[EV] Vanilla Multiball

[mith]

I'm going to attempt to get at least a good approximation in my spreadsheet this week, but if someone is able to run some simulation numbers, that would be awesome.

Setup:
X Mafia
X Werewolves
Y Townies

Obviously the first step in a simulation would be to assume town lynches (randomly) every day and scum teams shoot (randomly) every night. However, there are complications:

1. At some counts, town should probably no lynch. (The obvious one is 1:1:1. Town cannot win if there is a lynch.)
2. If there is an imbalance in the scum teams, it can be valid for a member of the weaker team to claim prior to being lynched, and the town leave that member alive. That player is guaranteed to die the following night, whether they are telling the truth or are a member of the stronger team fake claiming, and it gives town an opportunity to lynch from the stronger group to balance things.

[mith]

Here's a start with the 1:1:Y setups; after this point I should be able to write a (very complicated) formula to recursively calculate it, but I haven't gotten around to it yet.

1:1:1 (Night)
1/4 - Town
1/4 - Mafia
1/4 - Werewolf
1/4 - Mafia/Werewolf

1:1:2 (Night)
1/9 - Town
2/9 - Mafia
2/9 - Werewolf
2/9 - Mafia/Werewolf
2/9 - 1:1:1
1/18 - Town
1/18 - Mafia
1/18 - Werewolf
1/18 - Mafia/Werewolf

1:1:2 (Day) - Town offer self-lynch to get to 1:1:1, EV 25%.

1:1:3 (Night)
1/16 - Town
3/16 - 1:2 Mafia
1/16 - Town
1/8 - Mafia
3/16 - 1:2 Werewolf
1/16 - Town
1/8 - Werewolf
3/16 - 1:1:2
3/64 - Town
3/8 - 1:1:1
3/32 - Town

EV 21/64 = ~33%

1:1:3 (Day)

Lynch:
1/5 - 1:2 Mafia (1/3)
1/5 - 1:2 Werewolf (1/3)
3/5 - 1:1:2 (1/6)
EV 7/30

No lynch, EV 21/64 = ~33%

[mith]

You can potentially do some tricksy Prisoner's Dilemma shenanigans in the 1:1:2 (Day) case. The logic is as follows:

Town prefers a town lynch, absent the shenanigans we're trying to set up, and they can force that through if desired (a player offers self-lynch, and this player must be town - scum lose if they are lynched - while a player not agreeing to hammer the offered self-lynch is also confirmed scum and will be killed by the opposing scum faction).

You have to follow through on this offer often enough to prevent scum from being able to offer and be "confirmed town". However, if you don't follow through some small fraction of the time, you can consider that player confirmed (it is still not in the scum's best interest to offer).

If you didn't follow through, you can solicit a self-lynch offer from the remaining players. You can follow through on this one 100% of the time. Now we go to night 1:1:1, with a crucial difference from the above numbers: the townie is confirmed!

From each scum's perspective, it is better to kill the (known) other scum: If the Mafia kills the Werewolf, then the Werewolf loses either way; but if the Mafia kills the Townie, the Werewolf gets a draw by also killing the Townie, and a win by killing the Mafia. The reverse also holds.

The game theoretical result of this situation is an EV of 1 for the Town - although of course in practice it doesn't always work (I abused this in a large number of early games, which were often multiball).

The question then is what fraction of the time you should skip the first self-lynch offer. EV is 1*p + 1/4*(1-p), for p = skip first offer probability.

[mith]

It's more complicated though, because if we assume the Mafia and Werewolf offer self-lynch at the same rate, they jointly maximize their win probabilities by making that rate as high as possible (as high as the Town rate)!. This is because when Mafia is lynched, it's a Werewolf win, and if the Werewolf is lynched, it's a Mafia win.

However, if Mafia knows Werewolf is going to offer that often, Mafia shouldn't offer at all. And vice versa. So it's another Prisoner's Dilemma situation within the problem.

This is going to give me a headache.

[mith]

Lynch (probability p):
m/(2+m+w): Lynch Mafia, Werewolf win
w/(2+m+w): Lynch Werewolf, Mafia win
2/(2+m+w): Lynch Town, 1:1:1 (EV 1/4)
Town EV 1/2(2+m+w); Mafia EV w/(2+m+w) + 3/4(2+m+w) = (3+4w)/4(2+m+w); Werewolf EV (3+4m)/4(2+m+w)

That means if we were always going to lynch, m->0 and w->0, as expected.

[mith]

Rate of self-lynch offer:
Mafia: m
Werewolf: w
Town: 1

Lynch (probability p):
m/(2+m+w): Lynch Mafia, Werewolf win
w/(2+m+w): Lynch Werewolf, Mafia win
2/(2+m+w): Lynch Town, 1:1:1 (EV 1/4)
Town EV 1/2(2+m+w); Mafia EV w/(2+m+w) + 3/4(2+m+w) = (3+4w)/4(2+m+w), counting the draw as 1/2; Werewolf EV (3+4m)/4(2+m+w)

Don't lynch offer (probability 1-p):

After the second lynch offer (which is carried out), the probabilities are:
m/(1+m+w): Mafia "confirmed"
w/(1+m+w): Werewolf "confirmed"
1/(1+m+w): Town "confirmed"

Going into night, consider the perspective of the Mafia player if not in the "confirmed" position (probability (1+w)/(1+m+w)). From this player's perspective, there is a 1/(1+w) chance the Werewolf is "not confirmed" and a w/(1+w) chance the Werewolf is "confirmed". It is at least as good for the Mafia to shoot the "not confirmed" player (it is the same if w=1).

The same goes for the Werewolf player in a "not confirmed" position.

If a scum player is in the "confirmed" position, the choice between the two "not confirmed" players is random. This our EVs are:
m/2(2+m+w): Mafia win (confirmed and shoots Werewolf)
m/2(2+m+w): Mafia/Werewolf draw (Mafia confirmed and shoots Town)
w/2(2+m+w): Werewolf win (confirmed and shoots Mafia)
w/2(2+m+w): Mafia/Werewolf draw (Werewolf confirmed and shoots Town)
2/(2+m+w): Town win

The Town's overall EV is: p/2(2+m+w) + 2(1-p)/(2+m+w).

Optimal p appears to be 2/3, with EV of 1/2 - there is actually a second "Prisoner's Dilemma" situation in terms of the m and w values. At 2/3, if w=0, EV for Mafia is 1/4 for any m. If m=w=1, EV for each is 5/12... but if w=1, Mafia does better with m=0 (17/36). For any w>0, Mafia EV is max at m=0, and vice versa, so both should pick 0.

(This isn't quite a proof that p=2/3 is optimal, but for p>=2/3, the above argument applies, and m=0 and w=0 are optimal for the scum factions; given the town EV is better when the self-lynch offer is declined, you want p to be as small as possible for fixed m and w.

For p<2/3, optimal m and w are greater than 0, and the resulting EV seems to always be worse for town. For example, at p=1/2, optimal m=w=3/4, and town EV is only 1/4 again.

For p between 6/13 and 2/3, optimal m=w is (6-9p)/4p, and resulting town EV is (4p-3p^2)/(6-5p) which is increasing over this interval. For p<6/13, m=w=1 and the argument for who they shoot at night breaks down; note that at p=0, m=w=1, town EV is again 1/2 in the above formula - more on this in next post. For p>2/3, EV is (4-3p)/4, so we want p to be small.)

Town also has the option of choosing between the first self-lynch offer and no lynch, and in fact I think this is exactly as good - the cases at night are:

Cross-kill (1/4) - Town win
Mafia kill Werewolf kill Townie - Mafia win
Werewolf kill Mafia kill Townie - Werewolf win
Both kill Townie - 1:1:1 with remaining townie confirmed, Prisoner's Dilemma Town win.

You could also try having two "simultaneous" self-lynch offers, and then randomly choose one to lynch and the other to consider confirmed. In this case (since p=1/2, in some sense), optimal m and w are greater than 0, but the change in strategy may counteract the loss of control.

Anyway, if the 1/2 EV result holds, this is in some sense the perfectly balanced setup. (In another sense, 1:1:3 is, as all factions have an EV of almost exactly 1/3.)

[mith]

As noted above, the argument breaks down if p=0. However, I think the calculated EV is actually still correct, and this results in a simplification of the method:

1:1:2
Pick a player at random (effectively every player offers with equal probability, knowing that p=0 and there won't be a self-lynch accepted). Then have a different player offer a (true) self-lynch (this player will always be town).
It would seem at first glance that we are left with 1:1:1 and the factions are equally likely to have been picked, but this is actually an application of the Monty Hall problem - the probability of a town player being picked randomly is 1/2 because there were two townies to start with. Effectively, the fact that the player offering the self-lynch will always be a different player, we have gained information about the likelihood of that player being town.

From here, the argument proceeds as above - both Mafia and Werewolf (if not picked) will reason that the not-picked player is more likely to be scum, so if Town is picked scum kill each other and otherwise the picked faction wins or draws. EV 1/2.

[mith]

In short:

1:1:1 - EV 1/4
1:1:2 - EV 1/2
1:1:3 - EV ~1/3

Which means it's one of those weird situations where the town would benefit from a non-day-ending modkill on a townie (though of course that townie would not benefit).

(In fact! Since the EV of 1:1:3 is slightly less than 1/3, it would actually be beneficial on average for the town to agree to randomly pick a player who will get modkilled. If scum is picked, it reduces to 1:2 after no lynch and town EV is 1/3; if town is picked, there is a 2/3 chance of a particular townie not being picked and a 1/2 EV at 1:1:2, so their personal EV is now 1/3 exactly as well. Of course, once the player has been randomly picked, they are now playing against their win condition to actually follow through, regardless of their alignment. But amusing situation, anyway.)

[mith]

1:1:4

Lynch:
1/6 - EV 1/3
1/6 - EV 1/3
2/3 - 1:1:3 Night EV 3/8
13/36

1:1:3 Night
1/16 - EV 1
3/16 - EV 1/3
3/16 - EV 1/3
6/16 - 1:1:1 EV 1/4
3/16 - 1:1:2 EV 1/2

6/16 = 3/8

Self-lynch:
1:1:3 Night EV 3/8

No Lynch:

1:1:4 Night
1/25 - EV 1
4/25 - EV 1/3
4/25 - EV 1/3
12/25 - 1:1:2 EV 1/2
4/25 - 1:1:3 EV 21/64
EV 527/1200 = ~44%

So No Lynch is again better, unless we can improve by a self-lynch offer strategy again. (Which we might be able to do. I suspect p is lower here - scum offering and being lynched is way more damaging to their chances than town offering and being lynched. And in the 1-p where we don't lynch the offer, that "confirmed" townie is really strong.)

[mith]

Finally put the 1:1:X in a spreadsheet, and lynching does become viable after this point.

The pattern is interesting (parity considerations, basically):

1:1:1 - EV 25% (No Lynch)
1:1:2 - EV 50% (Other, see above)
1:1:3 - EV 37.5% (No Lynch; note, this is improved, I forgot to take into account the improved 1:1:2 result)
1:1:4 - EV 44.7% (No Lynch; ditto for the now improved 1:1:3)
1:1:5 - EV 45.2% (Lynch)
1:1:6 - EV 46.6% (No Lynch)
1:1:7 - EV 48.2% (No Lynch)
1:1:8 - EV 49.4% (Lynch)
1:1:9 - EV 50.8% (No Lynch)
1:1:10 - EV 52.3% (Lynch)
And lynching is always better after this point.

Note that this assumes there is no "other" strategy after 1:1:2; given how long it takes to get back to 50% (1:1:2 is better than 1:1:8!), I'd be surprised if there weren't some shenanigans for town to take advantage of at higher counts.

[mith]

Strategy for 1:1:3...

Pick two players at random. Have a self-lynch offer from the other three, and carry out the lynch.

3/10: Town and Town were picked
3/10: Mafia and Town were picked
3/10: Werewolf and Town were picked
1/10: Mafia and Werewolf were picked

If Mafia was picked, there is a 1/4 probability of the Werewolf being the other picked player, leaving a 3/8 for each of the other slots. Likewise for the Werewolf picked case.

If Mafia was not picked, there is a 1/2 probability of the Werewolf not being picked, vs. 1/4 for each of the picked slots.

In each case, Mafia and Werewolf are better off shooting among the not picked players. This means:

3/10: Town and Town were picked - Cross-kill, Town wins
3/10: Mafia and Town were picked - Werewolf kills Town, Mafia kills Werewolf or Town with equal probability; 1/2 1:1:1, 1/2 Mafia win.
3/10: Werewolf and Town were picked - Mafia kills Town, Werewolf kills Mafia or Town with equal probability; 1/2 1:1:1, 1/2 Werewolf win.
1/10: Mafia and Werewolf were picked - 1/2 Mafia/Werewolf draw, 1/2 1:1:1

If we get to 1:1:1, it is still more likely that one of the picked players is Town (from the perspective of picked scum, 3/4 again), so the picked scum should shoot not-picked. If Town was picked, the not-picked scum has a 1/2 chance of hitting the other scum (Town win) and 1/2 of hitting Town (picked scum win):

3/10: Town and Town were picked - Cross-kill, Town wins
3/10: Mafia and Town were picked - Werewolf kills Town, Mafia kills Werewolf or Town with equal probability; 1/2 Mafia win, 1/2 1:1:1 -> 1/2 Town win, 1/2 Mafia win.
3/10: Werewolf and Town were picked - Mafia kills Town, Werewolf kills Mafia or Town with equal probability; 1/2 Mafia win, 1/2 1:1:1 -> 1/2 Town win, 1/2 Werewolf win.
1/10: Mafia and Werewolf were picked - 1/2 Mafia/Werewolf draw, 1/2 1:1:1 -> Mafia/Werewolf draw

So the overall EV of the middle cases is 1/4, and the total EV is 3/10 + 6/10*1/4 = 9/20 (45%).

[mith]

It occurs to me that the X:1:Y EVs should be the same for Mafia as the corresponding setup with a Vigilante instead of a Werewolf.

The above 1:1:3 strategy has a Mafia EV of 11/40 (27.5%), which would mean a 72.5% Town EV in the 1:1:3 Vigilante setup; I had it above that at some point (for a no lynch strategy), so I'll have to figure out where the mismatch is.

This is almost true. It works out in every case except 1:1:1, because town can lynch in the Vigilante setup but can't lynch in the Werewolf setup.

I think it works if the Vigilante setup considers 1 Mafia vs. 1 Vigilante a Mafia win (50%) rather than a draw? Anyway, not really relevant at this point, just thought it was interesting. The strategy for 1:1:2 is basically equivalent whether it's a Werewolf or Mafia (the key at this count is that we never end up with 1:1:1 day with no information).

[mith]

Vanilla Multiball Nightless:

1:1:4 - 40%
1:1:5 - 48%
1:1:6 - 54%
1:1:7 - 58%

2:2:6 - 31%
2:2:7 - 36%
2:2:8 - 41%
2:2:9 - 44%
2:2:10 - 48%

3:3:9 - 31%
3:3:10 - 35%
3:3:11 - 38%
3:3:12 - 41%
3:3:13 - 43%
3:3:14 - 46%
3:3:15 - 48%
3:3:16 - 50%

4:4:15 - 39%
4:4:16 - 41%
4:4:17 - 43%

Any setup with only 1 Townie is a Town loss; initially I had thought 1:1:1 would have to be a draw (EV 1/3 for each), but Mafia and Werewolf both have incentive to lynch - 1/3 of the time they lose, 1/3 of the time they draw with other scum (counting as EV 1/2), 1/3 of the time they win outright, for an EV of 1/2. Basically they are risking losing in order to cut the Town out of the EV pie entirely.

[mith]

(Interestingly, X:X:cX has roughly the same balance for X=1 to 4; 40% exactly at 1:1:4, up to 40.7% at 4:4:16, for c=4. I suspect X:X:cX setups will converge to the 2X:cX Nightless value for large X., so X:X:4X would converge to 50% exactly. Far too lazy to prove that right now, though.)

[mith]

(Not sure what X:X:2X is converging to though. Possibly 20%, which is the EV for 2X:3X; 2X:2X has EV of 0 of course. X:X:X is also greater than 0 for X>1.)

[mith]

2:1:2 looks like 5/36 after no lynch (vs. 1/15 after lynch; 1:1:2 is only 1/6 going into Night, can't take advantage of that sweet self-lynch action).

When there is only 1 of the weaker faction, there is no benefit to claiming; that faction loses either way once they are on the chopping block. This will change at 3:2:X, though.

[mith]

2:1:3 is 49/240 (20.4%) for no lynch, vs. 7/36 (19.4%) for lynch.

[mith]

If my spreadsheet is correct (it at least matches what I got by hand for the two above), no lynch is better at every count except 2:1:5.

For 2:2:X, no lynch is only better at 2:2:2 and 2:2:7. Don't hit 50% town EV until 2:2:33, which I find somewhat surprising. 2:2:11 is 34%.

[mith]

One easy-to-make adjustment is for the 2:2:2 case. (Or any X:X:2, actually.) The strategy is the same as 1:1:2 - pick a player at random, and then have someone else offer themselves as the lynch. Both scum groups will have better luck shooting at the non-picked player in the resulting Prisoner's Dilemma, so EV is 1/3 (or 1/(X+1), for the more general case).

Doesn't have much effect on the overall balance at higher counts. But 2:2:3 is currently sitting at a mere 17.9%, and I'm sure that can be improved as well (see post 10).

[mith]

Vengescum Multiball (last scum of a group can't vengekill):

2:2:7 - 32%
2:2:8 - 36%
2:2:9 - 40%
2:2:10 - 44%
2:2:11 - 47%
2:2:12 - 50%

3:3:11 - 28%
3:3:12 - 30%
3:3:13 - 33%
3:3:14 - 36%
3:3:15 - 38%
3:3:16 - 41%
3:3:17 - 43%
3:3:18 - 45%
3:3:19 - 47%

Not wildly different from Nightless Multiball, which makes sense.

[mith]

See the Vanilla Variant thread for the Nightless formula - I never added the proof, it's worth doing as an exercise.

I'm not sure what you mean about reaction times - you did remind me that 1:1:1 should be considered a scum draw, because they can agree to try to lynch the Townie rather than going to night; Town can go along with it and basically serves as kingmaker as you mention later. (Anyway, 1 Mafia and 1 Werewolf going into night should be considered a draw between those factions; any ruleset saying otherwise has issues. And I say that having argued in game a long time ago as town in a 1:1:1 that if they agreed to lynch me they should be forced to play a night and only get the draw if they both refused to kill.)

I agree that 1:1:2 is a draw if alignments are known. However, if alignments are not known, scum have a sort of asynchronous Prisoner's Dilemma situation. If they both claimed, alignments would be known. However, if one claims first (say Mafia), it is now in the Werewolf's interest to go to night. Werewolf would prefer to no lynch (only 1/3 for Mafia to get the cross kill, town win), but a town self-lynch makes the EV for both sides 1/2 and Werewolf doing anything other than going along with this plan makes his alignment known and we're back to a draw, so town self-lynch will happen and 1/2 Town, 1/2 Werewolf. So, returning to the claim, neither can claim and they have to go along with the best Town plan for fear of being outed and losing outright, so my pick-a-player-and-then-lynch-town-from-the-others plan should succeed with optimal play all around, 1/2 Town, 1/4 Mafia, 1/4 Werewolf.

(Man, that is one complicated 4 player setup. I'm tempted to run a series of them as a psychology experiment.)

I'll adjust my numbers based on the 1:1:1 EV 0 tomorrow. Regardless, I definitely don't think 1:1:1 (or 1:1:2) should be considered town wins (unless it's White Flag; but then you probably have similar issues at 2:2:1 and 2:2:2?).

[mith]

Ah, yeah, that's clever. And also not necessarily applicable to FTF, if scum can just vote simultaneously. I don't think I've ever modded or played in a game where this wouldn't have been allowed.

Anyway, applying that, if alignments are unknown scum can't claim because of the same issue as 1:1:2; in this case, town can threaten to vote the first claim. So yeah, everyone pretends to be town and we go to night. I'm happy with 1/4 being correct again.

[mith]

Finally put together a script to do this for higher counts. These are the results with the 1:1:2, 1:1:3, and 2:2:2 improvements included (I'm including the first setup with EV > 1/3, the closest setups to EV = 2/5 and EV = 3/7, and the last setup with EV < 1/2):

2:2:9 (33.6%)
2:2:15 (40.2%)
2:2:18 (42.9%)
2:2:28 (49.5%)

3:3:20 (33.8%)
3:3:30 (40.0%)
3:3:36 (42.9%)
3:3:56 (50.0%)

4:4:32 (33.7%)
4:4:48 (39.9%)
4:4:57 (42.7%)
4:4:88 (49.9%)

5:5:46 (33.5%)
5:5:69 (40.1%)
5:5:82 (42.9%)
5:5:125 (49.9%)

[mith]

I think my next run of the script is going to be something like Jungle Republic - multiball setups where each group plays with "Vanilla Variant" rules but not necessarily the same rule.

[mith]

One interesting thing I found generating the above numbers is that for a general setup X:Y:Z where 0<=Y<=X, town EV is not necessarily maximized at Y=0 or Y=X. For example:

2:0:9 - 0.3520923521
2:1:9 - 0.3790586343
2:2:9 - 0.3362550955

This isn't universally the case though:

2:0:28 - 0.5730082052
2:1:28 - 0.5546413745
2:2:28 - 0.4952180666

[mith]

Posting to answer shos' old question:

I may make my script available at some point (I haven't looked at it in a while, and would want to make it more user-friendly). It is only useful for setups with no role abilities, or with very limited (both in number and in options) role abilities. For example here is an example of caluclating EVs with a single Vigilante with the Innocent Child modifier (to avoid fake/counter claiming strategies); a subsequent post allows fake claims but makes some assumptions about both Mafia and Town strategy in the process.

The more significant improvements in EV calculation come with the really small setups that the script isn't useful for anyway (such as 2 Mafia, 1 Vigilante, 2 Townies). It's hard to prove that a particular claim/lynch strategy is optimal, but improvements affect larger setups that can reduce to the small setup.

[mith]

I think the confusion here is from me being lazy with brackets and order of operations.

In the 1:1:2 setup:

If scum is lynched first (which has probability of w/(2+m+w) or m/(2+m+w), where m and w are the likelihood that Mafia and Werewolf would offer themselves as the lynch), the other scum wins (EV 0 for town, EV 1 for that scum group).
If town is lynched first (which has a probability of 2/(2+m+w), where m and w are the likelihood that Mafia and Werewolf would offer themselves as the lynch), town's EV is 1/4 (it's a 1:1:1 Night with unknown alignments, and town only wins if Mafia and Werewolf shoot each other). Mafia and Werewolf each win 3/8 of the time (counting the both-shoot-town as a draw and EV 1/2). So town EV is 1/4 * 2/(2+m+w) = 1/(2(2+m+w)), since this is the only way they can win.

Mafia EV is 1 * w/(2+m+w) + 3/8 * 2/(2+m+w), with the first part coming from Werewolf lynch and the second from town lynch, this reduces to (3+4w)/(4(2+m+w)). Similarly for the Werewolf EV.