[OLD] The Balance Quiz

[mith]

After numerous discussions of EV balance in various Mafia setups, I started wondering last night about perception vs. reality when it comes to balance. So, I've decided to put together a little quiz. I have a spreadsheet which currently has 7 different Vanilla-variants (games with no "roles" except for Mafia and Townie, but with various way of handling lynches and the day/night cycle).

To take the quiz, read the definitions, read the questions, and write down your answers

without doing any calculations or looking at other threads

. Feel free to explain your thought process, particularly if your answer is based on something you remember being discussed, or on calculations you have personally done previously. When you're done, post your answers/discussion in this thread (in spoiler tags, preferably).

All calculations assume random lynches (theoretical "optimal play" by both sides); in practice, we expect the Town to, on average, outperform the EV by some degree.

Definitions

In all setups with Night phases, Mafia must kill; Town has the option of No Lynch when applicable. All games start in Day. Except where otherwise noted, the Town's Win Condition is to eliminate all the Mafia, and the Mafia's Win Condition is to eliminate the Town.

X:Y [VARIANT] - A game of [VARIANT] type with X Mafia and Y Townies.

Vanilla - The basic game of Mafia; Town chooses someone to lynch, Mafia chooses someone to kill.

Nightless - A game with no Night phase, only lynches. The Mafia's Win Condition in these games is to comprise 50% of the Town.

PairedLovers - A game in which the Mafia are in Lover pairs; if one member of the pair is lynched, the other commits suicide.

PolyLovers - A game in which all the Mafia are Lovers with each other; if one is lynched, the rest all die.

Treestump - A game in which the game only moves to Night if a Mafia member is lynched (this is equivalent to a game in which all the pro-Town players have the Tree Stump role).

White Flag - A game with alternate Win Conditions: The Town needs to lynch all-but-one of the Mafia, and the Mafia needs to comprise one third of the Town.

Double Day - A game in which the Town gets (up to) two lynches each Day.

Questions

1. What is the Town EV (probability of the Town winning) for each setup?

a. 1:11 Vanilla
b. 2:10 Nightless
c. 3:9 Double Day
d. 4:8 PairedLovers

2. In each setup, what does X need to be in order for a 10:X setup to be as close to a 50-50 balance (each side has a 50% chance of winning) as possible?

a. Vanilla
b. Nightless
c. PairedLovers
d. PairedLovers Nightless
e. PolyLovers
f. Treestump
g. White Flag
h. Double Day

3. Put the following setups in order from lowest Town EV to greatest Town EV (least likely for the Town to win to most likely):

a. 2:5 PairedLovers
b. 2:10 Double Day
c. 3:9 Treestump
d. 3:32 Vanilla
e. 4:7 Nightless
f. 4:8 PairedLovers Nightless
g. 5:12 PolyLovers
h. 5:64 White Flag

[Empking]

44%
94%
44%
30%

250
500
250
125
100
125
125

Just in case its a trick, I'll say town has already lost.
aecbfgdh

[mith]

Empking, you're getting the numbers backwards; the first number is the number of Mafia, the second is the number of Townies. So question 2 is asking how many Townies you need in each variant to balance a group of 10 Mafia.

[Xdaamno]

What does "without doing any calculations" mean? You can mentally run through the possible outcomes, right?

[mith]

I mean, don't open up Excel or get out pencil and paper and calculate the exact probabilities. I want to know what your gut says; I know what the actual answer is. (You can sit there and think about it all you like, though.)

[Xylthixlm]

Gut says...

1a: 35%
1b: 65%
1c: 45%
1d: 30%

2a: 200
2b: 30
2c: 60
2d: 20
2e: 11
2f: 45
2g: 300
2h: 80

3: e, d, h, b, c, f, a, g

[Empking]

You did G twice.

[Shanba]

Gut, eh?

1 A - town get a lot of chances of winning, but I seem to recall some thread that suggested that adding lots more townies actually doesn't help town's EV that much. I'd guess about 65 %

B - Ouch. I'd guess about 70 %. Wouldn't like to be scum, there, though I doubt it's

that

much higher than in A

C - That's probably not massively unbalanced - I'd guess it's about 50/50. Conventional wisdom suggest 3/9 vanilla in unbalanced in favour of town, but doulbe day would probably even that out.

D - Short game, I'd guess about balanced - actually, this is pretty similar to 2/10 vanilla, but the extra scum for endgaming and vote purposes probably ups their winning percentage, plus town have two less townies. 55%

2 - A) we do 2 - 10 vanilla as roughly balanced. Wonder whether the amount of townies needed exponentially or geometrically increases. Go with geometrically plus some - 75

B) Far less. 40

C pairedlovers is similar to having half the amount of scum - a good approximation probably suggests about 30 town.

D both factors require less town, but town still needs to outnumber scum by a fair margin. 20

E Heh. 11 gives a 10/21 chance of town winning, 13 gives a (1- 13/23*11/21) chance. I reckon 11 is probably closer to balanced, without doing the calcs in my head, as 13 will tend to around 3/4 since both are approx 1/2. This wouldn't be balanced in practice, though, since scum have no need to ever vote for scum.

F Treestump is slightly worse than nightless from a pure maths perspective (though prob better in other, intangible ways.) 45.

G) Urgh. Lots. 80.

H Worse than nightless, better than nilla. 55.

3 - not sure. d,b,a,f,h,c,e,g.

[PokerFace]

<<As a guy who has never moderated or calculated these thing before I obstain from this test

<<As a guy that absolutly loves power roles with irc and does like some swingy as fuck setups, I further abstain

[Max]

1. What is the Town EV (probability of the Town winning) for each setup?

a. 1:11 Vanilla - 55%
b. 2:10 Nightless - 70%
c. 3:9 Double Day - 80%
d. 4:8 PairedLovers - 40%

2. In each setup, what does X need to be in order for a 10:X setup to be as close to a 50-50 balance (each side has a 50% chance of winning) as possible?

a. Vanilla - 60
b. Nightless - 50
c. PairedLovers - 40
d. PairedLovers Nightless - 30
e. PolyLovers - 12 (Nightful I presumed (Chance of town D1=12/22 only one day no matter who gets lynched)
f. Treestump - 16
g. White Flag - 15
h. Double Day - ???

3. Put the following setups in order from lowest Town EV to greatest Town EV (least likely for the Town to win to most likely):

b. 2:10 Double Day
e. 4:7 Nightless
d. 3:32 Vanilla
f. 4:8 PairedLovers Nightless
g. 5:12 PolyLovers
c. 3:9 Treestump
h. 5:64 White Flag
a. 2:5 PairedLovers

[petroleumjelly]

1.)

a.) 1:11 vanilla. I would probably suggest a No-Lynch to start and then lynch randomly 5 times. Town probably wins this about 65% of the time. I personally see little point in playing mafia when there is only one member of the mafia so I can’t imagine there would be much actual scum-hunting. (There are exceptions, such as Assassins in the Palace.)
b.) 2:10 nightless. Town gets to lynch at least 8 times and possibly 10 times. I think town will win this 77% of the time. That is – simply put – a lot of lynches to avoid. The saving grace here is that if one scum can convince everybody s/he is town, they do not have to deal with the “why haven’t you been nightkilled?” argument. On that same note, though, if actual

townspeople

attain that status then the scum are going to have a tough battle. I threw in an extra 2% win chance for the town on the presumption that the mafia never gets to coordinate, since they will never get to talk with each other except before the game begins.
c.) 3:9 double-day. This depends on whether alignments are revealed with each lynch or at the end of each day. In the case of instant alignment-reveal, I would say town has 50% chance of winning. In delayed alignment-reveal, I would say 40% chance.
d.) 4:8 paired lovers. In a very real sense this set-up prevents a lot of “bussing” strategies; for every mafia member there is a partner they simply cannot afford to bus. This game almost necessitates the town into wanting

looong

days to look for this exact type of reluctance/avoidance. This is a strange set-up, however, because the town could be in LYLO by Day Two. If the town lynches a mafia on D1 or D2, I think town wins 70% of the time. This is contrasted with the 0% for failing to lynch mafia D1 or D2. I think overall the town wins 35% of the time.

2.)

a.) Vanilla. 10:80.
b.) Nightless. 10:40.
c.) Pairedlovers. 10:35. (Assuming 10 members of the mafia with 5 ties).
d.) Pairedlovers nightless. 10:25.
e.) PolyLovers. 10:13.
f.) Treestump. 10: 33.
g.) White flag. 10:70
h.) Double day. (a) 10:25 for immediate reveal. (b) 10:30 for delayed reveal.

3.) Lowest Town EV to greatest Town EV.

e. 4:7 Nightless.
g. 5:12 PolyLovers
f. 4:8 PairedLoversNightless.
c. 3:9 Treestump.
b. 2:10 DoubleDay. (Regardless of which variant).
a. 2:5 PairedLovers.
d. 3:32 Vanilla.
h. 5:64 White Flag.

Big big big note: This is all assuming play

on a forum

where people do not have internet access at all times. I think a PolyLovers game in particular changes quite drastically when considering forum v. face-to-face, and is also very contingent on how votes are counted. That type of set-up almost necessarily results in the mafia wanting to speedlynch.

[mith]

For Double Day, my numbers assume that lynches are revealed as they happen; in the random EVs, it only potentially matters in situations where the town might decide to just lynch once depending on whether they hit town or scum the first time (not sure if this ever happens, didn't check yet).

Interesting responses so far, particularly in the "how would this setup realistically play out?" department. Keep in mind that the "correct" answers are assuming random lynches.

[petroleumjelly]

I was trying to hit what I thought the chances are in practice. I'm too lazy to bother to change my answers, but for others' sake:

Would you please clarify by what you mean by "random lynch"? There seem to be a good deal of ways to compute that...

1.) Randomly generate who is lynched.
2.) Randomly generate who a person would vote other than themselves.
3.) Randomly generate who a person would vote other than themselves or partners.

... and so on.

[mith]

The first.

[GIEFF]

Questions

1. What is the Town EV (probability of the Town winning) for each setup?

a. 1:11 Vanilla -

60%.

b. 2:10 Nightless - Assuming 1 scum dies before it gets to 5-player lylo, I'd say 85% town,because 83% of the players are town to start with. To get to 5-player lylo, 5 townies in a row need to die, which is about 25% chance, and scum has a very good shot to win at that point, so I'll take 20% away from the 85% to reach

65%

.
c. 3:9 Double Day.

40%

. No idea.
d. 4:8 PairedLovers.

60%

. No idea.

2. In each setup, what does X need to be in order for a 10:X setup to be as close to a 50-50 balance (each side has a 50% chance of winning) as possible?

a. Vanilla - 300 (practically much less, as buddies will be found)
b. Nightless - 50 (practically much less, as buddies will be found)
c. PairedLovers - 100
d. PairedLovers Nightless - 25
e. PolyLovers - 11, technically, but it would play differently. Probably 13 with practical considerations (i.e. it is less likely than 10/21 that a scum will be lynched even if there are 10 scum out ouf 21 players, because scum-lynch = scum-loss)
f. Treestump - 55 (based on nightless guess of 50). If town wins, there will be 9 more dead townies than in nightless(9 more night phases)
g. White Flag - 250. No idea. I'm assuming that town needing to lynch one fewer scum more than offsets the ~5 fewer townies the scum need to get rid of to win.
h. Double Day - 150. Probably halfway between Vanilla and Nightless.

3. Put the following setups in order from lowest Town EV to greatest Town EV (least likely for the Town to win to most likely):

g
h
f
c
a
b
d
e

[mith]

Will post answers tomorrow.

[mith]

Alright, here are the answers...

1a. 1:11 Vanilla EV = 63.06%. Most did pretty good on this one, no surprises there.
1b. 2:10 Nightless EV = 66.67% (2/3). General formula for Nightless is [T-M]/[T+M].
1c. 3:9 Double Day EV = 29.31%. Nightless would be 50-50, Double Day is between Vanilla and Nightless.
1d. 4:8 PairedLovers EV = 37.46%.

2a. 10:421 Vanilla has an EV of 49.99%. Some of you guessed that it was high, no one guessed that high. The number of Townies needed to balance a Vanilla game is O(M^2) (that is, it grows quadratically).
2b. 10:30 Nightless has an EV of 50% exactly.
2c. 10:31 PairedLovers has an EV of 49.97%. I was surprised these two were as close as they are... PairedLovers looks roughly linear, like Nightless, with a slightly different ratio. Want to investigate this a bit further.
2d. 10:13 PairedLovers Nightless has an EV of 55.76%; 10:12 has an EV of 44.17%. Probably the most surprising of the bunch to me. With X (even) Mafia, you need X+2 or X+3 Townies to get EVs on either side of 50% for X>6. For smaller X, 2:3 (40%), 2:4 (60%), and 4:6 (50.48%) are the best fits. For a given "Townie Surplus", the EV drops until X = 12, and then starts to climb again; I didn't track it out far enough to see whether it converges or if eventually X:X+1 is the most balanced.
2e. 10:11 PolyLovers has an EV of 47.62% (10/21); 10:13 has an EV of 70.39%. Several of you saw that since the Town only has to lynch correctly once, giving them more than one chance will swing this in their favor.
2f. 10:68 Treestump has an EV of 50.05%. Quadratic growth, but much slower than Vanilla.
2g. 10:227 White Flag has an EV of 50.14%; 10:225 has an EV of 49.86%. When Scum win big, they will have to lynch much fewer Townies, but most of the time they'll only get a small bonus; not enough to outweigh the benefit the Town gets from not having to lynch one of the Scum. Still quadratic.
2h. 10:118 Double Day has an EV of 50.05%. Still quadratic.

3. ECDBHAFG

27.27% - 4:7 Nightless
38.69% - 3:9 Treestump
45.47% - 3:32 Vanilla
49.88% - 2:10 Double Day
54.08% - 5:64 White Flag
57.14% - 2:5 PairedLovers
70.51% - 4:8 PairedLovers Nightless
84.20% - 5:12 PolyLovers

Had to include 4:7 Nightless, since I've noticed a rather common misconception about the Town's chances in Nightless games. 3:9 Treestump has also been run before, and 2:5 PairedLovers has at least been suggested (has it been run?). 3:32 Vanilla again demonstrates how many Townies you need to get a 50-50 balance in a Vanilla game... I think most would argue, however (and I'd agree) that the Town would be favored in such a game realistically.

[Thok]

Had to include 4:7 Nightless, since I've noticed a rather common misconception about the Town's chances in Nightless games. 3:9 Treestump has also been run before, and 2:5 PairedLovers has at least been suggested (has it been run?). 3:32 Vanilla again demonstrates how many Townies you need to get a 50-50 balance in a Vanilla game... I think most would argue, however (and I'd agree) that the Town would be favored in such a game realistically.

Given my intuition, this suggests that town benefits from the lack of nightkills in games like 4:8 Nightless and 3:9 Tree Stump (and the continuing participation of dead players in actual Tree Stump.) Basically, if a couple people can convince the town that they are townies and that they shouldn't be lynched, there's nothing scum can do about that.

Of course, that's "real life" which this doesn't quite model accurately.

[petroleumjelly]

Gah; for some reason I read the 5:12 poly-lovers set-up to mean "5 out of 12" and thinking one wrong lynch meant town lost. Giving town

4

lynches makes it easily the most winnable.

Also, I went for 10:13 PolyLovers rather than 10:11 mostly because I figured town almost auto-loses in 10:11,

especially

in a face-to-face game with "simultaneous" voting... this caused me to bump it to 10:13, even though even in face-to-face simultaneous this obviously gives town the edge.

</needs to learn to read>

I would like to see some White Flag games run.

[Yosarian2]

2e. 10:11 PolyLovers has an EV of 47.62% (10/21); 10:13 has an EV of 70.39%. Several of you saw that since the Town only has to lynch correctly once, giving them more than one chance will swing this in their favor.

That's true with "random lynches". In practice, though, you really only need one townie voting for one other townie in order for scum to win.

If all 11 townies vote at random, the odds that NONE of them vote for another townie is very, very small. Of course, they're not going to be voting at random, but in any kind of real game it seems very, very likely for at least one of the scum to convince at least one of the town to sheep on him, and that's all the scum need to win. In practice, unless every single pro-town person in the game is at a very high skill level, I wouldn't expect town to win that setup very often at all.

In other words; there's more then just "random lynches" to consider, there's also the strength of the scum voting block.

[Thok]

I was surprised these two were as close as they are... PairedLovers looks roughly linear, like Nightless, with a slightly different ratio. Want to investigate this a bit further.

Having thought about that, it's not that surprising. Think of it in terms of the number of lynches town needs to win.

In Nightless, town needs 10 correct lynches, each townie lynch costs 1 mislynch and 1 townie, and each scum lynch gains back a town mislynch and kills 1 scum.

In Paired Lovers, town needs 5 correct lynches, each townie lynch costs 1 mislynch and 2 townies, and each scum lynch gains back a mislynch, kills two scum and loses 1 townie. You can sort of kind of think of 10 scum Paired Lovers as a double version of 5 scum Nightless if you look at it funny.

In general, I think there's a theorem floating around that says that if a scum lynch in a vanilla variant gives town a nonzero number of extra mislynches, then the number of townies needed for balance is linear. If town earns back 0 mislynches for a scum lynch, than the number of townies needed should be quadratic in scum.

I suspect that properly phrasing that theorem is harder than actually proving it.

[ac1983fan]

2e. 10:11 PolyLovers has an EV of 47.62% (10/21); 10:13 has an EV of 70.39%. Several of you saw that since the Town only has to lynch correctly once, giving them more than one chance will swing this in their favor.

That's true with "random lynches". In practice, though, you really only need one townie voting for one other townie in order for scum to win.

If all 11 townies vote at random, the odds that NONE of them vote for another townie is very, very small. Of course, they're not going to be voting at random, but in any kind of real game it seems very, very likely for at least one of the scum to convince at least one of the town to sheep on him, and that's all the scum need to win. In practice, unless every single pro-town person in the game is at a very high skill level, I wouldn't expect town to win that setup very often at all.

In other words; there's more then just "random lynches" to consider, there's also the strength of the scum voting block.

As if 10 scum could bandwagon one townie within a short enough time period for people to not notice that the scum were likely bandwagoning and for a townie to not unvote.

[Thok]

The trick to polylovers is to squeeze out as much information as you can before voting. You don't have to find all of the scum: you just need to find one of the weakest scum in the scum group and force him to crack under pressure.

[GIEFF]

Yes, but a scum would also notice townies happening to bandwagon on a scum and unvote. To lynch a scum in 10:11 polylovers, ALL ELEVEN TOWNIES must vote for the same scum. That is a very difficult thing to accomplish, and I think that even 10:13 polylovers might be scumsided, random-lynch EV aside.

[mith]

In other words; there's more then just "random lynches" to consider, there's also the strength of the scum voting block.

Here's the thing, though. If you found yourself playing that setup, and were town, you would know this to be the case. Given that it's very likely the town would screw up and lose if voting proceeded as usual, you would immediately suggest that the lynch happen in some truly random fashion, because that maximizes your chances of winning.

So, assuming it's possible to choose a random victim (and I'm sure there's plenty of ways to do it, even with the mod banning dice rolls), that's what would happen - the town's win percentage would be exactly the EV reported above. And that's why such a setup would never be played - it's broken, the town has an optimal strategy which is something other than "playing Mafia".

Thok: It's not quite that simple (they happen to be very close for 10 scum, but 20:60 Nightless is balanced vs. 20:70 for PairedLovers), but make sense as a rough estimate. And you're probably right about a general theorem.