An article with a proof on optimal mafia play.

[LicketyQuickety]

The balanced number of mafia for a mountainous game is very close to sqrt(num_players)/2

Sorry if I am misunderstanding, but are you saying the smaller the game, the more this formula of Scum:Town is not accurate?

[Kagami]

The exact relationship between the number of mafia for a balanced game and the number of total players in the game has an almost linear relationship with the square root of the number of players.

For non-massive games with X players, the optimal number of mafia, M, is about

M = sqrt(X) * .4833 - .1325

for massive games (over 1000 players), the above formula is still pretty close to accurate, but there's a very slight bend in the curve.

[LicketyQuickety]

31.6 is the sqrt of 1000. that looks like way too few mafia. I am not understanding you at all.

[Kagami]

You would think it's way too few, but it's not.

Even a single scum against 1000 town has a decent shot of winning mountainous (~4%). Throw in another 1000 town and his odds aren't radically worse at 2.8%.

[quadz08]

this is because mountainous is really fucking difficult for town to win, not even including the psychological effects of playing a game of that type of dullness

[LicketyQuickety]

You would think it's way too few, but it's not.

Even a single scum against 1000 town has a decent shot of winning mountainous (~4%). Throw in another 1000 town and his odds aren't radically worse at 2.8%.

this is because mountainous is really fucking difficult for town to win, not even including the psychological effects of playing a game of that type of dullness

There is something I am clearly missing then..

In a game with like 21 people you can have 5 scum.

But in a game with ~48X time that amount of people you only have like 1/6 the number of scum. I don't get it.

[Kagami]

You can have 5 scum in a 21 person game with no PRs, but town will lose with high likelihood unless the scumteam is composed of terrible players.

We often have more scum than mountainous would have because we typically include PRs, like the detectives discussed in the paper. As it mentions in the paper, even a single Cop role causes the optimal number of scum to be linear in R, instead of linear in sqrt(R). That relies on the possibility of private communication, though.

It's still probably the case that mods put too many scum in large games, even when the town has multiple PRs.

[Sajin]

For mafia the win condition for town becomes exponentially harder with the number of mafia. The mafia can lynch anyone, including their own faction as long as one remains, to achieve victory while using nightkills to remove problematic people from the game. Town must lynch specific people to win.

Pure random lynching for 1(M)v4(VT) gives town two lynches. One lynch has 1/5 shot of ending the game and the second has 1/3. This results in 20 percent town win+.8 (1/3) which results is a 46.7 percent town win rate. The number of VT needed in larger mafia sizes grows very quickly.

Cops are actually very hard to calculate because they will claim at different times with different amounts of innocents/guilties. They can be killed off early or never full claim and just crumb results in case they are killed. They can also be counter claimed or not believed by town or a non standard mafia variant can be played which restricts crumbing or does not reveal roles on death which alters power and thus balance. Informational roles have high variance. Cops have extremely high variance which is why other informational roles have become more popular to use in setups.

[quadz08]

In any mafia game, scum have two primary advantages: knowledge/communication, and a nightkill. Knowledge and communication are harder to quantify, and can be used against the mafia sometimes (associative tells), but it's an advantage nonetheless. Night action coordination and the ability to talk to other players about the correct decision / play / etc is extremely valuable. The nightkill is extraordinarily powerful, though - you have a near-guaranteed capability to remove any player in the game, and it is, in most cases, definitely going to advance your wincon (exceptions include killable traitors, shooting a PGO). The other methods of removing players from the game (that is, methods available to not-mafia) are much less likely to be beneficial for the town, making the scum kill a huuuuuge weapon. You put those two advantages in a mountainous game, where there is NO power to counter that from the town, and the only advantage you can possibly give town is sheer numbers - and you need a LOT of fucking numbers to counter those advantages.

[callforjudgement]

OK, for people who are confused about what the paper's saying, let me try to translate.

The first point it makes is that in a mountainous, if you approximately double the number of scum, you need to approximately quadruple the number of town in order to compensate and keep the game approximately balanced (i.e. it's a quadratic relationship). It doesn't say that scum = sqrt(players) is balanced (just that the relationship has a form looking much like that; Kagami posts a formula above). The assumptions it makes aren't that unrealistic; although it requires moves that would be illegal here on Mafiascum (to do with randomness and the like), effectively random lynching is something that towns could easily find a way to do (e.g. a method in which each day's lynch picks a lynch order for the rest of the game, and you follow the lynch order of the most recent townie to flip; it needs to be the most recent to ensure the list constantly changes, or else the scum can kill townies low down on the list to prevent town benefiting from the smaller lynch pool).

The next point it makes is that when there's even a single Cop in the game (and no other power roles), a doubling in the size of the scum only needs an approximate doubling in the size of the town to compensate and maintain balance.

They first give the following strategy for the situation: wait until a given day (depending on the size of the game) so that the Cop can compile a townblock; then the Cop tells every member of the townblock about the other members via PM (effectively forming a masonry, and something that's illegal on MS); the Cop is lynched to prove no fakeclaims; then the masonry anonymously chooses players outside the masonry to lynch until everyone else is dead (hiding its identity using anonymous votes, and everyone else abstains so as to not interfere with the masonry). This strategy requires a pretty odd and townsided collection of game mechanics (PMs that are identified from coming as a particular player + anonymous votes), and I'm not sure it would apply to a ruleset like ours. (There are also obvious improvements under that ruleset; e.g. the Cop can fullclaim each day to players that they've confirmed as via PM, meaning that you're less screwed if the Cop happens to get lynched or NKed.)

They also analyze a Mafiascum-legal version of the strategy: get the Cop to keep investigating until over half the remaining players are confirmed Town, then reveal and line up lynches for the win. This is a pretty simple proof that there is (or at least can be) a linear relationship between town and scum win rates with a Cop. It doesn't imply what the correct town/scum ratio is, though. (Also, that setup would suck to play unless it were very small, as it'd all come down to when the Cop got NKed).

[Ythan]

There are many confounding factors pushing the scales in both directions, when it comes to balance.

[LicketyQuickety]

OK, for people who are confused about what the paper's saying, let me try to translate.

The first point it makes is that in a mountainous, if you approximately double the number of scum, you need to approximately quadruple the number of town in order to compensate and keep the game approximately balanced (i.e. it's a quadratic relationship). It doesn't say that scum = sqrt(players) is balanced (just that the relationship has a form looking much like that; Kagami posts a formula above). The assumptions it makes aren't that unrealistic; although it requires moves that would be illegal here on Mafiascum (to do with randomness and the like), effectively random lynching is something that towns could easily find a way to do (e.g. a method in which each day's lynch picks a lynch order for the rest of the game, and you follow the lynch order of the most recent townie to flip; it needs to be the most recent to ensure the list constantly changes, or else the scum can kill townies low down on the list to prevent town benefiting from the smaller lynch pool).

The next point it makes is that when there's even a single Cop in the game (and no other power roles), a doubling in the size of the scum only needs an approximate doubling in the size of the town to compensate and maintain balance.

They first give the following strategy for the situation: wait until a given day (depending on the size of the game) so that the Cop can compile a townblock; then the Cop tells every member of the townblock about the other members via PM (effectively forming a masonry, and something that's illegal on MS); the Cop is lynched to prove no fakeclaims; then the masonry anonymously chooses players outside the masonry to lynch until everyone else is dead (hiding its identity using anonymous votes, and everyone else abstains so as to not interfere with the masonry). This strategy requires a pretty odd and townsided collection of game mechanics (PMs that are identified from coming as a particular player + anonymous votes), and I'm not sure it would apply to a ruleset like ours. (There are also obvious improvements under that ruleset; e.g. the Cop can fullclaim each day to players that they've confirmed as via PM, meaning that you're less screwed if the Cop happens to get lynched or NKed.)

They also analyze a Mafiascum-legal version of the strategy: get the Cop to keep investigating until over half the remaining players are confirmed Town, then reveal and line up lynches for the win. This is a pretty simple proof that there is (or at least can be) a linear relationship between town and scum win rates with a Cop. It doesn't imply what the correct town/scum ratio is, though. (Also, that setup would suck to play unless it were very small, as it'd all come down to when the Cop got NKed).

Nice semi-necro!

[callforjudgement]

I've been V/LA for ages. Was my first time reading the thread (and it's still on the front page), because I've mostly been living off egosearch recently rather than looking at MD directly.

[LicketyQuickety]

I've been V/LA for ages. Was my first time reading the thread (and it's still on the front page), because I've mostly been living off egosearch recently rather than looking at MD directly.

You're weird... but cool.

[Rob14]

Is this actually a good journal or minor journal? I'm not familiar with the reputations of math journals.

[LicketyQuickety]

Is this actually a good journal or minor journal? I'm not familiar with the reputations of math journals.

I have no idea \o/

[Majiffy]

Did anyone analyze how much it hurts town when you add a vig yet?

Best anti-utility role ever. All town vigs are dumb vigs.

[LicketyQuickety]

Did anyone analyze how much it hurts town when you add a vig yet?

Best anti-utility role ever. All town vigs are dumb vigs.

Maybe it should be an extra vote count.

[Ythan]

An extra lynch?

[LicketyQuickety]

An extra lynch?

Not exactly. You see it would be kinda like having a president. Everyone votes on who to use the Vig on, but ultimately its up the the Vig president.

[Ythan]

I always see people pitch in their two cents on who should get vigged when they think they smell one.

[LicketyQuickety]

I always see people pitch in their two cents on who should get vigged when they think they smell one.

Bombs away.

[Majiffy]

An extra lynch?

Not exactly. You see it would be kinda like having a president. Everyone votes on who to use the Vig on, but ultimately its up the the Vig president.

A vig governor.

Interesting.

Obviously the vote would have to be secret, or else it would out the vig.

[Ythan]

I don't think he means it as a literal role, but as a way to handle vig decisions, though I'm not certain.

[LicketyQuickety]

Obviously, we are not going to kill people irl...