Conspiracy Theory

[Klick]

So this isn't exactly about Mafia, but Conspiracy has been run many times in the Marathon forums so I figure it's fine to post about it here.

I'm up in the air about the 'proper' way to balance randomising roles for this game.

Option 1 - 1/2 chance of No Conspiracy and 1/2 chance of Conspiracy.
Option 2 - 1/

x

chance of No Conspiracy and 1/

x

chance of each individual player being Conspired against (where

x

is the number of players + 1).

Without thinking too hard about it, I thought Option 1 was the obvious choice. But if someone draws a Town Role PM under Option 1, the odds suddenly heavily favor there being No Conspiracy at all. Although Option 2 means there's a very high likelihood of there being a Conspiracy, there are equal odds of there being a Conspiracy when you've drawn a Town Role PM vs No Conspiracy.

So right now I'm leaning Option 2. But have I gone wrong somewhere? Is the actual answer somewhere in the middle?

(title changed because it sounds better)

[Nero Cain]

Boon ran this as a mini-game in one (or more?) of his games. Can't say I'm the biggest fan though

[Awoo]

Option 2 is correct because of the conditional probability thing.

[Xtoxm]

option 2

[Auro]

Option 2 would mean with a large enough player size, they're all mostly going to be conspiracies. The prior of receiving a town role PM is low. So while 2) is mathematically correct to avoid Bayesian gaming, it's worth trying to find the "middle ground" where all-town priors are also not super unlikely.

Option 3: Hide the protocol. Or say you're following 2), but actually follow 1) or some other 1/y with y<x. Probably after multiple runs town would realize what you're up to though, with the skew in all-town rolls.

Option 4: Explore the math with 2 players receiving town PMs in a conspiracy instead of just one. This would slightly skew the prior Conspiracy chance to n/(n+3) instead, so a slightly larger proportion of games are Non-Conspiracies.

Option 5: Realize that you can lower the Conspiracy rate and still have P(Conspiracy | received town PM) at like 30%, strictly mathematically players should Bayesian but part of the value is figuring out when you're part of that 30%, so people would play as intended. If it's too low (like in Option 1) it's a bit silly to entertain the idea, but this range would be fine.

You can probably combine the above. Having two players roll town in a Conspiracy opens up potential mechanics (like them "finding" each other for example) which could be fun to explore.

[Auro]

Doubles' Conspiracy

If it's a Conspiracy, 2 players out of N are town, and are informed that the other is town.
If it's a Non-Conspiracy, everyone is town, and each person is informed of one person being town (random pairings)

Win as town by guessing accurately if there's a Conspiracy or not.
Win as scum by making town guess there's no Conspiracy.

Math:
P(C|T) = P(T|C)P(C)/{P(T|C)P(C) + P(NC)}. Here, P(T|C)=2/n, P(NC)=1-P(C). Let P(C) be x.
P(C|T) = (2x/n)/{(2x/n)+ 1-x} = 2x/(2x+n-nx).

Adjusting P(C|T) as 1/3

, we have:
3 = 1+n/(2x) - n/2 => x=n/(n+4)

Which means for a size of 8 players, you have Conspiracies only around 66.6% of the time!

(Btw, math in my Option 4 in previous post was wrong)

[Xtoxm]

pairs could be a nice way of leveling the odds when you have a large number of players

[Klick]

I knew something was bothering me about Option 2 - thanks for putting it into words Auro.

I like the pairs idea. I'd also be interested in finding a potential ideal N number of town per X total players. A lot of it would probably come down to personal balance preference, but I could see 3 or 4 town being sensible with enough players. Maybe playing around with the number of potential town players in a conspiracy being made public would be interesting as well.

[Auro]

Auro#6810, feel free to PM me if you wanna discuss there! I can write a small script to run calculations for these scenarios.

[OkaPoka]

option 1 except you run multiple rounds and if it is no conspiracy, the more people who guess no conspiracy, the less value each correct answer is

[Krazy]

Conspiracy is significantly better if you do not reveal the odds. That's the main thing I've learned from running it

[Auro]

People can figure out the odds after a few rounds. In any case where the frequency of Non Conspiracies is significantly more than 1/(n+1) it becomes wise to play Bayesian

[Ythan]

Without thinking too hard about it, I thought Option 1 was the obvious choice. But if someone draws a Town Role PM under Option 1, the odds suddenly heavily favor there being No Conspiracy at all. Although Option 2 means there's a very high likelihood of there being a Conspiracy, there are equal odds of there being a Conspiracy when you've drawn a Town Role PM vs No Conspiracy.

I'm not sure you're correct that being town implying probably no conspiracy is a bad thing.

Edited for clarity.

[Ythan]

Also after playing over a decade of Mafia, Conspiracy is a better game I could really go on about how much I love it and why.

[Awoo]

Do it, I wanna hear it.

[Ythan]

Well for one it isn't confronted by any of the balance issues that come up when designing a mafia setup. It's essentially intrinsically balanced at any size. And speaking of size, it's perfectly functional with as few as two players. Did we ever even come up with a good five player mafia setup?

[Ythan]

And, I find it to simply be more fun to decide conspiracy vs no conspiracy as opposed to whom to lynch.

[Auro]

I'm not sure you're correct that being town implying probably no conspiracy is a bad thing.

If people begin to guess "No Conspiracy" as a policy just for overall winrate, it will be. The bias from knowing the probability skew will make it harder to believe you're in a Conspiracy as well.

Well for one it isn't confronted by any of the balance issues that come up when designing a mafia setup. It's essentially intrinsically balanced at any size. And speaking of size, it's perfectly functional with as few as two players. Did we ever even come up with a good five player mafia setup?

Different kinds of balance issues, for example: either a town role PM

heavily

implies No Conspiracy, or your games are rarely Conspiracies at larger sizes anyway. The former case I've described above, and the latter would be boring.

I agree the Conspiracy mechanic is elegant. I did propose an experimental mafia-conspiracy fusion setup, I'll paste it back here later.

[Ythan]

Tbf I only play it in person and it's honestly a nonissue. Not sure how that might change online.

[Ythan]

I've never once seen anyone just vote no conspiracy straight away for the odds, is what I mean.

[Auro]

Of course, but I'd ideally want to eliminate the bias anyway. If my town role PM means a 75% chance of No Conspiracy, I wouldn't personally be able to detach that from the usual sorting process.

[Ythan]

I actually like that it makes town complacent but that's a personal preference.

[Blair]

In any case where the frequency of Non Conspiracies is significantly more than 1/(n+1) it becomes wise to play Bayesian

This sounds exactly like what a conspirer would say.

VOTE: Conspiracy

[Auro]

That's a useful note, perhaps when it's not a conspiracy I can just mechanics to make people paranoid

[word321]

on a similar note
would it be applicable to apply a condition over a game of conspiracy?
like, if k scum out of n votes for no conspiracy, scum looses the game (limiting scum max voting)
that could actually balance the game in favor of scum in a "1/2 1/2" probability; if u roll town, and ppl arrive at the k barrier, now it is not up to a single town to make a decision if this is scum or not, but several that would be in the fence; either way, that would be a good spice to add