There's no such thing as an overeager vig

[Adele]

i agree with pie

that is all

I am shocked. Truly shocked. Fritzler, I would never have imagined this of you. In the games I've played with you and the comments I've seen you make, I've never seen any evidence that you'd be anything but a most

reluctant

vig. In fact, I don't believe you. I think this is a Yank's attempt at irony.

[Pie_is_good]

blah blah blah Since the C and D didn't change in these two scenarios blah blah blah

I disagree - you explained yourself why C and D each change. C and D certainly have the power to change over the course of the game - a cop, for example, is more valuable late-game when he has information, so his C goes up as the game goes on.

C and D certainly have the power to be dynamic variables. I would argue that that fact doesn't change the math attatched, because there's no reason to suppose that C and D don't go up proportionally to each other.

Pie, in a mountainous game, what is the "power" of the town and scum? Please come up for a formula in terms of the # of town and # of scum at the beginning with a day start.

I have no idea. Personally, I'm not big on mountainous theory. I'd guess the ideal equation would use exponents, as the wiki numbers seem to suggest that higher numbers in the beginning favor town (a trend that can easily be balanced out by mods).

Your mathematical model is inadequate, as it's based on the naive concept of random lynchings. It disregards

information

, and therefore cannot possibly account for my counter-contention that the desirable strategy is not the one that maximises the

number

of kills that are town-controlled, but the one that maximises the

quality

of the kills (with random town kills being superior to scum kills, but inferior to informed kills). You can't prove

anything

about Mafia strategy from first principles.

It also assumes that balance is linear, which is not the case (for example, four mafia is more than 33% more powerful than three mafia).

I certainly agree that balance is not linear, although I believe the equation does reflect that in the changing of C and D (again, the only way to invalidate the equation based on the fluctuating C and D variables is to show that power of the scum goes up as numbers get lower).

Which brings me to the second argument. Let's say each vig kill yields 1/3 of the information that a lynch would (certainly up for contention, but not entirely unreasonable - I'd say more than half the information of a lynch comes from the information gained from the player's death). In that case, overeager vigging would be the play if two or more kills are happening per night (really not that uncommon).

[Pie_is_good]

i agree with pie

that is all

:goodposting:

[Fritzler]

i agree with pie

that is all

I am shocked. Truly shocked. Fritzler, I would never have imagined this of you. In the games I've played with you and the comments I've seen you make, I've never seen any evidence that you'd be anything but a most

reluctant

vig. In fact, I don't believe you. I think this is a Yank's attempt at irony.

yea, we;re not that good at it sorry

but we're the best around at dick and fart jokes

[Seol]

Which brings me to the second argument. Let's say each vig kill yields 1/3 of the information that a lynch would (certainly up for contention, but not entirely unreasonable - I'd say more than half the information of a lynch comes from the information gained from the player's death). In that case, overeager vigging would be the play if two or more kills are happening per night (really not that uncommon).

Does it yield 1/3 of the information? That's a tricky one - there's no voting record or arguing for or against, you don't learn anything from anyone apart from the target. How do you quantify information - I sure can't. Does it yield the information you

need?

Sometimes - quite often, in fact - it is the play, but you can't say that "in general scenario X, vigs should all be overeager".

Sometimes, if you're having two nightkills per night, you need as many opportunities to take out scum as possible. Sometimes you need to slow the game down - after all, it's not just vigs that have town nightchoices.

[mith]

I haven't seen this amount of screwy math in one thread since... well, since whatever .999...!=1 thread I came across, whenever/wherever that was.

[Yosarian2]

I haven't seen this amount of screwy math in one thread since... well, since whatever .999...!=1 thread I came across, whenever/wherever that was.

Heh...on mietings, probably. That was a frustrating one.

Anyway, if the game is balanced to start with, completly random kills are probably bad for the town, becuase they give much less information then lynches and shorten the game. 2 random vig kills=1 less lynch in a standard mafia/town game, and I would think that 2 random kills yield a lot less information then 1 lynch. Pie mentioned the "two or more kills a night" scenerio, but that's still not clear, as the scum kills themselves are semi-random (IE:they will sometimes cross-kill scum) so it's even less clear that shorting the game by adding some complety random kills helped the odds.

The only time random kills might be a good idea is if the game starts with a major pro-scum advantage, because introducing more random factors in that case might make it at least possible for the town to win a game they probably shouldn't. Games like that are rare IMO, games that start out unbalanced towards the town seem to be much more common. And even then, random kills later in the game would be better as random kills night zero (even if you manage to hit a scum night zero, that gives you no information; if you random kill later and get lucky and hit a scum, you might be able to make connections and see who attacked that person and so on.)

[Mr. Flay]

You mean http://en.wikipedia.org/wiki/0.999... ?

[Thesp]

Anyway, if the game is balanced to start with, completly random kills are probably bad for the town, becuase they give much less information then lynches and shorten the game.

This is another reason why mods should seriously reconsider starting in night without allowing players to influence each other in some fashion. (In any games where I'll start w/Night, I'll require everyone to post at least twice before we move to night, sort of a pseudo-Day w/no lynch.) Random killings can wreck the careful balance a mod puts into a game.

[Pie_is_good]

Does it yield 1/3 of the information? That's a tricky one - there's no voting record or arguing for or against, you don't learn anything from anyone apart from the target. How do you quantify information - I sure can't. Does it yield the information you

need?

Sometimes - quite often, in fact - it is the play, but you can't say that "in general scenario X, vigs should all be overeager".

I honestly have no idea if it yields 1/3 of the information; I pulled that number out of nowhere. It seemed to be erring on the low side to me, but I don't know for sure. I don't claim to be able to conceretely quantify information - just as I can't concretely come up with formulas, units to put C and D in, or anything of the like. That doesn't mean those concepts don't exist, though, and it's not futile to make reasonable guesses.

Sometimes, if you're having two nightkills per night, you need as many opportunities to take out scum as possible. Sometimes you need to slow the game down - after all, it's not just vigs that have town nightchoices.

It should be noted that this whole string of "vig kills are worth 1/3 the information" was not an argument unto itself - it was intended as a refutation of the "Vig kills shorten the game" line. I was pointing out that a) they really don't shorten the game all that much, and b) the information that they give is sometimes worth shortening the game for.

It should also be noted that vigkills can overlap with scum kills as well as scum/scum overlaps.

I should also make it clear that I don't claim that

vigges should kill every night no matter what.

I'm just saying that making kills should be the default, and a vig needs a good reason

not

to.

Maybe only .999... percent of the time.

[mathcam]

You mean http://en.wikipedia.org/wiki/0.999... ?

Nope, that's a pretty reasonable set of explanations. The thread being referred to, assuming I stumbled across the same one, was spirit-crushing hair-pulling failure by a blogger to convince her readers of this fact. (She did a fine job of explaining, but the responders were immune to reason of any form, and weren't very polite about expressing there wayward ideas...)

Cam

[mith]

What I meant to say was:

since whatever .999...!=1 thread I came across

most recently

These pop up from time to time pretty much anywhere. Probably the most recent one I was involved in was this one on the GL.

(It should be noted that there are systems where .999... doesn't equal 1. In the reals, they're equal.)

[/aside]

[Yosarian2]

I should also make it clear that I don't claim that

vigges should kill every night no matter what.

I'm just saying that making kills should be the default, and a vig needs a good reason

not

to.

IMHO, vigging every night becomes the default after you've had a cop claim and have a few confirmed innocents from him, and then start picking off the unconfirmed. That way, the town moves closer to a situation where half of all townies are confirmed, which is a win.

Before that, the risk of killing the cop or killing cop-confirmed innocents is quite high, and vigging should only be done if you're got a reason to do that. Killing off one cop-confirmed innocent in a mini game can actually hurt the town more in some situations then killing off one scum would help the town.

[Fuldu]

(AC/BD)(1+(1/(A+B)(B-1)))

...and we know that 1/(A+B)(B-1) + 1 is always a number greater than 1, meaning that the new game balance = the old game balance x some number greater than 1. Therefore, the new game balance is more favorable to the town than the old one.

I disagree - you explained yourself why C and D each change. C and D certainly have the power to change over the course of the game - a cop, for example, is more valuable late-game when he has information, so his C goes up as the game goes on.

But if C and D change in consequence of the vig's kill, then the conclusion you've drawn about the new balance necessarily being greater than the old balance isn't necessarily true. If your final term increased by, say, 10% and C fell by 20% while D remained unchanged (I believe that's a plausible set of assumptions for when the vig kills a strong pro-town role), then the overall balance would be worse for town.

And, I repeat for the second or third time, leave C and D out of it entirely by making all pro-town roles the same and all the scum roles the same and construct some game scenarios where the A/B ratio at the beginning of the game is the same. I promise you that the games don't have the same balance. I gave the example of 5:1 and 10:2 from the Wiki numbers, as well as an easily analyzed scenario of 2:1 and 4:2. In neither case is the expected win probability for town the same in each case. I note however, that you haven't tried to refute either of those arguments.

[Pie_is_good]

But if C and D change in consequence of the vig's kill, then the conclusion you've drawn about the new balance necessarily being greater than the old balance isn't necessarily true.

Trueish (the statement is a bit ambiguous). I agree that it's not necessarily true in the sense that

there's a possibility the kill could hurt the town

. But that doesn't matter, because

C and D are mostly unpredictable, as are the overall results of the random vig kill.

So sure, it's not necessarily going to wind up in your favor, in the same way that getting 3:1 odds that a coin will come up heads isn't neccesarily going to wind up in your favor.

If your final term increased by, say, 10% and C fell by 20% while D remained unchanged (I believe that's a plausible set of assumptions for when the vig kills a strong pro-town role), then the overall balance would be worse for town.

True, and certainly plausible. The coin you bet on could come up tails, but that doesn't mean you made a bad decision.

And, I repeat for the second or third time, leave C and D out of it entirely by making all pro-town roles the same and all the scum roles the same and construct some game scenarios where the A/B ratio at the beginning of the game is the same.

I've already explained why this isn't an accurate refutation. You can show me all you want that C and D change with the number of players left alive; I'm with you there. The point I'm making is that it's near-impossible to predict beforehand the change of C and D, so it's rational to make plays assuming the average change (0, unless you'd like to show otherwise).

I promise you that the games don't have the same balance. I gave the example of 5:1 and 10:2 from the Wiki numbers, as well as an easily analyzed scenario of 2:1 and 4:2. In neither case is the expected win probability for town the same in each case. I note however, that you haven't tried to refute either of those arguments.

The reason I haven't directly refuted them is because I agree with the conclusions they yield (although they don't yield the conclusions you think they do). What those models show is that C and D change based on A and B. I have already agreed to this. But C and D could change either way.

[Thok]

I've already explained why this isn't an accurate refutation. You can show me all you want that C and D change with the number of players left alive; I'm with you there. The point I'm making is that it's near-impossible to predict beforehand the change of C and D, so it's rational to make plays assuming the average change (0, unless you'd like to show otherwise).

C and D are not "unpredictable". They are proxies for the relative power of the town/scum, which is clearly directly correlated to the size of town/scum. If town is killed, C will go down and D will go up. If Scum is killed the reverse is true.

You're the one arguing the unconvetional point of view. It's your duty to convince us.

If you want to make this a semialid argument, at the very least throw out C and D, come up with some rough function p(A,B) that determines the probability of town winning give the size of A and B, and use that probability to determine the likelihood of a vig kill being useful. You've basically told us that you think p(A,B)=AC/BD which is just plain wrong from am empirical point of view.

[Pie_is_good]

You're the one arguing the unconvetional point of view. It's your duty to convince us.

I certainly agree, and that's what I spent the first post doing mathematically.

Now a refutation was made, saying that C and D

do

change. I made the claim that, while C and D do change, average change in C and average change in D both equal 0. If that statement is true, the mathematical model is still accurate. If someone would like to claim that that statement is not true, they're welcome to, but they now hold the unconventional point of view, and it is now their duty to convince me.

[mith]

I was hoping the joke comment would suffice to express my thoughts on this, but I just can't help myself.

Pie:

a. Your initial mathematical model is horribly naïve. Mafia can not be modelled with a function like that.

b. Even if you could come up with functions C and D to make P = AB/CD hold (which is doubtful), at the very least they would clearly depend on A and B, as pointed out by Thok right at the start, and so the whole construction is dubious as you treat them as independent variables.

ci. Even ignoring a. and b., claiming that they average change of either C or D is 0 is ridiculous when you don't even know what C and D are.

cii. Further, the onus is still very much on you. You have claimed to have a case backed by mathematical arguments. People have pointed out these mathematical arguments are flawed. To then turn around and make a completely unsupported claim to fill the hole and claim it is somehow now our duty to convince you you're wrong is stupid. I wish I could get away with stuff like that all the time, it would make my work a lot easier. It's one of those things that can't be argued against, because the claim is based on an entirely misguided foundation - the most I can say about C and D is that I don't believe they exist in any useful form.

d. Even if everything else is fine, which it isn't, you still can't treat C and D as constants just because on average they don't change. For a very simple example, say P = 1/D, and D is initially 2. Say a coin flip happens and D either goes up by 1 or down by 1. Clearly the average value of D is still 2, but now the average value of P is (1/2)(1/1 + 1/3) = 2/3.

[Fuldu]

The reason I haven't directly refuted them is because I agree with the conclusions they yield (although they don't yield the conclusions you think they do). What those models show is that C and D change based on A and B. I have already agreed to this. But C and D could change either way.

But you're not taking the next logical step in this chain. If C and D change based on A and B, then there's no mathematical guarantee that (AC/BD)(term) increases simply because (term) increases. And unless you want to systematically discuss the claim that, in addition to C and D not changing on average, that when they do change, they change in rough proportion to one another, this means that your conclusion just doesn't hold. I don't believe the "not changing on average" claim, but don't quite see how to refute it, given that these are theoretical constructs rather than calculable variables. But I would expect that, in general, there would be either zero or a negative correlation between the change in C and D, since what's good for one side is bad for the other.

[Pie_is_good]

a. Your initial mathematical model is horribly naïve. Mafia can not be modelled with a function like that.

...begging the question of

why?

Some parts of mafia certainly can be modelled mathematically, so I'm not sure what you're basing that claim on.

b. Even if you could come up with functions C and D to make P = AB/CD hold (which is doubtful), at the very least they would clearly depend on A and B, as pointed out by Thok right at the start, and so the whole construction is dubious as you treat them as independent variables.

ci. Even ignoring a. and b., claiming that they average change of either C or D is 0 is ridiculous when you don't even know what C and D are.

These can both be answered by defending the

average change = 0

thing.

While I certainly accept that C and D depend on A and B, they are still effectively independent until a correlation is established. It's like this:

I choose a random integer 1-20, then add another random integer between -3 and 3. I call the second random integer A and the end result C. While A and C are certainly not independent variables, because I could not tell whether A was more likely to make C bigger or smaller, the average of C is the same as it would be without introducing A in the first place.

If you want to convince me that my randomizer is broken and A is more likely to subtract from C, you can certainly do that - but the onus is on you to show me how my randomizer is broken.

cii. Further, the onus is still very much on you. You have claimed to have a case backed by mathematical arguments. People have pointed out these mathematical arguments are flawed. To then turn around and make a completely unsupported claim to fill the hole and claim it is somehow now our duty to convince you you're wrong is stupid.

The All-Knowing Razor states that the person making the positive claim has the burden of proof. I initially had the burden of proof, which I fulfilled. The refutation of my argument took the form of a positive claim, though (

as A goes down, C goes down

, whereas

No notable correlation between A and C

is the default). I don't need support of my assumption that A and C have no correlation; you need support that A and C

do

have some correlation.

I wish I could get away with stuff like that all the time, it would make my work a lot easier. It's one of those things that can't be argued against

I believe that's called "being right"

because the claim is based on an entirely misguided foundation - the most I can say about C and D is that I don't believe they exist in any useful form.

C and D are abstract representations of town and scum power. The concepts of town and scum power

do

exist. They certainly don't have measurable units (the town is not 3 meters more powerful than the scum), but that's immaterial as they express a ratio rather than individual value.

d. Even if everything else is fine, which it isn't, you still can't treat C and D as constants just because on average they don't change. For a very simple example, say P = 1/D, and D is initially 2. Say a coin flip happens and D either goes up by 1 or down by 1. Clearly the average value of D is still 2, but now the average value of P is (1/2)(1/1 + 1/3) = 2/3.

...except that the whole point of my argument is that as long as we don't have a simple P=1/D formula to plug in, we have to assume P its average. I'm saying that we don't have a P=1/D formula to use (the applicable formula is just as likely P=D^1/2, in which case the coinflip will, on average, lower the value of P)

[Fuldu]

These can both be answered by defending the average change = 0 thing.

I don't see anywhere that you've

defended

this, so much as just claimed that since these variables are incalculable, it is reasonable to assume. I can't speak for mith, but if I were to try to point to a positive claim that he still feels is your burden to prove, I think this is it.

...except that the whole point of my argument is that as long as we don't have a simple P=1/D formula to plug in, we have to assume P its average. I'm saying that we don't have a P=1/D formula to use (the applicable formula is just as likely P=D^1/2, in which case the coinflip will, on average, lower the value of P)

Except that your defense of your assumption of the independence of C and D from A and B that's just above that depends on the assumption of a particular mathematical formula for the relationship between them. It's true that if C is an equiprobable integer from [-3, 3], then the expected values of A and (A+C) are the same. But the expected value of A*C would now be 0, which is not likely the expected value of A. You've chosen an additive formula, but many others exist for which your claim is false.

Also, in the quote just above, even the formula you've selected illustrates mith's point. In the scenario he's described, the expected value of D doesn't change, but the expected value of P changes from sqrt2 to (1+sqrt3)/2. The fact that it has gone down is immaterial to his point. The fact that it changed at all is the problem with your argument.

More broadly, though, you can't pick a particular formula when it illustrates your point, but say that it could be any formula, so we have to assume the average, when a particular formula doesn't illustrate your point.