11)=1-(5C2)/(11C2)=9/11 ~ 81.818...%

(Assuming I'm not making some crude mistake, which tbh it's not even particularly unlikely)

[biancospino]

Also, note that 2:N will never have an ev of 1/2 with this approach. In fact that isn't surprising; the mafia ev of 1:N would be (N+2)/(2N+2) (the chance that the mafia is not in the bottom half, which is of course ~1/2 for N very large); and the ev of 2:N is going to be better than that still.

In fact, the mafia ev of 2:N approaches 3/4 (from above) when N grows to infinity