So I'm gonna go ahead and lay this out now. Frequently in this situation, it makes more sense to no lynch, go down from 3v1 to 2v1 over the night, and improve your odds of correctly lynching scum. Here's our RNGesus chances if we don't do that.
Day 1 outcomes:
Hit scum - 1/4 - Town victory gg
Hit the doctor - 1/4 - Mafia victory
Hit a townie - 1/2 - Proceed to night 1
Night 1*:
Townie/doctor dies - 3/4 - Mafia victory
No one dies - 1/4 - Proceed to day 2
Day 2 sub optimal:
Hit mafia - 1/3 - Town victory
Hit town - 2/3 - Mafia victory
OR
Day 2 where everyone claims like last game to be a bunch of cheeky bastards:
Hit mafia - 1/2 - Town victory
Hit town - 1/2 - Mafia victory
So playing "sub optimal" we have a (1/4) + (1/2 * 1/4 * 1/3) = 7/24 = ~29.2% chance to win
And with claiming we have a (1/4) + (1/2 * 1/4 * 1/2) = 5/16 = ~31.2% chance to win
Assuming we are all incompetent and RNGesus everything.
*Assuming no self protection, the doctor could mistakenly protect the mafia, so only 1/2 chance of even protecting the townie, then a 1/2 chance mafia tries to hit the townie instead of the doctor. Of course if the doctor can protect themself they could just self protect and turn this into a 50/50 chance overall, which brings up town success odds by a few percentage points. This brings up our chances of winning to 1/3 and 3/8 for each strategy.
So, I'm actually going to propose we not do that, specifically because this is day 1 and someone is likely going to just not get to even attempt to play the game in that scenario. We'd only get a tiny benefit from doing that, at the cost of 1/4 players not getting to play the game. And under optimal doctor mechanics we get the same odds or better anyways. I could totally factor in a 1/2 chance of getting optimal doctor mechanics, but I'm done doing math.
OH WAIT SHIT NO THERE IS A COMPLICATION IN THE NO LYNCH STRATEGY.
Day 1 - No lynch
Night 1*:
Doctor:
Protect scum - 1/3 - Somebody dies
Protect not scum - 2/3 - Somebody maybe dies
Mafia:
Hit doctor - 1/3 - Double townie day 2
Hit unprotected townie - 1/3 - Role claim BS opportunity
Hit protected townie - 1/3 - Doctor gets info?
If protected scum, hit either townie - 2/3 - Role claim BS opportunity
Total:
Double townie outcome - (1/3 * 1/3) + (2/3 * 1/3) = 1/3
Role claim BS - (1/3 * 2/3) + (2/3 * 1/3) = 4/9
Doctor info - (2/3 * 1/3) = 2/9
Double townie (or forgo BS - only thing that matters if planning on forgoing BS):
Hit scum - 1/3 - Town victory
Hit town - 2/3 - Mafia victory
Total - (1/3 * 1/3) = 1/9 = ~11.1% town victory by this path
Role claim BS:
Hit scum - 1/2 - Town victory
Hit town - 1/2 - Mafia victory
Total - (4/9 * 1/2) = 2/9 = ~22.2% town victory by this path
Oh god, the branching, why am I doing this?
Doctor info:
Doctor or protected townie is targeted for lynch - 1/2 - Doctor claims
Hit scum immediately - 1/4 - Town victory
Hit other townie immediately - 1/4 - Mafia victory
Doctor claims (Mafia counter claim is irrelevant, counter or not it's now a 1v1 situation with two confirmed town spectators):
Hit scum - 1/2 - Town victory
Hit town - 1/2 - Mafia victory
Total - (2/9) * (1/4 + (1/2 * 1/2)) = 1/9 = ~11.1% town victory by this path
*If doctor can self protect, this could force the role claim BS scenario, where the doctor only self protects every night until the mafia hits someone else (technically we could be assholes and have an endless game this way?), giving us a guaranteed 1/2 chance right from the get go if we're willing to do role claim BS. This just feels extremely cheesy though, so I hope the doctor doesn't try to force this situation if we go this route. Alternatively, the doctor self protects until a townie dies and we just play the "double townie" scenario without BS claiming, in which case total town victory odds are still just 1/3.
Overall (with no conscientious objection to the role claim BS where everyone claims) - 1/9 + 2/9 + 1/9 = 4/9 = ~44.4% town victory
So actually, if we're willing to do cheesy role claim BS, we improve our odds from ~31.2% or 37.5% (depending on doctor mechanics) to ~44.4% or 50% by doing a no lynch today. I don't really know how I feel about that revelation and this just got SO MUCH MORE META than I set out for this to be.