Charisma Model of EV Calculation

[Mathdino]

The first question for any economist or forecaster is "What's your model?" The second question is often "What are your assumptions?"

The current MafiaScum model assumes players are monkeys at typewriters, randomly lynching and randomly killing so long as they don't gamethrow. I've programmed a weighted democratic model where each hypothetical player gains a little accuracy and can plurality lynch shared scumreads. Every model is necessarily flawed, with incorrect assumptions, but ultimately the goal is more accuracy for the simulation.

I'm proposing and demonstrating a Charisma Model of Mafia.

My assumptions:

• Gamestates result from predetermined charisma levels, where charisma is defined as "ability to avoid being lynched".
  
• Town will always attempt to lynch the least charismatic (scummiest) player. Town will never no-lynch.
  
• Scum will always attempt to kill the most charismatic (towniest) town player.
  
• PRs will claim at L-1. If uncc'd, they become most charismatic.
  
• Scum will fakeclaim confirmable roles at L-1 in order to out TPRs, and will be lynched if counterclaimed.
  
• Scum are aware of the charisma list, and will counterclaim PRs if they are more charismatic and if doing so would not lose them the game.

Therefore games can be modeled by listing out players in descending order of charisma, and determining where mafia needs to be in order to win.

Spoiler: A quick demo if you're still confused

Suppose there are 9 players. VT1, VT2, VT3, VT4, VT5, VT6, Named Townie (NT), M1, and M2. Randomly arranged in charisma, I got:
VT4
VT1
VT6
M1
VT5
NT
VT3
M2
VT2

Game proceeds as follows:
D1: VT2 lynched.
N1: VT4 shot.
D2: M2 lynched, claims Named, NT counterclaims and becomes most charismatic.
N2: NT shot.
D3: VT3 lynched.
N3: VT6 shot.
D4 (LyLo): VT5 lynched, Mafia Win.

Through this lens, mafia wins if at least one member clears the top half of charisma/PRs. Town wins if all mafia end up in the bottom half.

First, looking at 1 scum Mountainous.

1:2 - Town wins if scum is scummiest, so town EV is 1/3 (33%), matching Random Lynch Model.
1:3 - See above. EV of 1/4 (25%).
1:4 - Town has 2 lynches and wins if scum is in bottom 2 charisma. So town EV is 2/5 (40%), down from Random Lynch Model's 7/15 (46.7%).
1:5 - Same but worse in evens. Town EV is 2/6 (33%), down from RLM's 9/24 (37.5%).
1:6 - Town has 3 lynches, scum must be in bottom 3. Town EV is 3/7 (42.9%), down from RLM's 19/35 (54.3%).

1:N (for even N town) - Town gets N/2 lynches, so scum must avoid being in the bottom N/2 players. Town EV is (N/2)/(1 + N), again matching Random Lynch.

This means that as the number of town increases forever, town EV approaches but is never over 50%.

Mountainous, 1 scum
Scum:Town	Random Lynch EV	Charisma EV
1:2	33.3%	33.3%
1:3	25.0%	25.0%
1:4	46.7%	40.0%
1:5	37.5%	33.3%
1:6	54.3%	42.9%
1:7	45.3%	37.5%
1:8	59.4%	44.4%
1:9	50.8%	40.0%
1:10	63.0%	45.5%

Town EV is always lower than what would be expected from random lynching. This can be attributed to scum's unique ability to remove people they know will not be lynched, while getting to choose who to go to LyLo with.

For Mountainous with more scum, it's bleaker. BOTH scum must be in the bottom half of the charisma list for town to win, making the problem combinatorial rather than recursive. For simplicity's sake, I'll only consider odd numbers of players.

Mountainous, 2 scum
Scum:Town	Random Lynch EV	Charisma EV
2:3	13.3%	10.0%
2:5	22.9%	14.3%
2:7	29.8%	16.7%
2:9	35.2%	18.2%
2:11	39.5%	19.2%
2:13	45.3%	20.0%
2:15	48.0%	20.6%
2:17	50.3%	21.1%

For here, as we continue to add players, RLM EV approaches 100% but Charisma EV slowly caps out at 25%. And for 3 scum, Charisma EV should approach 12.5%, the chance that all 3 scum have below average charisma.

I believe this model accounts for why Mountainous is so difficult. I'm going to continue updating this for White Flag, Named Townies, Innocent Children, and Doctors. If I'm right, this version of EV calculation should predict winrate more consistently than taking Random Lynch EV and shooting for ~40% EV.

If I'm wrong, it's because this model assumes town are morons and won't question why a high charisma player has been left behind at LyLo. Feel free to discuss!

Credit to my conversation with RadiantCowbells for inspiring this methodology.

[chennisden]

This is a lot closer to an accurate sim of mafia

Kudos

[popsofctown]

Charisma is not a great name for this. These sims are localized to a single game so they don't need to assume career-wide charisma at all. The thing that prevents you from getting lynched in mafia games isn't charisma, but it isn't not-charisma, also, I see lots of players perform very well without being particularly charismatic, except for when what that english term is understood to mean gets really warped, "arguing so virulently against confirmed scum means that player is charismatic".
But also if this terminology gains any traction I really don't want people to think that forum mafia is an online game that doesn't work very well if you have difficulties socializing IRL because that is so so the opposite of what forum mafia can and has offered to people before.
I would recommend a name like "alibi", "townreadness", "trust", or at least soften charisma down to "popularity".

Second, I would argue that you don't need percentages. Period. Random lynch EV percentages needs them because otherwise you have no numerical representation of your simulation, if you don't grab the EV percentages, you say that lovers mafia is a setup "town wins sometimes and scum wins sometimes" but C9++ on the other hand is a setup "town wins sometimes and scum wins sometimes" and obviously that's not useful. You already have a numerical representation from this, though. I can say, "Death list mafia was batshit rigged because one mafia had to be second most townread to win that 9p" and I can say, "to compare that to a setup I believe is easier for scum, the mafia have to be the 5th most townread in a 9p in 2:7 mountainous." The probability of chimpanzees on typewriters producing these lists isn't necessary. It was only necessary with random lynching. I don't think you should actually be aiming for a specific observed win percentage, you should be aiming for specific observed group subjective experiences, and you can tally those from games with a matching required mafia pole position. I think even my polar opposites on friendliness to the scum would agree that requiring scum to be the most townread slot in a 9p is too rough, and even I can surely admit that if there's a 9p where scum only need one player to be at least 5th highest townread, we have a problem, and will probably have townies who are frustrated by that goalpost

even if they win through it

. Whether the ideal value for a 9p is 2, 3, or 4 is what merits exploring, the chimpanzee typewriter % on whichever of 2, 3, or 4 has the most positive player feedback is irrelevant.

I mean, you

can

certainly get these theoretical EVs, but like I wonder, what are we doing with them? Have you decided what percentages you want to see and do you have an argument for why? Plus framing the dialogue in terms of the alignment of the most popular player is pretty convenient. I can actually roll scum and join a 9p and read the popularity analysis for that particular setup says I need to be 3rd most townread, and watch three different townies exhibit a "textbook towntell", and then go, "well, I can definitively say I need to undermine the consensus townreads on one of these three slots or I lose." I think it's interesting to get something that tangible.

Lastly, I feel like nightless should be one of the things you look at as you go to look at how different mechanics interact with this. Nightlessness is a very valuable knob/lever in setup design. It's becoming increasingly popular and I think semi-nightless setups in particular are really underexplored design space.

[RadiantCowbells]

Credit to my conversation with RadiantCowbells for inspiring this methodology.

I feel like you're undercrediting me here, particularly given that you said that you're proposing and demonstrating.

[RadiantCowbells]

Like others can weigh in, maybe I'm overly touchy, but it's not at all the impression I think that I would get reading it that you're presenting to the general public something I made years ago and have been using for years.
The phrasing "inspiring this methodology" implies that I like said something that helped you along the way as opposed to it literally being something I told you about.

[Alisae]

RC giving Mathdino free content

[Alisae]

Also like, yeah its free content
but did you ask RC if he wanted his content posted?

RC had being using this for years.
Ya'd think that if he wanted this public he would post this publicly right?

[Ramcius]

It's just barebones and to apply to actual game as a scum you still need a lot of work to do, but hopefully it will help to balance games better, cause EV was terrible way to do so

[popsofctown]

Like others can weigh in, maybe I'm overly touchy, but it's not at all the impression I think that I would get reading it that you're presenting to the general public something I made years ago and have been using for years.
The phrasing "inspiring this methodology" implies that I like said something that helped you along the way as opposed to it literally being something I told you about.

Also like, yeah its free content
but did you ask RC if he wanted his content posted?

RC had being using this for years.
Ya'd think that if he wanted this public he would post this publicly right?

"Inspired by" looked gross and was surprising because I thought mathdino was a standup guy.

That said I think Ali's position that RC can keep it as secret sauce forbidden from public discourse indefinitely if he wants to is unreasonable, because it's a mathematical/scientific discovery rather than a creative creation. Someone else would have discovered it eventually. That's just the standard way discoveries versus inventions are treated. I could see how someone could maybe decide they think this is more of an invention than a discovery, but I think they'd be incorrect and it's a discovery.

I think in a perfect world mathdino prompts RC and requests he makes a thread where RC is OP, and if RC refuses to do so then mathdino makes a thread anyway, describing the model as being developed by RC and not using the phrase "inspired by"

[Mathdino]

So if I change it to "developed by", are we good? And given that, if RC feels there's a better name, I'll change that too.

I didn't want to explicitly call it the RC method because I didn't want to make it look like I was hiding behind RC or throwing him under the bus if there were any flaws with it or disagreements. Like I didn't call the other model mith's method because it's not about him, it's about numbers. I'm not trying to take credit, I'm literally trying to make balance predictions easier to do.

Here's the thread where RC told me it and where I ask RC if he'd posted the numbers yet, which I'll edit into the OP. I'll respond to other objections shortly, along with more calculations.

Edit: I also literally didn't see anything like this until that thread itself, so if RC has posted anything proposing/demoing it himself, I'll edit that in too and change the verbiage.

[Mathdino]

snip

1. Like I said, I'll defer to RC on the name. I do think charisma varies on a game to game and on an alignment basis but if the word isn't communicating lynchability right I understand.

2. I think we might have different goals in modeling. I'm doing this purely as an intellectual exercise to see if this model predicts game outcomes better, and for that purpose I do need percentages. I'm not doing this to make any kind of argument over what a good or bad setup is. I kinda see this as a mafia economics discussion.

3. I actually started doing Nightless EV in the OP, before realizing this model creates the exact same outcomes as random lynching under Nightless Mountainous. If scum don't have any ability to mechanically influence the game, it's functionally the same as randomly ordering people in line to get lynched.

Notably, this means 2:4 Nightless Lover's Mafia still retains its 60% EV despite being scumsided in practice. My ultimate goal is to somehow combine this with my own model that biases lynches based on reads accuracy.

[popsofctown]

snip

... I'm doing this purely as an intellectual exercise to see if this model predicts game outcomes better, and for that purpose I do need percentages.
... My ultimate goal is to somehow combine this with my own model that biases lynches based on reads accuracy.

Yeah I didn't really get that you were looking for predictive value, because you hadn't gotten to the reads accuracy step yet. Because before you add the reads accuracy factor, this system is kind of fundamentally flawed as a winner-predictor. The EVs are lower than random lynching, so why isn't the town just random lynching? (you can't lynch 100% randomly but you can approximate it a lot better than this model if you toss in some retributive votes for scum that never bus). The reason why is that the reads accuracy is sometimes better than rand so it is correct for town to play this way, usually. So you really want to include that piece to get accurate predictions.

You would want to use empirical data to find a reads accuracy value that reflects the winrates of completed games, then plug it in when you use the model to predict the outcome of future games.

My position is just the percentage value on games before you add that to your sim is just a reddit karma value, not an actual win percentage, so you might as well use the scum pole position ordinal at that stage.

[Mathdino]

I think to some degree the fact that town has >rand reads in practice combined with scum's ability to influence the lynch in practice will cancel itself out in most setups except ones with too much scum (where scum just gains the advantage). I'm interested to see if it does achieve close to the actual win percentage for things like Mason games, without having to do much fine tuning.

Tbh I think it might be easier to do that final model by assuming scum has a certain chance of taking a hit to charisma. To do plurality lynching would definitely be a whole coding exercise in its own.

[RadiantCowbells]

Step 2 for setups like lovers nightless requires actually sorting out individual players willingness to vote and realizing that every single player has to vote scum together to win. Same reason that naively LyLo EV in 3 way should be closer to 1/4 than 1/3; the scum who is willing to instahammer anyone has higher odds of getting the win because both of the other townies have to be right to win whereas only one has to be wrong for him to win.

These voting mechanics matter significantly more with larger town to scum ratio such as lylo whereas they don't really influence the early game.

[skitter30]

huh this is interesting

i think intuitively this is basically how i play scum (murderize everyone townier than me and/or who will undermine my towniness + be townier than remaining {number of mislynches} people) but i've never tried to articulate it just this way

i think for me this is the winning formula: be the towniest person alive and never be the lowest person on the lynch totem pole
(of course this is easier said than done but theoretically this is what you need to do to win scumgames imo)

also i agree that 'charisma' is the wrong word; i don't think 'charisma' == 'ability to avoid being lynched' == 'towniness'
i'm not even sure 'towniness' == 'ability to avoid being lynched'

[Mathdino]

For Mafia Lovers (all mafia are lovers with each other), town has (T-S+1)/2 lynches. So we calculate the probability that all avoid that many of bottom slots, and subtract it from 1 to get town EV.

Mafia Lovers, multiple scum
Scum:Town	Random Lynch EV	Charisma EV
2:3	40.0%	40.0%
2:5	57.1%	52.4%
2:7	66.7%	58.3%
2:9	72.2%	61.8%
2:11	76.9%	64.1%
2:13	80.0%	65.7%
2:15	82.4%	66.9%
2:17	84.2%	67.8%
3:4	42.8%	42.8%
3:6	61.9%	58.3%
3:8	72.3%	66.1%
3:10	78.7%	70.6%
3:12	83.0%	73.6%
3:14	86.0%	75.7%
3:16	88.2%	77.3%
3:18	89.9%	78.5%

I heavily suspect that with 2:T as town increases, this model's EV caps out at 75%. 3:T seems to roughly cap out at 87.3%. Under random lynching, both 2:T and 3:T can go up to 100% with more and more townies.

Note that Lovers Mafia is Nightless, meaning both models should predict the same EV. In Nightless games, scum cannot influence the gamestate mechanically, so deaths result purely from the already random player list.

[Awoo]

I think this model drastically overshoots scum's EV for setups with a lot of scum, but also a lot of guaranteed scum flips. To put it to a specific setup that breaks this model: Extradition mafia, and to a lesser extent maybe Purgatory.

Extradition mafia under this model is almost totally unwinnable for town because the mafia at the bottom of the charisma list escape first, leaving town with abysmal chances of catching the mid to higher charisma goons hiding in the rest. However in practice what ends up happening is that the associative tells from the weaker flipped mafia tend to give damning evidence for the stronger unflipped mafia which ends in town winning.

Purgatory has a similar problem. Scum that would otherwise be voted to heaven often get boxed closer to endgame out due to assosciative tells with their many flipped partners.

[RadiantCowbells]

i do feel like the model is flawed in games where scum have specific motivations to defend each other like that that put all of scum's performances a little closer to the mean

[Mathdino]

As Purgatory is Nightless, the scum have no control over the playerlist outside of voting. This model produces the same EV as Random Lynch for that and similar setups.

[Not Known 15]

This model fails to take one thing into account, a very important one.
"Charisma" isn't static. Just because someone won't get lynched Day 1 that doesn't mean that they are still viewed as town Day 4.

[OkaPoka]

the important thing is how should setups be balanced right

with random lynching gen consensus was ev ~40% because we assume town can be big brain to help out wrs

do we start balancing @50% using this model or ... ?

[Kagami]

I think this is showing two ends of the extreme in what is very much like the Monty Hall problem.

If players' reads are completely static, you get your "charisma EV." If they fully re-evaluate on each flip, you get the random EV.

The real takeaway here is that players should always take a step back and reconsider their positions following a flip, even if it isn't surprising. Typical scum strategy strongly assumes that players will be fairly static in their reads, which is also why replacements are so scary; deviating from that is powerfully pro-town.

[popsofctown]

Re evaluating on "why wasn't X killed" is the most info-rich kind of flip.

[Kagami]

the important thing is how should setups be balanced right

with random lynching gen consensus was ev ~40% because we assume town can be big brain to help out wrs

do we start balancing @50% using this model or ... ?

You shouldn't balance to 50%, imo, but that aside, you should balance to random EV or greater.

This metric is just describing a phenomenon that is somewhat prevalent and the OP demonstrates that it's wildly suboptimal.

It's trivially possible to perform at random EV levels, so any balancing exercise should treat that as the lower bound even if poorer strategies exist.

[Psyche]

ah finally a good theory thread