[OLD] The Numbers Thread

[mith]

Now consider a 1:3+Cop setup.

Assume we select a lynch target at random (as we did in the scum-can't-fake-claim-cop setup), with the Cop only coming out if chosen. We have:

1/5 - Mafia lynched (EV 1)
1/4*1/5 - Cop outed, Mafia lynched (EV 1)
3/4*1/5 - Cop outed, Townie lynched (EV 1/3)
3/5 - Townie lynched (EV 26/45)

EV = 97/150 = 64.6~%

[Fishythefish]

I'd misunderstood "California" - I thought it was a general name for a setup with 1 cop and nothing else interesting.

Thinking about it, I wouldn't be at all surprised if the upper bound of "mafia can't fakeclaim" and the lower bound of "don't let claims derail lynches" (but do get information from wagoned cop) were pretty close for large numbers of players. I'm away for the weekend, but I'll test this next week.

[mith]

(This is where the random claim order makes things interesting - if Mafia claims second, he now knows which two cases to avoid.)

I realized this morning that this should say "If Mafia claims

third

, he now knows which two cases to avoid." If either the Cop or the Townie claims before him, he knows the setup.

I suspected a random claim order would be better for the scum anyway, but now I'm pretty sure. Of course, a simultaneous claiming would be pretty difficult to do on the forums, so perhaps I need to run those numbers anyway.

---

The problem with a Doctor-only setup is the handling of "no kill". So far I have assumed the scum must always kill, and there's no reason for them not to as long as the "happily ever after" resolution goes in the town's favor. Obviously, with a Cop in the setup the scum can't just give away investigations.

However, with a Doctor, there's the strategy of no-killing in hopes that the Doctor protects you and "confirms" you that way. I can't imagine it being that effective... but I expect it's optimal for the scum to no-kill sometimes, in order to keep the Doctor from confirming anyone.

Except near endgame, the town would probably want to lynch anyway - either they started with even numbers and have now gained a lynch, or they started with odd numbers and still need another protection/no-kill to gain a lynch. And I can always throw in a rule regarding "happily-ever-after" - if the scum no-kills and the town responds with a no-lynch, scum must try to kill the next night. Still something annoying to consider. Maybe I should go with a Vig next instead.

[Fishythefish]

So, I've implemented two strategies for setups with 1 cop, otherwise mountainous. I think the program is working properly.

1. Lynch all cop claims immediately (then believe them if they were real). The town can do at least as well as this.

2. Scum never claim cop; cop claims are always believed. The scum can do at least as well as this.

This gives us a range for the true answer. When the town is large, it tends to be a very small range (basically, the cop claims at some point. Lynching him immediately and having him NK'd immediately aren't that different). I'm looking to improve strategy 1 to a still simple (it should be possible to deal with all possible scum strategies easily), but better, town strategy. My hunch is strongly that strategy 2 is near the truth; in general, the cop will claim somewhere in the middle of the game, at which point scum counterclaiming his is silly - it wastes a scum for no reason.

Balance ranges for various numbers of scum (ie. if we have fewer townies than the bottom number, scum certainly have an advantage. If we have more than the top number, scum certainly have a disadvantage):
1 - 1 townie
2 - 6-10
3 - 13-17
4 - 20-26
5 - 28-34
6 - 36-42
7 - 44-51
8 - 53-60
9 - 62-69

[mith]

Scum will probably never counterclaim (for large enough setups), but I'd guess it's better for them to fake claim are significant fraction of the time when they are the lynch target - with rule 1 in effect, it doesn't matter what they claim, of course, but if we're trying to improve rule 1 we'd need to take that into account.

Anyway, we don't *need* exact numbers to get a feel for balance... and those numbers certainly look reasonable. What EVs do you get for California with those strategies? (Strategy is going to matter much more for small setups.)

[Fishythefish]

For California I got the same 43.55% that you got for "no fakeclaim". For "lynch all claims" you get 26.78% (for small setups like this, it's going to be a disaster of a strategy - here it hardly improves on vanilla).

Yeah, the good thing about rule 1 is that you don't need to ask "can the mafia exploit this by fakeclaiming some of the time?". My next plan is to design a rule which they can exploit in quite simple ways.

[mith]

Part of an argument about how towns would actually do in a Vanilla setup... some numbers of town win rates given some assumptions about how well they lynch: Post.

[MichelSableheart]

(Correct play in F11 is for the Cop to come out D2, if there is one, and then for the Doc to come out D3, if still alive. This ensures that the Cop survives to the endgame, and confirms the Doc even if the Cop hasn't investigated him, and catches lying scum if he fakes in a no-power-role setup.)

Suppose the roleblocker is lynched day 1, and a vanilla dies night 1. The cop claims with an innocent investigation. You are the doc, and you weren't investigated. If you claim, you have a guaranteed win right there. Both you, the cop, and the cop investigation will be known innocent, which leaves four players unknown. Town has 3 lynches and an investigation available to find the 1 scum in those four players.

If you don't claim, the town can miss that guaranteed victory if town is lynched, then during night 2 the cop investigates you and the goon kills you. That would leave 3 unknowns, with only 2 lynches to find the scum there.

In the abstract, claiming day 3 may give the town better chances of a win. However, you are playing to win this game. If the cop claims day 2, with an innocent result on a living player other then me, and I am a doc, surely I should claim?

[mith]

In that situation, absolutely. That comment should have read "then for the Doc to come out D3 at the latest".

The only potential problem I see is the question of whether the Cop should claim his investigation D1. Let's see...

It's D2, the RB was lynched D1, the Goon shot a Townie, and the Cop has an innocent on a living player. If the Cop didn't investigate the Doctor, then it is a certain win for the Cop to claim the investigation and then for the Doctor to come out. However, if the Cop has investigated the Doctor, the Doctor will basically be revealed, and the Goon has a chance (though small: 1/5 - the chances of being the "last" to be investigated/lynched amongst the 5 unknowns). So, EV for revealing the claim is 96%.

If he doesn't reveal the claim, we'll let the Cop pick the lynch target at random from the non-investigated. If he didn't investigate the Doctor, and he picks the Doctor as the lynch target, the Doc will claim and it's a win. If he picks the Goon, it's also a win. Otherwise (3/5), the Goon has to hit the Doctor (1/4) *and* the Cop has to investigate the Doctor (also 1/4), in which case the Cop only has one confirmed innocent and the Goon as a 1/3 chance of escaping. Put all that together, and the Goon only has a 1/80 chance of winning if the Cop has an innocent on a Townie (vs. 0 if the Cop has revealed to begin with).

If he did investigate the Doctor in the first place, the Goon needs to survive the D2 lynch (4/5), hit the Doctor (1/4), not get investigated (3/4), and again has a 1/3 chance of winning from the Goon+Cop+ConfirmedInnocent+2Townies endgame. So the Goon has a 1/16 chance here (vs. 1/5 if the Cop revealed the investigation).

So, assuming I haven't messed up somewhere along the way (which is entirely possible, it's early), the EV for not revealing the investigation is 97.75%, and the Cop should keep that information to himself.

[mith]

(It shouldn't be *too* hard to figure out the overall EV for a D1 RB lynch from there, assuming no additional information was revealed D1. Most interesting scenario is probably the one where the Doctor stopped the N1 kill.)

[Ether]

I never thought about the cop directing the lynch there; that's interesting. What's the EV for the cop not claiming its result, and the town no-lynching until it can force a guaranteed win?

[mith]

Not as good, though it would take a bit longer to work through that whole tree.

If Scum can survive N2 (not get investigated: 4/5) and hit the Doctor (1/5), then even in the worst case of the Cop having two innocents (he'll only have one if one was on the Doctor) the Scum wins 1/3. That's already over 5% worth of EV for the Goon.

[Ether]

Sweet. I can live with 97.75%.

[Ythan]

In other words, balance = EV, deviation from EV = skill.

This is more than just a useful way to shirk any responsibility to refine your process. I came to this same conclusion trying to balance non-vanilla games and I think it is an important premise to maintain even if you can easily alter the numbers to reflect real world results.

[Nikanor]

Would it be possible to re-upload the spreadsheet in the first post? It looks like some ninja snuck into the server during the move.
EDIT: I checked the old forum and it isn't there, either. Those ninjas are thorough.

[Max]

Yeah... I was thinking that

[Mr. Flay]

Try this one: Spreadsheet for Vanilla Variants

[Max]

Thank You

[Nikanor]

Thanks!

[Fishythefish]

I'm sure nobody will care, but I saw the rather nice formula for the town EV in a Nightless game - (T-M)/(T+M) - and thought I'd prove it. It's easy by induction, of course, but I also found a direct way.

A game is determined by a "lynch list" - the order in which the players will be lynched, until the town wins or loses. However, we may as well go on making the list after the game ends, to get a list of (M+T) players. So if the list goes ABCDE..., we intend to lynch A then B then C....

Now, consider that (M+T) cyclic permutations of this list. Which ones result in town wins?

Firstly, the last player on the list must be town. But if the list ends .....MT, we still lose. To find which townies shouldn't be in the last position on the list, run the following:
Select each mafioso (in any order), and go right on the lynch list until you find an unmarked townie. Mark that townie.

After a little thought, we can see that the marked townies are exactly those who will make the town lose if they go in the last place on the list, and obviously M townies have been marked. So the number of the permutations of this list that win for town is (T-M), and so the EV is (T-M)/(T+M).

[Ythan]

I'm not exactly sure what T and M represent.

[Fishythefish]

Oh sorry, T is the number of townies and M is the number of mafia.

[zoraster]

I can do it, but I have to do it by long-hand because I know of no other way so it'd take forever... if anyone has a moment and can do it in a way that is quick (I don't want to burden your time), can you tell me what the EV of 9 town vs. 3 mafia is with the town winning if they lynch correctly twice?

If not, I'll do it, but if someone could help me out, I'd be in your debt.

[Ythan]

Oh sorry, T is the number of townies and M is the number of mafia.

This seems obvious in retrospect.

[mith]

zoraster, I get an EV of 40.35%. (Pretty simple modification of the Vanilla or White Flag tables in the spreadsheet - just changing "boundary" conditions.)

At some point, I'll upload an expanded version of that spreadsheet, with things like Census and this variation on White Flag (which perhaps we should just call White Flag, since I don't think anyone has used the other win condition part of the original White Flag suggestion). Maybe some Mason stuff. Anyway, I've saved the change I made, so it'll be there when I update.