[OLD] The Numbers Thread

[mith]

This is a thread for calculating EVs (expectation values - how often each side is expected to win given random play) for various (simple, Open and maybe Semi-Open) setups.

Placeholderish for now, but if there's a setup you are particularly interested in let me know, and if you want to run your own numbers feel free. At the moment, I've got a spreadsheet for Vanilla, and can easily modify it for several no-power-role variants (Nightless, Treestump, Lovers, Nightless Lovers, Treestump Lovers, etc.), so I'd like to get that up in a viewable form first.

Spreadsheet for Vanilla Variants

[mith]

It's not really something we could effectively model, and I'm not sure we would want to. The scum can play so that the chances of a day 1 scum lynch are lower, but in doing so they risk linking themselves and giving the town better odds on later days; on they other hand, they could bus/distance more than they should, and up their day 1 lynch odds while improving their chances on later days. I would think it would all come out about evenly.

Anyway, the EV is calculated for some theoretical "optimal play" by both sides. If there *were* a clear advantage to be gained by the scum in playing a certain way, the town can always counter it by lynching in a truly random manner (by "random" here I mean "lynch independent of argumentation", not "lynch any living player with equal odds", since clearly there will be some situations where we want power roles to out themselves before being lynched, or we want to lynch from a subset of the living players, or whatever). And the assumption that the scum are playing optimally ensures that the town can never do better than random, either.

In practice, it doesn't actually work out this way; but that's because players don't play optimally. In other words, balance = EV, deviation from EV = skill.

So, yes, random lynches and NKs (depending on optimal strategies which depend only on the information that we the outside observers have, and not on in-game tells and so forth), justified by the above hand-wavy pseudo-math.

[mith]

Yeah, I want to go through my original-newbie numbers again sometime. The breaking strategy of the Cop coming out D1 is explored in the wiki article linked on my page, but I can't remember if I ever ran the numbers on scum countering that claim (and if I did, I don't have it anywhere).

It's still a broken setup in some sense (and unsuitable for Newbie games), but I'm now wondering if the following setup is more balanced EV-wise:

1 Mafia Goon
1 Devil (Mafia Goon, but see below)
1 Angel (Cop - Angel was the original name for this role)
1 Doctor
3 Townies

Day Start, at the beginning of the game, the mod reveals who the Angel and Devil are, but not which is which.

Basically, it's the original newbie setup in which the Cop and one of the scum have both claimed Cop.

[mith]

I think it is useless to lynch someone else, but have not calculated it.

When I was discussing the Death Miller variation a few days ago, I was actually operating under the assumption that the town

shouldn't

lynch one of the Cop claims, because I had a feeling that I had analyzed both branches before and that one was better. But I may be remembering wrong. Either way, need to do the numbers on both to have a full grasp on the setup... unfortunately, the other strategy is

much

more complicated.

Anyway...

Case 1 (Devil lynched D1) isn't entirely correct... you aren't taking into account that the Cop could investigate the Doc (in which case there is no extra confirmed player). I think you're correct that the scum must kill the Doc, though (D2 Doc claims, scum can't counter, and now Cop can investigate one of the other two players after the lynch to be sure he knows who the scum is D3).

So... 1/4 (Doc killed) * [3/5 (Townie Investigated) * 2/3 (Townie Lynched) * 1/2 (Townie Lynched) + 1/5 (Doc Investigated) * 3/4 (Townie Lynched) * 2/3 (Townie Lynched)] = 3/40 = 7.5%.

Case 2 is definitely off. If the Cop is lynched D1, the town knows the Devil is scum. Best case for the scum, it's a 1 Mafia, 2 Townie endgame (2/3 scum win), worst case the Doc survives, everyone claims, and it's a 50-50. I think those cases are equal probability (2 shots at the Doc, 4 possibilities), so the overall probability for Case 2 is 7/12 for the scum, 5/12 for the town.

Total EV for the Town would then be 5/24 + 37/80 = 161/240 = 67%.

So scum are better off not countering in original newbie. I may have just been remembering wrong regarding the optimal strategy in case of a claim, but I'd like to verify the other way later anyway.

[mith]

Heh, I forgot about the Double Day variant (just caught it going through setups in the certification thread). That makes 9 Vanilla variants total that I want to run numbers on (including Double Day Treestump Lovers!).

[mith]

Not to mention the so-called "Multiball" variants (which are a bit more complicated, but possibly simpler than any setup with a Cop).

[mith]

I should perhaps make it clear that it's only feasible to run the numbers on very simplistic setups. Things get messy fast, even with just a Cop and a Doc. I could probably come up with an EV for an 8 player version of that setup (with just two sets of four), after some careful thinking wrt claims and what not.

Were I to actually find myself in that setup, however, I would take my chances on a massclaim. You're guaranteed at least 12 confirmed innocents (maybe more, if the scum happen to claim the same role, which is definitely possible, though I suspect mostly they would claim the miller-of-their-set). The scum would need to mostly shoot confirmeds at first, though there would be the temptation to shoot at non-confirmeds in hopes of getting a jump on the other scum, and it would be impossible for them to coordinate their kills to guarantee they get the full spread of kills. And even if the scum were successful in reducing the town's numbers quickly, then you've got some Prisoner's Dilemma fun.

I'd be mildly interested in a reduced version being run in scumchat just to see what happened, but it's not really Mafia - there's no groups at all, too many named roles, and would end up being either broken or a mostly random shootout.

[mith]

Ok, my spreadsheet currently has 8 Vanilla variants; but before I put it online, go take The Balance Quiz

[mith]

Answers are up, I'll get my spreadsheet up soon. I want to put together a few graphs to demonstrate some trends.

[mith]

Ok, haven't gotten around to doing any graphs, but I put the spreadsheet up. Link in the first post.

I'd like to play with California next. I'll probably make some simplifying assumptions to start with (Cop will always claim if first-choice for lynching, Townie claim will always be lynched). With those assumptions, Scum will always claim Cop if first-choice for lynching, and then we need to figure out whether the Cop should counter, whether the Scum should ever counter the Cop, and whether the Town should ever lynch a Cop claim.

[mith]

If I were using a simulator, probably. I'd rather find an exact solution, though.

[mith]

(There's also the rather interesting problem of what the scum should do if a townie is lynched D1 and they don't hit the Cop N1; it's probably optimal for one of them to claim an investigation, but you've got to deal with the question of which claim you are more likely to believe (first or second), and how often scum should claim to have gotten guilty/innocent.)

[mith]

Ok, assuming the cop always counters a scum fake claim (pretty sure this is correct play), and assuming the town always lynches a claimed cop if there are two such claims (see below), I get that the scum should *always* counterclaim the cop. Obviously the town should always lynch the first "cop" to claim (since it's twice as likely to be scum first), and so the EV in the 3/7 where we have a cop claim D1 is 19/105. Not very good for the town.

If those numbers are correct, the town's optimal strategy can't be to lynch a cop: they're probability of winning if they lynch one of the other five players is clearly at least 1/5 (if they hit the other scum D1, they have two lynches to find the fake cop). So I'll need to calculate that probability, and then rerun the numbers on scum countering. It's a bit more tricky, because there's the investigations to consider (what the scum claims, as well as what the town can deduce from the two claims; they might be able to eliminate certain pairings or confirm players).

[mith]

Oops, I see my problem. I'm multiplying by the wrong thing. EV *given a cop claim* is 19/45 (with a 3/7 chance of a cop claim - 19/105 is the contribution of that branch to the overall EV).

Still possible the town's strategy should be to not lynch a cop claim, though.

[mith]

Eesh, messy. Here's the problem, for those confused by my ramblings:

There are two cop claims D1. The town lynches someone else, who comes up townie. We go to night with 2 claimed cops, 1 other scum, and 3 townies. One option for the scum is to kill the real cop, which reduces the whole thing to a 1:4 vanilla game, basically (town has two chances to hit the other cop, they win 7/15 of these).

The second option is to kill someone else. D2, there are now 56 cases to wade through:

Which cop has to claim his investigation first? (2)
Who did the real cop investigate? (4)
Who did the fake cop claim to investigate, and what did he claim to get? (7 - he'll obviously claim innocent on the dead player if he goes with that claim, but could claim either on anyone else)

If the scum claims first, he has (essentially) five options - rat out buddy, protect buddy, frame townie (2 options for which), buddy up to townie (same), claim dead player. But if the real cop claims first, the scum has either five or seven options... and the probabilities for how he should claim could potentially be different for each different situation he finds himself in.

Then the town has to sort out what they've learned (in some cases, they can reduce it to a 50-50)... and they can still take into account how much more likely it was for the scum to have claimed first D1.

What a mess.

[mith]

We're discussing multi-group vanilla setups in the certification thread, thought I'd move some of that over here.

I think I've shown that at 1-1-2, the town's best play is for one of the townies to offer himself up as a sacrifice. This reduces it to 1-1-1, which has an EV for the town of either 1/3 or 1/4, depending on how draws are counted.

Now the question is: What is the best play in 1-1-3?

The sacrifice plan now puts us in 1-1-2 the next night, giving us 7/27 (assuming my interpretation of the draw rules).

Lynching at random is a slight improvement: 2/5 chance of hitting scum (taking us to 1-2, 1/3 EV), 3/5 chance of hitting town (7/27, as above). Overall: 13/45

No Lynching is better still. There's a 1/16 chance of a cross-kill (town win); otherwise, either one scum gets hit (1-2, 1/3 EV) or one/two townies get hit, putting us in either 1-1-1 or 1-1-2 which have the same 1/3 EV as shown above. Overall: 3/8

[mith]

1-1-4... Lynching scum only gets us to 1-3 (NL down to 1-2, EV 1/3), so that's no good.

NL gives 1/25 cross-kill, one scum hit puts us at 1-3 (NL down to 1-2, EV 1/3), two townies hit put us at 1-1-2 (EV 1/3), but one townie double killed takes us to 1-1-3 (3/8); 4/25 of the latter happening. Overall we have... 1/25+4/15+3/50 = 55/150 = 11/30 = 36.666%.

3/8 = 37.5%, so the best play is actually to sacrifice a townie again and reduce to 1-1-3 going to night.

1-1-5 lynching: 2/7 chance of hitting scum, with a EV of 7/15 if we do; 5/7 chance of a 1-1-4 night, with the EV of 11/30. Overall: 2/15+11/42 = 83/210 = 39.5%. Better, but:

NL gives 1/36 cross-kill (EV 1), 10/36 one scum dead (1-4, EV 7/15), anything else is 1-1-4 or 1-1-3 (3/8). Overall: 1/36+7/54+25/96 = 361/864 = 41.8%.

[mith]

Simulation is fine... the main issue is that it's difficult to know which strategy is optimal in certain situations, and in order to run a simulation you either need to decide beforehand how it will treat those situations (so you are just guessing what is optimal, and may not be finding the true EV... and in some cases, a non-obvious strategy may drastically shift the balance of the game), or you need to brute force it and try all the different possible strategies (at which point, might as well find an exact solution, IMO).

[mith]

No idea what the "Vamp" role is, but if it's something that can be claimed, the random lynch EV is bumped to 33% (or 50%, if scum are stupid and counter).

(An example of the difficulties of simulation - either you need to anticipate all such situations where the town might have a strategy beyond "lynch someone at random or don't lynch at all", or you need to include branches for various claims and so forth.)

Tenchi, the formula in the vanilla spreadsheet is basically:

EV[M:T] = MAX[EV[M:T-1]; M/(M+T) * EV[M-1:T-1] + T/(M+T) * EV[M:T-2]]

In plain english: The EV given M Mafia and T Townies is the maximum of two probabilities. The first is the EV given M Mafia and T-1 Townies (no lynch, followed by a nightkill). The second is EV after lynching a random player, followed by a nighkill, which is the sum of (the EV after lynching scum + nightkill times the probability of that happening) and (the EV after lynching town + nightkill times the probability of that happening).

It's recursive, so you need boundary conditions (if M is 0, EV = 1; if M >= T, EV = 0).

[mith]

Alright, bumping this to remind myself that I want to play with this stuff again.

After glancing at the Open discussion thread, I want to look at a vanilla+masonpair setup. And then get back to California again. The mason setup ought to be simpler, and hopefully figuring out how claims might work there will give some insight on the more complicated Cop-claim mess.

[mith]

Actually, perhaps I should start with a "NamedTownie" variation first - one innocent has a non-vanilla role which can be claimed (but doesn't have any other ability). The Mason setup reduces to this if a Mason is NKed before a claim happens, anyway.

Two possible starting point strategies for the Mason game (while both are still alive):

1. Town should always force a full Mason claim. This takes away the Mason claim from the scum (if they claim a player as their partner who is innocent, that innocent just denies it and the scum gets lynched without revealing any info on the Masons; if they claim their scum partner, the real Mason pair comes out and that's a win unless it's lylo, and a 50-50 even then).

2. Town shouldn't force a full claim. This improves the power of the role slightly, and it may be that scum still can't claim effectively (since one of the Masons can just counter). Would have to check for a Nash equilibrium to determine that, though.

[mith]

Ok, I've got NamedTownie done for 1 scum, and I think this is the first example of a situation where adding a townie hurts the town.

1 Mafia, 1 Townie, 1 NamedTownie = 50%
1 Mafia, 2 Townies, 1 NamedTownie = 44.4~%

1 Mafia, 3 Townies, 1 NamedTownie = 56.6~%
1 Mafia, 4 Townies, 1 NamedTownie = 56.2~%

For 1:2:1, obviously NoLynch is still correct, but you might lose your NT and go to 1:2 rather than 1:1:1. For 1:4:1, NoLynch is no longer correct, since you want to make use of the NT's presence during the day as much as possible - a bit counterintuitive, though. With more townies, the probabilities strictly increase.

I think for multiple scum we can safely assume that the first scum to be a lynch target will claim and be countered, but I don't know if we can assume scum will never counter an NT claim. Should be easy enough to check what happens if they do counter (assuming they do so with a small probability, town will always lynch the first claim, since it that player is more likely to be scum, so for M:T-1:1 we're looking at either M-1:T-3 or M:T-1 with a lost nightkill on the NT; the latter is almost certainly better for scum if there are lots of townies alive, but maybe not close to lylo).

[mith]

Looks like with 2 Mafia, NoLynch is always correct for even numbers (and thus adding a townie to go from odd to even is always bad for the town... which is unfortunate, because I rather like the idea of an even-numbered setup where the correct play is to lynch until the NT dies or it reaches lylo).

The value of the NT is pretty small - 2:9:1 vs. 2:10 gains the town about 2%.

[mith]

A further variant on this would be what I'm going to call "Census". The setup is:

X Mafia
Y Townie Type 1
Z Townie Type 2

where Y and Z are known at the start of the game. Correct strategy for Y and Z approximately equal is likely to mass claim at the beginning of the game, forcing the scum into one of the two types, and turning the game into two subgames. NamedTownie is the case where Z=1 - but note that it's not correct play to mass claim at the beginning there, since the NamedTownie becomes more valuable the longer he survives.

(I feel like this sort of setup has been discussed before somewhere, but I don't remember where or when.)

[mith]

It's very rare indeed that 4 mafia should claim 3-1.

Can you post some examples of this? My intuition would have been that the scum should claim so as to make the proportion of scum in each group as equal as possible.

Re: California, the potential for fake claims definitely makes any analytical treatment difficult - but I don't doubt that it's doable, it's just messy (and I would hope that solving California would shed some insight on how to go about solving the more general case). I think simulations with some reasonable strategies should give a starting point (and then you can add more strategies to try to improve the result on either side).

[mith]

I thought I posted here last night. Guess I got distracted before I pressed submit.

Not that I didn't trust your numbers, but I got Census into my spreadsheet and our numbers agree. I'll format it nicely and reupload the spreadsheet at some point. (Or you could email me what you've got and I could use it, as it's probably more elegant than the mess I made. )

I think the 3-1 split is so rare because of the nonlinear effect of the two underlying factors controlling whether 4-0 is best. First, the strength of the Mafia group is quadratic with size (for vanilla games, but also for games with confirmed townies). Second, the strength of confirmed townies (in the case where the Mafia goes with a 4-0 split) is pretty small until we get close to X=Y (and it climbs rapidly to hit an EV of 1 wherever the confirmed group is a majority, though obviously once we're past X=Y we'd be looking at a 0-4 split, not 4-0). So, when the X/Y split is uneven, the benefit gained by having an extra member in the bigger group is more significant than that gained by having a member in the small group and keeping that group unconfirmed - the confirmed townies aren't that strong anyway, and early on the Mafia will have to take out townies in the smaller group anyway, until the ratios are more even. Because of the nonlinear trends, the breakeven point between 4-0 and 3-1 isn't when the ratio X:Y is closer to 3:1, but rather when it's closest to 2:2 (or nearly there, anyway), at which point 2-2 has already overtaken 3-1 in most cases.

[mith]

The divide between 3-1 and 2-2 looks a bit more like I might have expected (though not at the ratio I would have expected). The boundary stays around a ratio of 0.57-0.58 for most of the range I'm looking at (up to X=Y=25). That said, I wouldn't be surprised if the boundary stays essentially linear plotted as X vs. Y, rather than staying roughly constant by ratio.

The boundary between 3-1 and 4-0 on the other hand essentially remains parallel to the X=Y line, only stepping over at 11-8.

So as far as why 11-8 and 12-9 are the only setups where 3-1 is the best split for the scum, it's because step in the 3-1 vs. 4-0 boundary happens to coincide with a step in the 2-2 vs. 3-1 boundary. The boundaries are adjacent for low X-Y, and diverge for large X-Y, but the jump in the former trendline is enough to make them overlap briefly.

[mith]

Alright, back to California...

From the NamedTownie variant, we have a lower bound - if the Cop has no actual investigation (or doesn't use it, or is known to be Naive/Paranoid, etc.), the EV is 163/630 (~25.87%).

For an upper bound, consider the case where the scum can't fake claim or counterclaim the Cop at all (even in an endgame). Now the problem is simply one of figuring out when the Cop should claim and reveal his results. For a larger game, this might actually be a tricky question (though I don't doubt we could throw together a recursive spreadsheet or a program to solve it). However, for this setup, it turns out that the Cop should almost never claim until it's down to three, unless he is the lynch target.

If the (randomly selected) lynch target is the Cop, the Cop claims. Pick a different lynch target, Cop will be nightkilled, game becomes vanilla.

If a scum is lynched Day 1, obviously the Cop should reveal if he gets a guilty after that. If he gets an innocent Night 1, however, he should stay hidden - revealing gives an EV of 2/3, but if instead we select a lynch target at random with the Cop only claiming if the chosen lynch target is the Cop or the investigation choice, the EV jumps to 7/9 (because either the scum is the lynch target, or if an innocent is lynched there is still a good chance the Cop will survive and win the game). In fact, it doesn't matter whether the Cop steps in to save the investigated innocent if that player is chosen as the lynch target (2/3 either way).

If a townie is lynched Day 1, the Cop should follow a similar plan with an innocent investigation - protect himself or the confirmed innocent, but otherwise stay hidden. The benefits of staying hidden to endgame outweigh the risk of getting killed before confirming the single investigation.

The less obvious case is that if the Cop gets a guilty, he should *still* stay hidden, at least temporarily. The instinct is "I have a guilty, it's lylo, better claim", but in fact the correct play in this case is to pick a target at random, and have the Cop come out only if the lynch target is not the player he got a guilty on. It's likely the Cop will be forced to come out, but if the lynch randomly lands on the investigated scum, the EV doubles.

Anyway, putting it all together I get an EV of 6859/15750 (~43.55%).

I suspect we can push the upper bound down some by adding the following rule: If the Day 1 lynch is on scum, the scum are told who the Cop is (this is equivalent to allowing the scum to fake claim, forcing the Cop to counter). I suppose it's possible that the Cop should stay hidden, but I think it's unlikely. Anyway, if we add in that rule just for Day 1, it pushes the EV down to ~35.78%. Adding it in across the board would push the EV lower still (may go back through and calculate this, have other things to do at the moment).

There are more complicated things going on in the actual California setup. As was pointed out earlier in the thread, the scum shouldn't claim Cop 100% of the time, or a Townie claim is confirmed innocent - rather, they should claim Cop some large fraction of the time, in order to draw out the Cop most of the time but ensure that the town must lynch Townie claims. The Cop may be better off staying hidden if scum claim Cop - if the Cop is NKed, the scum gets lynched the next day, and the real Cop gets an investigation, but on the other hand the odds of a successful D1 lynch go way down and when the Cop does come out D2 he won't necessarily be believed (since if this strategy was viable, it may also be viable for scum to leave the Cop alive N1 and counterclaim him D2). Then you've got the whole "which claim is more likely to be true" problem, in all cases (both claim D1, both claim D2, one claims D1 and the other D2). It's a mess.

I think I'll tackle 2 Mafia, 1 Doctor, 4 Townies next. Tired of dealing with Cops.

[mith]

Oh, I certainly wasn't suggesting that particular assumption would apply for larger setups; I don't even think it could be strictly applied for California, for the reasons mentioned.

The 43.55% is for the "no fake claim" version; does your recursion agree with that? 35.78% is with the "remaining scum is told who the cop is if D1 is a scum lynch", which I don't think we can say is necessarily an upper bound (because Cop might want to stay hidden) - but it probably is higher than the actual EV (because allowing the scum to fake claim more generally should be a significant improvement for them, while allowing the cop to stay hidden will only help the town a little, if at all, and the rate at which scum should claim townie should be pretty low).

(In the 10:100+Cop setup, I'd expect the strategy would be for scum to claim Cop a large percentage of the time until the real Cop is dead, and for the real Cop to remain hidden until he's caught at least one unclaimed scum. Maybe two, but that might be pushing it. If the Cop survives long enough, he would eventually consider coming out with a lot of innocent results, of course. Probably some more tricky stuff to consider for when the Cop actually does come out. But that's pretty irrelevant if we can't even solve California.)

Actually, I think my next attempt with Cops may be a 1:2+Cop Night Start. Small enough to be managable, but it still has some interesting strategic considerations (what investigation the scum should claim, and how the town should handle claim order).

[mith]

Ok, I need to go through these again at some point because they're very much back-of-the-envelope calculations, but here's what I've got.

Obviously, if the Mafia hits the Cop (1/3 chance), it's a 1:2 endgame with an EV of 1/3.

Otherwise:

Mafia will always claim Cop. (Why? If he doesn't, the Cop is confirmed innocent; even in the Cop investigated the dead townie, it's a 50-50, and there's a 2/3 chance he investigated a living player, so the EV is 5/6.)

Assumption for simplicity: Mafia and Cop claim investigations at the same time. (I'll have to run through it later to see how things change if there's a random claim order.)

Mafia essentially has three choices: Claim a guilty on someone; Claim an innocent on someone; Claim to have investigated the dead player. Because of symmetry, it turns out that it doesn't matter wheather he claims a guilty or an innocent; so let's say he does one of the first four options (guilty on the Cop, guilty on the townie, innocent on the Cop, innocent on the townie) with probability p/4, and claims a dead player investigation the other 1-p.

The real Cop, of course, claims his actual result, and there is a 1/3 probability for each possibility.

In two of the five Mafia options, Mafia loses instantly - if he claims a guilty on the townie or an innocent on the Cop, he's dead with the Cop out. (This is where the random claim order makes things interesting - if Mafia claims second, he now knows which two cases to avoid.)

That leaves nine pairs of claims. Three are obviously symmetric - they claim guilty on each other, they both claim innocent on the townie, they both claim to have investigated the dead player. Two others are also symmetric, since there is no reason to favor a guilty or innocent from either perspective - one claims innocent on the townie, the other claims guilty on the first. In each of these five cases, it's a coin flip.

The four remaining cases:

Mafia claims Cop is Guilty, Cop investigated Dead - p/12
Mafia claims Townie is Innocent, Cop investigated Dead - p/12
Mafia claims Dead, Cop investigated Mafia - (1-p)/3
Mafia claims Dead, Cop investigated Townie - (1-p)/3

In each case, the Townie now has a choice: lynch the player who claimed a result, or lynch the player who claimed to investigate the dead. Let's call the probability of lynching the dead claim x.

Town win percentage is:

6*p/12 (the won cases)
+ 1/2*4*p/12 + 1/2*(1-p)/3 (the symmetric cases)
+ (1-x)*2*p/12 (the cases where the Cop investigated the Dead)
+ x*2*(1-p)/3 (the cases where the Cop has a result)

= 1/6 + 2p/3 + 2x/3 - 5px/6

To find the equilbrium values, we take the derivative with respect to each variable:

dEV/dx = 2/3 - 5p/6 => p = 4/5
dEV/dp = 2/3 - 5x/6 => x = 4/5

(The first says that if the Townie chooses p to be 4/5, the EV is constant with respect to x - even knowing p, the Mafia can't manipulate x to improve his odds. Likewise with the second.)

Thus: EV = 7/10.

The overall EV for the 1:2+Cop NS setup is then 1/3*1/3 + 2/3*7/10 = 26/45 = 57.7~%.

[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~%

[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.

[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.)

[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.

[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.)

[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.

[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.

[mith]

Note that for a Vanilla game, correct play with day start 3:9 is to no lynch to start the game; so 3:8 has the same EV.