[EV] Vanilla Multiball

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Post Post #0  (ISO)  » Mon Sep 12, 2016 12:09 pm

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.

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Post Post #1  (ISO)  » Tue Sep 13, 2016 5:41 pm

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%

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Post Post #2  (ISO)  » Tue Sep 13, 2016 6:06 pm

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.

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Post Post #3  (ISO)  » Tue Sep 13, 2016 6:23 pm

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.

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Post Post #4  (ISO)  » Tue Sep 13, 2016 6:27 pm

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.

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Post Post #5  (ISO)  » Wed Sep 14, 2016 3:21 pm

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.)

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Post Post #6  (ISO)  » Wed Sep 14, 2016 3:44 pm

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.

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Post Post #7  (ISO)  » Wed Sep 14, 2016 3:49 pm

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.)

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Post Post #8  (ISO)  » Wed Sep 14, 2016 4:04 pm

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.)

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Post Post #9  (ISO)  » Wed Sep 14, 2016 6:41 pm

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.

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Post Post #10  (ISO)  » Thu Sep 15, 2016 10:54 am

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%).

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Post Post #11  (ISO)  » Thu Sep 15, 2016 1:52 pm

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).

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Post Post #12  (ISO)  » Thu Sep 15, 2016 3:19 pm

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.

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Post Post #13  (ISO)  » Thu Sep 15, 2016 3:24 pm

(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.)

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Post Post #14  (ISO)  » Thu Sep 15, 2016 3:34 pm

(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.)

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Post Post #15  (ISO)  » Thu Sep 15, 2016 5:13 pm

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.

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Post Post #16  (ISO)  » Thu Sep 15, 2016 5:18 pm

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

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Post Post #17  (ISO)  » Thu Sep 15, 2016 5:51 pm

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%.

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Post Post #18  (ISO)  » Thu Sep 15, 2016 6:23 pm

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).

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Post Post #19  (ISO)  » Thu Sep 15, 2016 6:41 pm

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.

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Post Post #20  (ISO)  » Thu Sep 15, 2016 9:12 pm

I missed this thread earlier.

I believe 2:1:1 is mathematically a 3 faction draw if the players' alignments are known; either scum can no-kill to force the other to no-kill (because if the other scum shot, then town would lynch them in response), and town doesn't want to lynch scum and scum don't want to lynch anyone, thus nothing can happen during the day (theoretically, a townie would benefit from being lynched, but the scum won't vote for it and 2 votes are not enough).

1:1:1, on the other hand, is broken. There's some debate as to how it's broken, and it depends a lot on the details of the ruleset. (In particular, in face-to-face play that uses mafiascum.net hammer rules, any attempt by scum to vote for town will cause the game to be decided by the players' reaction times.)

I think it may be best to explicitly rule 1:1:1 and 2:1:1 town wins in multiball in order to avoid the problems with endgame breakage, in which case this will throw off the probabilities somewhat.

Nightless and Vengescum setups don't have these issues, so analysing the EV is more meaningful. (That said, in nightless 1:1:1, the townie has no chance to win and kingmaker ability, which is normally considered broken.)

By the way, although I haven't proven it yet, I suspect that the EV for a vanilla Nightless always stays much the same if you multiply every faction's member count by the same factor. This probably holds at least approximately true even in multiball. As such, you can approximate the various faction EVs via dividing the setup down to a :1 setup (e.g. 3x:x has a 50% EV because 3:1 does).
scum · scam · seam · team · term · tern · torn · town

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Post Post #21  (ISO)  » Thu Sep 15, 2016 11:28 pm

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?).

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Post Post #22  (ISO)  » Thu Sep 15, 2016 11:58 pm

In post 12, mith wrote: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.


Surely this isn't right given that it's never right for town to vote? The only person that can be lynched is the town which causes an auto draw.
Taking a break from the site.

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Post Post #23  (ISO)  » Fri Sep 16, 2016 5:22 am

In 1:1:1 Nightless, town has no mathematical chance to win. Therefore, it can't be wrong to vote or not vote; whatever they do doesn't affect their own victory chances.

The reaction time argument for 1:1:1 daystart goes like this: town can prevent scum A voting for them by voting for scum A in response (under the assumption that if the townie gets lynched, they lose anyway, and therefore they have no reason not to vote for someone to force them to unvote). If scum A unvotes, the townie then wants to unvote/vote no-lynch; however, if scum B is fast enough, they'll hammer scum A before the townie has a chance to unvote.

I believe that in 1:1:1 daystart with unknown alignments, everyone's best option is to pretend to be town (and thus will vote no-lynch and crossvote a vote on them, both optimal moves as town). Thus, you'd expect no lynch to be hammered very quickly, as not doing so would be a scumclaim. If alignments are known, I can see scum attempting to go for the draw, but doing so is very risky because if the other scum's reactions are faster than either yours or the townie's, you'll end up losing as a result.
scum · scam · seam · team · term · tern · torn · town

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Post Post #24  (ISO)  » Fri Sep 16, 2016 8:24 am

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.

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