The most common WIFOM situation in Mafia is "do I nightkill someone who suspects me, or someone who doesn't suspect me?", but this is a hard situation to analyse because it's based on reads, which aren't something that can be easily mathematically quantified. Instead, I wanted to find the simplest setup I could in which WIFOM situations fell naturally out of the game mechanics.
The setup I'd like to focus on is (as specified in the title) a Doctor plus 3 VTs versus 2 Goons. (This is a typical Normal Doctor, who can't self-target and can save one player every night.) I'll also make the setup must-lynch, in order to avoid issues with happily-ever-after/RFED situations at 3:1 mylo with a living doctor (the way these situations are resolved depends heavily on the moderator, so aren't really ideal for an EV discussion). If we analyse this from the point of view of random actions, in which town don't try to break the setup, we get this:
1/6 chance of Doctor lynched → scum win (EV 0)
1/2 chance of VT lynched → Doctor needs to save N1 (1/6 chance), town need to win 3:2 endgame (2/15 chance) → EV 1/45
1/3 chance of scum lynched → if Doctor misses (13/16 chance), town can win via scum lynch (1/4) or N2 save + D3 scum lynch (3/4×2/9×1/3), if Doctor hits (3/16 chance) then town need to win a 4:1 endgame (just over 2/5 chance; I'll treat the possibility of making two more saves as negligible) → EV 931/2880 (~32.3%)
(Note that in the absence of claims, the Doctor's chance of hitting is
t
/(s
+t
)(t
+1) where t
is the number of VTs and s
the number of living scum; if scum shoot the Doctor blind (1/(t
+1) chance) then the Doctor can't save, if they shoot a different townie (t
/(t
+1) chance) then the Doctor has a (1/(s
+t
)) chance of saving.)As we can see, random behaviour in this setup is not very good for town at all: it's (just over) 1027/8640, or about 11.9%. As such, town are probably going to want to break it.
There are several standard techniques that are seen in, say, Mini Normals all the time, that town might try here to help out their win chances. For example, one possibility is to ask for a claim at L-1, and not to lynch a claimed Doctor unless there's a counterclaim (and even then, randomize between the Doctor and the counterclaim; if you don't do that, then scum would just always counterclaim a Doctor claim for the win). In this case, the EV for the lynched-doctor case increases to 12.5% (after a scum counterclaim, 50% chance of lynching the fakeclaimer × 25% chance of winning a 3:1 must-lynch mylo the next day); if scum don't counterclaim, the EV is a little higher, at 2/15 or about 13.3% (the EV of winning the resulting 3:2 setup, as the Doctor is necessarily going to claim N1), meaning that scum should always counterclaim in this situation. However, the EV for the lynched-scum case might at first seem to drop to 12.5% for the same reason (because scum can just fakeclaim Doctor). It's at this point that our first WIFOM situation comes up: if scum
always
claim Doctor when run up to a lynch, then a VT claim at L-1 confirms you as town!Instead, we need to find a balance. Assuming that VTs and Doctors always trueclaim, let
x
be the chance that town lynches a VT claim (treating VT claims as confirmed town the rest of the time), and y
be the chance that scum fakeclaims Doctor rather than VT if run up to L-1. We'll also use z
for the probability of town winning a 4:2 with a Doctor and an Innocent Child (a setup that's WIFOMy in its own right, and thus needs separate analysis). If town play randomly other than never lynching an uncounterclaimed doctor, and lynching among players who make contradictory claims, the EV is:1/6 (Doctor run up) +
y
/3 (scum run up) chance of a counterclaim situation → 12.5% EV1/2 (town run up) →
x
/60 (town lynched) + z
(1-x
) (town assumed innocent) EV(1-
y
)/3 scum claim VT → 931x
/2880 (scum lynched) + 0 (scum assumed innocent) EVThe total EV here is:
(1/8)(1/6 +
y
/3) + (1/2)(x
/60 + z
(1-x
)) + ((1-y
)/3)(931x
/2880)which simplifies to:
1/48 + 1003
x
/8640 + y
/24 + z
/2 - 931xy
/8640 - xz
/2Town should choose
x
and scum choose y
so that neither faction can improve their victory chances with a small, in-bounds (i.e. not below 0 or above 1 because probabilities don't work like that) change to x
or y
. In the case of town, increasing x
is helpful so long as (1003-931y
)/4320 > z
(and decreasing x
is helpful if that inequailty is reversed). Meanwhile, scum want to increase y
as long as 931x
/360 > 1. Note that increasing x
means that scum will want a higher y
, and increasing y
means that town will want a lower x
, leading to a natural negative feedback loop that tends to an equilibrium. Thus, if it's in-bounds, an equilibrium is found where (1003-931y
)/4320 = z
and 931x
/360 = 1, i.e. x
= 360/931 and y
= (1003 - 4320z
)/931. (The feedback loop is broken if the value of z
is such that y
is out of range; this might cause x
to vary from the 360/931 figure.)My conclusions from this are thus that town should lynch a VT claim around 38.7% of the time; this value is high enough that scum don't benefit from a VT fakeclaim, but low enough that town get as much advantage as possible after running up a VT to L-1. (I doubt this is what most people would expect in a setup like this one; given that it means that a Doctor claim is
more
likely to get lynched than a VT claim.) The total EV from the strategy, assuming z
is in range, is1/48 + 1003(360/931)/8640 + (1003 - 4320
z
)/931/24 + z
/2 - 931(360/931)(1003 - 4320z
)/931/8640 - (360/931)z
/2= 1/48 + 1003/22344 + 1003/22344 - 180
z
/931 + z
/2 - 1003/22344 + 180z
/931 - 180z
/931= 979/14896 + 571
z
/1862or approximately 6.6% + 30.7%×
z
.This strategy is thus better than the original one (which has an 11.9% EV) if the 30.7%×
z
term makes up at least (11.9-6.6)% = 5.3% of the EV; 5.3% / 30.7% is around 17.3%, which is the minmum z
value town require for claiming to be worth it. It seems likely that the IC+Doctor+2 VT vs. 2 Goon setup has an EV of at least 17.3% (simply treating the power roles as named townies and massclaiming gives a town EV of 16.7% in that setup, and town can do better than this by lynching from the pool that doesn't contain the Doctor first), meaning that town probably should be forcing a claim at L-1 (and not lynching uncounterclaimed Doctors, lynching from among counterclaimants, and lynching a VT claim only about 40% of the time, treating it as confirmed town the rest of the time).There are plenty more WIFOM situations in this setup. For example, if a player is being treated as confirmed, and there's an unknown Doctor, do scum shoot the confirmed player or try to find the Doctor? (If they're trying to find the Doctor, the Doctor wants to protect someone other than the confirmed player.) Can town exploit the fact that a Doctor save is almost certainly town, or can scum screw that up with a no-kill? (If it's 3:2 going into night then the scum may as well kill – it's even better to outright win the game than to have scum confirmed as town – but this reasoning doesn't apply after a scum lynch D1.) There are possibly also other claiming strategies that town could use. As such, this is only a basic analysis, but should help to explain how WIFOM can be a useful tool for setup breaking and how it's analyzed .
