[EV] M Mafia, 1 Vigilante, T Townies

[mith]

This thread will be the first in what I hope to be a series of EV discussions. The setups presented are not necessarily intended to be run at any particular count - rather, the goal is to determine best play for both sides.

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.

1:1:0

- 1/2 (draw)

1:1:1

- 3/4 (1/2 Mafia lynch + 1/2*1/2 draw)

1:1:2

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.

2:1:2

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.

[mith]

I think best town can do in 1:1:2 is EV of 35/48.

I think so too, but town also has the option of never lynching the self-lynch offer (obviously if this is known to be the strategy Mafia should offer as often as a Townie), or lynching with some probability y. Wish I had more time to think about this today. Maybe tomorrow.

[mith]

And in your 2:1:2, the EVs you are giving are for Mafia winning, not town. First part should be:

Lynching among the VT claims:
2/3 VT lynch = Town loses, EV 0
1/3 Mafia lynch = Town wins, EV 1
EV 1/3.

Lynching among the Vig claims:
1/2 Vig lynch = Town loses, EV 0
1/2 Mafia lynch = if Vig doesn't shoot, EV 1/3,
============ if Vig shoots wrong (2/3), EV 0.
============ if Vig shoots right (1/3), EV 1.
============ (therefore it doesn't matter if Vig shoots or not, EV is 1/3)
EV 1/6.

Town should lynch among the Townie claims. (Or no lynch, which is just as effective. Vig claims shoot each other.)

Of course, this assumes Mafia will always claim Vig; or at least that Mafia will claim Vig on a first wagon as often as claiming Vig on a second wagon after the real Vig claims. The latter may be true (insofar as I don't think it matters how often Mafia claim Vig after the real Vig has already claimed - if they claim Townie and are lynched, EV is still 1/3; so Mafia can make the probabilities match up if they want). But it's complicated to prove.

[mith]

Ok, here's the numbers for 1:1:2:

Townies offer self-lynch at probability 1
Mafia offer self-lynch at probability x

Town lynch self-lynch at probability y
Town lynch other at probability 1-y

Self-lynch:
Is Mafia: x/(x+2) - EV 1
Is Townie: 2/(x+2) - EV 5/8

EV (8x+10)/(8x+16)

Non-Self:
Is Vigilante: 1/3 - EV 2/3
Is Mafia: 2/3*1/(x+2) - EV 1
Is Townie: 2/3*(x+1)/(x+2) - EV 5/8

EV (110+46x)/(72x+144)

Total:
EV (26xy-20y+46x+110)/(72x+144)

∂/∂y = (26x-20)/(72x+144)
Equilibrium at x=10/13, EV 35/48

∂/∂x = (5184y-1296)/(72x+144)^2
Equilibrium at y=1/4, EV 35/48

So town should lynch the offered self-lynch only 1/4 of the time, Mafia should offer the self-lynch 10/13 as often as a Townie, and none of this is better than just lynching normally.

[mith]

Actually, there's an additional complication I overlooked - if town doesn't lynch the self-lynch offer, and ends up lynching a townie, the two remaining players for the Vigilante to target are not equally likely to be scum (unless x=1). I'm screwing up somewhere in the calculation, I'll figure it out later.

[mith]

Non-Self:
Is Vigilante: 1/3 - EV 2/(2+x)
Is Mafia: 2/3*1/(x+2) - EV 1
Is Townie: 2/3*(x+1)/(x+2) - EV 1/(x+1) + x/(x+1)*1/2*1/2 = (x+4)/(4x+4)

EV (2x+24)/(12x+24)

Total:
EV (20xy-18y+4x+48)/(24x+48)

Equilibrium is at x=9/10, y=40/58, with EV 43/58 (74.14%).

That makes a little more sense anyway, lynching the self-lynch offer most of the time.

[mith]

And after all that, I thought of a way to do even better:

Two

non-Vigilante players offer to self-lynch. Lynch one of them; vig the player who didn't. There is a 2/3 chance of killing the scum. If you didn't, the Mafia has a 50-50 shot at hitting the Vigilante, or it's a draw again. 2/3 + 1/3*1/2*1/2 = 3/4 (75%).

Logistically, there is still the question of how to pull this off, of course. You'd want to arrange for everyone to be on at a particular time and then first two to post are the self-lynch claims. Vigilante just has to wait until two players have posted. (If you can't come up with a way to avoid the Mafia cheating and making sure they are one of the two, EV goes back down to 5/8; and if the Vigilante gives himself away by being way too slow, EV goes down to 2/3.)

[mith]

I was posting some stuff for 1:1:3 yesterday when the site went down, and forgot to save it. Bah. I'll recreate it at some point, but IIRC no lynch was about 74% and random lynch was about 73%; I'd be shocked if there isn't a way to get it to 75% here as well, but adding that extra townie does make things tricky - it adds an extra suspect without providing an extra lynch/vig.

[mith]

I modified my multiball script to run this setup. The following numbers are with the following assumptions:

1 Mafia, 1 Vigilante, 0 Townies is a draw (EV 50%).
1 Mafia, 1 Vigilante, 2 Townies is EV 75% per the strategy above.
The Vigilante in this run effectively has the "Innocent Child" modifier - it is assumed the Mafia never claims Vigilante. (In practice, Mafia will almost always* claim Vigilante; it then becomes more complicated to determine whether town should no lynch and let the Vigilante kill the fake claim, or try to lynch someone else.)

Here are setups in the range of 1/3 < EV < 1/2; the count is by total number of each alignment (so 2:3 means 2 Mafia, 1 Vigilante, 2 Townies), and the number in parenthesis is the improvement over the corresponding vanilla setup (2:1:2 vs. 2:3) - this is a relative improvement (relative to the absolute percentage it

could

improve, that is (Vig-Vanilla)/(1-Vanilla). As you can see, having a fake-claim-proof Vigilante is a huge benefit to the town.

2:3 - 34.44% (+24.36%)
2:4 - 46.15% (+37.86%)
2:5 - 44.94% (+28.63%)

3:8 - 35.98% (+23.38%)
3:9 - 38.63% (+26.54%)
3:10 - 39.68% (+23.86%)
3:11 - 42.18% (+27.01%)
3:12 - 43.61% (+25.29%)
3:13 - 44.61% (+26.62%)
3:14 - 46.18% (+25.47%)
3:15 - 47.48% (+27.27%)
3:16 - 48.05% (+25.09%)
3:17 - 49.46% (+27.11%)

4:14 - 33.75% (+21.98%)
4:15 - 34.63% (+20.51%)
4:16 - 36.23% (+22.46%)
4:17 - 37.33% (+21.45%)
4:18 - 38.16% (+22.49%)
4:19 - 39.36% (+21.78%)
4:20 - 40.36% (+23.07%)
4:21 - 40.95% (+21.73%)
4:22 - 42.06% (+23.2%)
4:23 - 42.82% (+22.22%)
4:24 - 43.41% (+23.02%)
4:25 - 44.29% (+22.34%)
4:26 - 45.01% (+23.34%)
4:27 - 45.47% (+22.18%)
4:28 - 46.3% (+23.37%)
4:29 - 46.87% (+22.47%)
4:30 - 47.34% (+23.15%)
4:31 - 48.03% (+22.52%)
4:32 - 48.58% (+23.34%)
4:33 - 48.96% (+22.34%)
4:34 - 49.61% (+23.34%)

5:24 - 34.16% (+19.23%)
5:25 - 34.96% (+20.22%)
5:26 - 35.5% (+19.31%)
5:27 - 36.41% (+20.46%)
5:28 - 37.06% (+19.77%)
5:29 - 37.59% (+20.44%)
5:30 - 38.35% (+19.97%)
5:31 - 38.97% (+20.78%)
5:32 - 39.41% (+19.94%)
5:33 - 40.14% (+20.91%)
5:34 - 40.65% (+20.24%)
5:35 - 41.09% (+20.83%)
5:36 - 41.72% (+20.37%)
5:37 - 42.22% (+21.06%)
5:38 - 42.59% (+20.3%)
5:39 - 43.19% (+21.14%)
5:40 - 43.62% (+20.52%)
5:41 - 43.99% (+21.04%)
5:42 - 44.52% (+20.6%)
5:43 - 44.93% (+21.2%)
5:44 - 45.25% (+20.52%)
5:45 - 45.76% (+21.26%)
5:46 - 46.13% (+20.68%)
5:47 - 46.45% (+21.15%)
5:48 - 46.9% (+20.74%)
5:49 - 47.26% (+21.27%)
5:50 - 47.54% (+20.65%)
5:51 - 47.98% (+21.31%)
5:52 - 48.3% (+20.77%)
5:53 - 48.58% (+21.2%)
5:54 - 48.97% (+20.82%)
5:55 - 49.29% (+21.3%)
5:56 - 49.54% (+20.73%)
5:57 - 49.92% (+21.33%)

6:35 - 33.59% (+18.09%)
6:85 - 49.97% (+19.71%)
7:49 - 33.58% (+17.76%)
7:117 - 49.84% (+19.24%)
8:65 - 33.4% (+16.67%)
8:155 - 49.87% (+18.49%)
9:84 - 33.54% (+16.3%)
9:199 - 49.97% (+18.26%)
10:104 - 33.36% (+16.12%)
10:247 - 49.95% (+17.83%)

[mith]

At least in 2:1:2, if there is a Vig claim it is strictly better for town to no lynch. I may adjust the script so that if Mafia is run up they always claim Vig and town goes no lynch, see how different the results are (probably not much, other than at small counts). This wouldn't be exactly right either, because if Mafia were really to always claim Vig then a Town claim would be confirmed innocent.

[mith]

Allowing for Mafia fake claims, with the assumption that they will always claim Vig and town will always no lynch after a Vig claim.

2:3 - 34.44% (+24.36%)
2:4 - 40.42% (+31.25%)
2:5 - 43.64% (+26.94%)
2:6 - 47.38% (+31.78%)
2:7 - 49.8% (+28.45%)

3:9 - 34.38% (+21.46%)
3:10 - 36.7% (+20.09%)
3:11 - 38.65% (+22.56%)
3:12 - 40.25% (+20.84%)
3:13 - 41.85% (+22.96%)
3:14 - 43.31% (+21.49%)
3:15 - 44.52% (+23.18%)
3:16 - 45.76% (+21.77%)
3:17 - 46.88% (+23.4%)
3:18 - 47.87% (+21.96%)
3:19 - 48.86% (+23.43%)
3:20 - 49.78% (+22.15%)

4:17 - 33.92% (+17.18%)
4:18 - 35.14% (+18.7%)
4:19 - 36.23% (+17.74%)
4:20 - 37.27% (+19.09%)
4:21 - 38.29% (+18.21%)
4:22 - 39.21% (+19.42%)
4:23 - 40.1% (+18.53%)
4:24 - 40.97% (+19.7%)
4:25 - 41.76% (+18.81%)
4:26 - 42.53% (+19.88%)
4:27 - 43.28% (+19.06%)
4:28 - 43.98% (+20.05%)
4:29 - 44.65% (+19.23%)
4:30 - 45.31% (+20.2%)
4:31 - 45.93% (+19.4%)
4:32 - 46.53% (+20.29%)
4:33 - 47.12% (+19.54%)
4:34 - 47.67% (+20.38%)
4:35 - 48.21% (+19.64%)
4:36 - 48.74% (+20.46%)
4:37 - 49.24% (+19.74%)
4:38 - 49.72% (+20.51%)

5:27 - 33.42% (+16.72%)
5:28 - 34.2% (+16.12%)
5:29 - 34.92% (+17.05%)
5:30 - 35.63% (+16.43%)
5:31 - 36.31% (+17.32%)
5:32 - 36.97% (+16.72%)
5:33 - 37.6% (+17.55%)
5:34 - 38.21% (+16.97%)
5:35 - 38.81% (+17.76%)
5:36 - 39.38% (+17.17%)
5:37 - 39.94% (+17.94%)
5:38 - 40.48% (+17.37%)
5:39 - 40.99% (+18.09%)
5:40 - 41.51% (+17.54%)
5:41 - 42% (+18.24%)
5:42 - 42.48% (+17.69%)
5:43 - 42.95% (+18.36%)
5:44 - 43.4% (+17.83%)
5:45 - 43.84% (+18.47%)
5:46 - 44.28% (+17.95%)
5:47 - 44.7% (+18.57%)
5:48 - 45.11% (+18.06%)
5:49 - 45.51% (+18.66%)
5:50 - 45.9% (+18.16%)
5:51 - 46.28% (+18.74%)
5:52 - 46.66% (+18.25%)
5:53 - 47.02% (+18.81%)
5:54 - 47.38% (+18.33%)
5:55 - 47.73% (+18.88%)
5:56 - 48.07% (+18.41%)
5:57 - 48.4% (+18.94%)
5:58 - 48.73% (+18.48%)
5:59 - 49.05% (+18.99%)
5:60 - 49.36% (+18.55%)
5:61 - 49.67% (+19.04%)
5:62 - 49.98% (+18.61%)

6:40 - 33.58% (+15.83%)
6:90 - 49.84% (+18.21%)
7:55 - 33.53% (+15.22%)
7:124 - 49.9% (+17.43%)
8:72 - 33.41% (+14.78%)
8:163 - 49.91% (+17.11%)
9:92 - 33.51% (+14.26%)
9:207 - 49.91% (+17.02%)
10:114 - 33.52% (+14.29%)
10:257 - 49.97% (+16.72%)

As a rough approximation, you need M-1 more townies to get the same EV as compared to the previous "IC Vig" results - this makes intuitive sense, as you are losing a lynch every time Mafia claims.

[mith]

The EV for 2:1:2 is the same in either case, and I believe this is the correct EV, but I haven't quite proven it. Here's some math on the two simple choices after an initial claim (lynch the claim or no lynch); this doesn't account for the third choice (pick someone else to force a claim), which makes things even more complicated.

2/5 - Town, Town claim
1/5 - Vig, Vig claim
2/5 - Mafia
x - Vig claim
1-x - Town claim

(1+2x)/5 - Vig claim
y - Lynch
1/(1+2x) - Vig, EV 0
2x/(1+2x) - Mafia, 1:1:2 Night, EV 11/18
1-y - No Lynch
1/(1+2x) - 2:1*:2, EV 1/6
2x/(1+2x) - 2*:1:2
1/3 - 1:2, EV 1/3
2/3 - 1:1:1, EV 3/4
EV 11/18

It doesn't matter in this case whether we lynch the fake claim or not; however, we are obviously better off not lynching a true claim. So no lynch is optimal here.
EV (3+22x)/(18+36x)

(4-2x)/5 - Town claim
z - Lynch
2/4-2x - Town, 2:1:1 Night, EV 1/6
2-2x/4-2x - Mafia, 1:1:2 Night, EV 11/18
EV (28-22x)/(72-36x)
1-z - No Lynch
2/(4-2x) - Town, 2:1:2* Night
1/3 - 1:0:2*, EV 1/2
1/3 - 1:1:1*, EV 1
1/3 - EV 0
EV 4/9
(2-2x)/(4-2x) - Mafia, 2*:1:2 Night
1/9 - 1*:0:2, EV 0
2/9 - 1*:1:1, EV 1/2
2/3 - EV 0
EV 1/9
EV (20-4x)/(72-36x)
EV (8z-18zx+20-4x)/(72-36x)

Total EV: (8z-18zx+23+18x)/90

d/dz = (8-18x)/90; x=4/9

If x=4/9, it doesn't matter what z is; EV is 31/90 (34.44%). For x>4/9, town is better off not lynching in the town claim case; scum have claimed Vig too often, and now the chances are good enough that a town claim is innocent to let them live. For x<4/9, town should always lynch a town claim.

d/dx = (-18z+18)/90; z=1

If z=1 (always lynch a town claim), it doesn't matter what x is; EV is 31/90 (34.44%). For z<1, scum is better off always claiming town (x=0, EV is (8z+23)/90).

Overall, if these are the only two options after a claim, scum should always claim town (in case town doesn't always lynch a town claim), and town should always lynch the town claim (which makes the EV constant in x).

[mith]

Adding in the "lynch other" possibility after a Vig claim:

Town should always lynch other after a Vig claim; it's complicated, but what it boils down to is that the partial d/dx now gives y'=36(1-z)/5 as the optimal relationship between the two town strategies, where z is still lynch-after-town-claim and y' is lynch-other-after-vig-claim, and subbing this in for y' gives an EV which has a negative correlation with z, so we want z to be as small as possible while keeping that equation valid, which happens at y'=1, z=31/36. I'm not sure what optimal x is in this case; somewhere between 4/9 and 3/5 is my best guess at the moment. Either way, the following town strategy:

Force claim. If claim is Vig, get another claim - if that claim is also Vig, no lynch, otherwise lynch the Town claim. If claim is Town, lynch the Town claim with probability 31/36 and no lynch with probability 5/36.

Gets the EV to 113/324 (34.88%), which is a mild improvement. I suspect there are more layers we could add to the strategy at this point (no lynch sometimes after the second claim is town; get a second claim sometimes after first claim is town), and I find it highly likely there is a strategy similar to the 1:1:2 one which is better still. I'd be shocked if something this messy were actually optimal.

[mith]

You mean you don't want to run a 10:257?

[mith]

It's not hard to see that having a Vig is better than not having a Vig, at least if the remainder of the game is Vanilla. Having a Vig gives 2 Town-controlled kills for every 1 Mafia-controlled kill. Mafia-controlled kills always hit town, so you want as few of those over the course of a game as possible. (A Vig kill is actually slightly better than a lynch, as has already been pointed out - the Vig kill is choosing from a smaller pool with the same number of Mafia, and so is more likely to hit Mafia if chosen randomly.)

The math is different if there are other town roles. But unless the game can be broken through claims or information roles, I am firmly in the "Vig should always shoot" camp.

[mith]

Yeah, that's another good way of looking at it (though that plan is actually demonstrably worse than letting the Vig make the decision randomly, if it's followed strictly; it tells Mafia who is likely to be killed, so they can kill someone else and avoid doubling up on a townie, and it also tells Mafia who the Vig is as soon as they don't die).

(If I weren't so busy, I'd put something together to calculate the EV of that. Actually might be an interesting setup to run, just having that as a weakened Vig role.)