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.

[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.

[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.

[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.

[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.