The setup for this thread:
M Mafia
1 Vigilante
T Townies
(Standard rules: Daystart, Mafia have factional kill. If last two remaining are Mafia and Vigilante, the result is a draw.)
When down to the last Mafia, Mafia no longer gets any benefit from claiming Vigilante.
If town picks a random lynch target:
There is a 1/4 chance they pick the Vigilante; in this case, the EV is 2/3 (they have the lynch and the vig, out of the three remaining targets).
There is a 1/4 chance they pick the Mafia, which wins.
There is a 1/2 chance they pick a Townie. In this case, there is a 1/2 that the vig hits, and if it misses there's still a 1/2 that the Mafia misses the Vigilante, for an EV of 5/8 (1/2 + 1/2*1/2*1/2).
The total EV then is: 1/4*2/3 + 1/4 + 1/2*5/8 = 35/48 (~73%)
Can the town do better? I think the answer is no; no lynch doesn't help - 1/3 + 2/3*1/3*3/4 + 2/3*1/3*1/2 = 11/18, ~61%.
The other possibility is that a Townie offers a self-lynch (would have to work out how this happens logistically, without outing the Vigilante); if Mafia never does this, that Townie would be confirmed and we'd make it to the 1:1:1 case, EV 3/4; if Mafia does it as often as a Townie, we now have a 1/3 chance of lynching Mafia and 2/3 Townie, so 1/3 + 2/3*5/8 = 3/4 again. But it seems likely that Mafia has some percentage strategy to keep the EV below 73%. I'll figure out the Nash equilibrium at some point.
I suspect the answer to this question will be the same for higher player counts, but that's something to explore later.
Now the strategy is potentially much more complicated. Assuming town lynches, Vigilante should obviously always claim to avoid the lynch. The question here is: how often should Mafia fake claim Vigilante vs. Townie? Here, the Vigilante fake claim doesn't ultimately prevent death, but it does keep the day going to potentially out the real Vigilante or lynch a Townie. Or perhaps the town should no lynch as soon as there is a Vigilante claim. Mafia can't fake claim Vigilante 100% of the time though, or a Townie claim would be confirmed, as above. My suspicion here is that the correct strategy is ultimately pretty straightforward, but proving it might be messy.