There's no such thing as an overeager vig

[Pie_is_good]

Alright, time for a second vig rant from Pie. The conclusion this time:

Vigilantes should kill every night they get the chance.

There's one intermediate conclusion to reach first, though - I'll also argue that

random deaths are good for the town.

That second conclusion seems a bit counter-intuitive, but its support is all mathematical.

Let A=The number of protown players alive
Let B=The number of proscum players alive
Let C=The average power of a protown player
Let D=The average power of a proscum player

Then AC=Power of the town and BD=Power of the scum. (AC)/(BD) expresses the balance of the game, where a higher number means the town has an advantage and a lower number means scum has the advantage (1 would mean that the game is perfectly balanced).

Now, let's say the mod were to decide to randomly off a player. The game balance after the decision is made (but before the decision is executed) is:

(odds of hitting a protowner)(New Gamebalance after hitting a protowner) + (odds of hitting a proscummer)(New Gamebalance after hitting a proscummer)

[ (A) / (A+B) ] [ ((A-1)(C)) / (BD) ] + [ (B) / (A+B) ] [ (AC) / ((B-1)(D)) ]

[ ((A)(A-1)(C)) / ((A+B)(B)(D)) ] + [ (ABC) / ((A+B)(B-1)(D)) ]

[ ((A)(A-1)(C)(B-1)) / ((A+B)(B)(D)(B-1)) ] + [ (ABsqrdC) / ((A+B)(B-1)(D)(B) ]

((A)(A-1)(C)(B-1) + ABsqrdC) / ((A+B)(B-1)(B)(D))

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

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

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

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

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

(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.

The obvious next step from this conclusion is that those who have the power to make random deaths happen and are protown should do so. The vig can do exactly that,

and

his kills will be better than random,

and

he knows for sure that he won't be killing himself. His odds of hitting scum over town are significantly better than just a random kill.

(Side note: The whole "random deaths are good for the town" thing is actually a good argument for suicide vigging, too - it shows that gaining a "Free" lynch is, in fact, always a good thing.)

So. There's no such thing as an overeager vig - All vigges should choose to kill every night, anyways.

The most ironic thing about all this is that overeager vigges are more powerful than normal vigges, as the player playing the role is forced to make the correct strategic decision every night.

Pie

[Thok]

Let A=The number of protown players alive
Let B=The number of proscum players alive
Let C=The average power of a protown player
Let D=The average power of a proscum player

Then AC=Power of the town and BD=Power of the scum. (AC)/(BD) expresses the balance of the game, where a higher number means the town has an advantage and a lower number means scum has the advantage (1 would mean that the game is perfectly balanced).

C and D clearly depend partially on A and B (compare 2 townie and 1 scum to 1 townie and 1 scum). To do this correctly, you'd have C_old, D_old and C_new, D_new (in fact you need C_scumkilled, D_scumkilled, C_townkilled, D_townkilled), and you need to define the dependance of the old variables on the new variables. Your AC/BD above is actually some weird mix of the new version of the variables, and thus need not be equal to AC_old/BD_old.

There are other issues (namely that not all townie deaths are the same) but that's the big issues with your argument.

[Kelly Chen]

In general I'd agree that vigs should kill. Just as in general I'd agree that the town should lynch.

(Side note: The whole "random deaths are good for the town" thing is actually a good argument for suicide vigging, too - it shows that gaining a "Free" lynch is, in fact, always a good thing.)

Is the vig's special knowledge about his own alignment unimportant?

Even if so, I don't think this argument does support that. The town has to weigh the possibility that the vig really is protown, in which case he can kill off a lot more random people than just himself.

[Pie_is_good]

At Kelly - I was referring to Seol's argument made in thread that an extra lynch is not always good. As for the other questions, I'd argue no to the rest of the statements, but those fall outside the scope of this thread.

At Thok - I'd certainly agree that C and D depend on A and B, and also that some roles have synergy with others (2 mafia are less than twice as good as 1 mafia, since the number of nightkills remains the same, but 4 mafia are more than twice as good as 2 mafia, as they can work together in lynches), but for your argument to be valid, you'd have to show that synergy between mafia roles < synergy between townie roles. Until then, because the changing of the C and D variables are just as likely to rise as they are to drop, it's rational to base numerical arguments off their current value. Similarly, the fact that not all townie deaths are the same is another non-issue, given that C represents the

average

value of a protown player. Given only the knowledge that a kill is going to fall on a protowner, C is how much you should expect that kill to hurt the town.

[MrBuddyLee]

If the math favors overeager vigging, it's good for gaining wins, but probably not so good for gaining friends. I'd hate to be the victim of a random vig if I'd played a good game to date.

If there's no one significantly scummy, I think no vig is a perfectly reasonable choice. I imagine it'd make the results of the game less random and more dependent on actual scumhunting skills.

[Fuldu]

I can think of several reasons to disagree with this conclusion, but they mostly fall under the category of "I don't think your numerical constructs C and D actually mean what you want them to." My main concern is that vig kills rob the town of important context that is vital to being able to determine who is on what side. When the vig kills scum, this is usually a tradeoff that is worthwhile, but when a vig kills a pro-town player, they're dead and the town learns very little from their death as compared to if they had been lynched.

Also, look at some simulation numbers for games with no roles other than plain scum and vanilla townies. A game with 5 townies and 1 scum gives the same (AC)/(BD) ratio as a game with 10 townies and 2 scum, but the actual balance of those games is different - 46.67% and 35.21%, according to the Wiki.

Finally, your math assumes that all players are equally likely to be killed. I suppose you might be recommending that all vigs should kill randomly, but I doubt that's what you'd actually recommend. And while I can't say exactly what happens to your math if you remove that assumption, I can say that I doubt the conclusion is quite as clear. Further, the generally highest powered scum player often has kill protection, which would mean that the actual value of targeting scum is lower than D, since you won't be able to kill the GF.

[Pie_is_good]

In terms of making the game more fun, I can see where you're coming from. If only gunning for the win [/badpun], you should always kill. If trying to make the game more fun, I could see where you would nokill, but remember - I'm not actually advocating random vigging. Who to vig is as much of a scumhunt as a lynch is, but only one man participates in it.

[Yosarian2]

I'm not sure I agree with that. Random kills can never be expected to change the town/scum ratio; if 2/3's of the people in the game are GG's, and 1/3 is scum, and 3 people are randomally killed, then on average one of them will be scum and 2 will be town, which means that the town/scum ratio will be the same as it was before.

What it will do is make the game shorter, which is bad for the town. Less votes, discussion, and bandwagons, and less night information from cops and such. That's why it's much harder for the town to win a newbie game then to win a 12 person game; simply because a longer game with more days and nights is much better for the town.

That being said, if your odds are better then average (say, you have a strong hunch person X is scum, or there are 3-4 people who are 100% cleared and so you take out one of the unconfirmed), then go for it and make the kill. But if it's completly random, like a night 1 vigging, then it hurts the town.

[Fiasco]

I'm not sure I agree with that. Random kills can never be expected to change the town/scum ratio; if 2/3's of the people in the game are GG's, and 1/3 is scum, and 3 people are randomally killed, then on average one of them will be scum and 2 will be town, which means that the town/scum ratio will be the same as it was before.

Averaging over the vig outcome and then calculating the change in ratio from that average (which is what you're doing) is not quite the same as calculating the change in ratio from each vig outcome, then taking the average ratio (which is what Pie is doing). In practice the difference will be very small, though. And if you choose to measure game balance with a difference rather than a ratio ("townie power - scum power" rather than "townie power / scum power"), random deaths won't change game balance. Using the expected difference seems more appropriate to my gut than using the expected ratio; I think the latter goes wrong when you try to figure out the effect of multiple days of vigging.

What it will do is make the game shorter, which is bad for the town. Less votes, discussion, and bandwagons, and less night information from cops and such.

Less scum nightkilling.

if your odds are better then average (say, you have a strong hunch person X is scum

I don't think strong hunches make an average player's odds better than average. According to the statistics, people in C9 games lynch very slightly

worse

than random. They could improve their game by using dice tags. Apparently, most of the things the average player thinks are scum tells, simply aren't. I don't think people here have assimilated this insight yet.

Not sure what all this means for vigs. I've never been a vig, but I'd probably be quite aggressive.

[Yosarian2]

Less scum nightkilling.

Well, that's true, but that's apparently not enough to make it worthwhle

I don't think strong hunches make an average player's odds better than average. According to the statistics, people in C9 games lynch very slightly

worse

than random. They could improve their game by using dice tags. Apparently, most of the things the average player thinks are scum tells, simply aren't. I don't think people here have assimilated this insight yet.

I disagree. You're forgetting that some of those votes are scum votes. If the results are close to average, and some of the voters are scum who don't want scum to be lynched, that would imply that townies who do want scum lynched vote better then avarage.

[Kelly Chen]

if your odds are better then average (say, you have a strong hunch person X is scum

I don't think strong hunches make an average player's odds better than average. According to the statistics, people in C9 games lynch very slightly

worse

than random. They could improve their game by using dice tags. Apparently, most of the things the average player thinks are scum tells, simply aren't. I don't think people here have assimilated this insight yet.

I guess you're referring to D1 lynches in newbie games. I don't know that I would generalize that to average players in any situation.

Not sure what all this means for vigs. I've never been a vig, but I'd probably be quite aggressive.

Are you an average player? Roll a die. If you get yourself, roll again. Vig that person, and as many other people as the mod allows. You're not vigging yourself, so your odds are better than the town's lynches.

[Pie_is_good]

Averaging over the vig outcome and then calculating the change in ratio from that average (which is what you're doing) is not quite the same as calculating the change in ratio from each vig outcome, then taking the average ratio (which is what Pie is doing). In practice the difference will be very small, though. And if you choose to measure game balance with a difference rather than a ratio ("townie power - scum power" rather than "townie power / scum power"), random deaths won't change game balance. Using the expected difference seems more appropriate to my gut than using the expected ratio; I think the latter goes wrong when you try to figure out the effect of multiple days of vigging.[/quote]

I certainly looked at the townie power-minus-scum power formula. The problem with that is that C and D have abstract units attatched. C=1 and D=2 is perfectly equivalent to C=2 and D=4. This throws some obvious problems into calculating by difference that can only be solved by calculating by ratios.

Although it also seemed counter-intuitive to me (I originally did the math setting out to prove that game odds were the

same

after a random kill, then checked my math about 7 times before deciding I was right), the following scenario made more sense in my mind:

Imagine a game of A=12, B=4, C=1, D=3. The game balance is perfectly even right now - each side has a 50% chance of winning. Now we start introducing random kills until A=3 and B=1. With more random kills introduced, balance is no longer even - If you were to finish out the game by killing randomly, the town would have a 75% chance of winning. Hence, as you get closer and closer to the end of the game, the random deaths help the town more and more (which is reflected in the formula - the number that the original ratio is multiplied by gets higher and higher as the numbers left alive go down).

[Kelly Chen]

If you're making the kills totally randomly (i.e. it's not even a vig doing it), I don't understand how the values of C and D make any difference.

How do you find that with random kills, the town should win 75% of the time with a 3:1 situation? It seems clear to me that this is still 50/50.

[Kelly Chen]

EBWODP

Looking at the math, I think C and D are a distraction. If you know A and B and take the setup to be in balance, you can determine whether another situation is similarly balanced just by the new ratio of A to B.

Say we start with 9 town and 3 scum. To preserve this same balance we need to end up with three times as many town as scum.

(9 / 12)((9 - 1) / 3) + (3 / 12)(9 / (3 - 1))
(3 / 4)(8 / 3) + (1 / 4)(9 / 2)
2 + 1.125
3.125 new ratio

Seems to work, although I don't understand why.

3 townies and 2 scum (ratio is 1.5):
new ratio seems to be 1.8.

2 townies and 1 scum (ratio is 2):
division by zero error.

This doesn't seem to be related to the obvious fact that the town has to lynch (even randomly) if it wants to win, since even if we found that random lynches generally worsen the town's situation, the town would still have no choice but to try.

[Thok]

Ratio of town to scum seem to be wrong thing to use; really it should be some measurement of winning percentage.

There's also an oppurtunity cost issue; making a random vig early on make prevent the vig from making a more informed choice later in the game.

That said, I do agree that there are people who would be better off trying to random vig then to trust there instincts.

[Pie_is_good]

C and D could certainly be distractions, but I wanted the math to cover all scenarios - not just the scenario where the game happened to be perfectly balanced.

To clarify my attempt at explaining the weirdness about the town gaining an advantage through random deaths...

Let's say 9 town and 3 scum. It's perfectly reasonable to assume that this is a balanced game - 50/50 chance of winning on either side.

Now, let's say the mod decides to make 11 random kills. Suddenly, the odds of the town winning are 75%. Conclusion being that each of those random kills helped the town 1/11 of the 25% gain the town saw.

[mikeburnfire]

Alright, time for a second vig rant from Pie. The conclusion this time:

Vigilantes should kill every night they get the chance.

There's one intermediate conclusion to reach first, though - I'll also argue that

random deaths are good for the town.

That second conclusion seems a bit counter-intuitive, but its support is all mathematical.
etc.. etc.

It all made sense... until you tried to do math with letters.

[Thok]

Now, let's say the mod decides to make 11 random kills. Suddenly, the odds of the town winning are 75%. Conclusion being that each of those random kills helped the town 1/11 of the 25% gain the town saw.

Sure, but mafia doesn't work that way. A more compelling scenario would be to have the mod start modkilling one person at a time until you either have a guaranteed town win or a guaranteed scum win; in that situation the odds of a mafia win are much higher then you suggest.

[Fuldu]

And, again, I would point to the math pages on the Wiki which point to the fact that a game with X pro-town players and Y anti-town players (in a single group) does not have the same balance as a game with 2X pro-town players and 2Y anti-town players. The ratio of pro-town to anti-town is a useful rule of thumb for balance, but it isn't anything even approaching a mathematical equivalence.

By way of illustration, consider a day-start game with 2 vanilla townies and 1 scum. Under anything even approaching good play by scum, it's hard to see how this could end as anything short of a random lynch, with a 33% chance of a win for town. Compare that to a day-start game with 4 vanilla townies and 2 scum. Now, it's possible that some degree of association between the two scum might be seen and pushed by a member of the town on Day One. I happen to think it's unlikely with decent scum play, but not impossible. So maybe the town has a slightly better chance than 33% of getting scum Day One. If they don't, scum will be able to force a no lynch Day Two and win that night. But assume that town finds scum Day One. Now it's 3 vanilla townies and 1 scum. If they tied the dead scum to the live scum, then they're in good shape. Otherwise, they may well be stuck with a random, lynch-or-lose choice (since a wrong lynch will still give them a tie the following day), this time with only a 25% chance of hitting the scum.

Since the C and D didn't change in these two scenarios, and the ratio of A to B is the same both ways, your math is requiring that town has the same chance of winning these two games. My analysis says that, while it isn't mathematically impossible that they'd be the same, it's mathematically very unlikely, relying basically on some outlier usages of phrases like "slightly better chance" meaning "a whole lot more."

[ShadowLurker]

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.

[Zindaras]

If the math favors overeager vigging, it's good for gaining wins, but probably not so good for gaining friends. I'd hate to be the victim of a random vig if I'd played a good game to date.

:goodposting:

If there's no one significantly scummy, I think no vig is a perfectly reasonable choice. I imagine it'd make the results of the game less random and more dependent on actual scumhunting skills.

I tend to vig on any Night but Night 0. I usually get rid of lurkers. I don't even care that much if they're town or not, but they're useless and need to go. Besides, most Cops tend to avoid lurkers, so you're reducing the number of unconfirmeds as well.

If my scumdar is tingling a lot regarding a player, I'll go ahead and vig him, too.

[Fritzler]

i agree with pie

that is all

[Seol]

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).

That said, I am in favour of aggressive vigging strategies.

[Axelrod]

Now, let's say the mod decides to make 11 random kills. Suddenly, the odds of the town winning are 75%. Conclusion being that each of those random kills helped the town 1/11 of the 25% gain the town saw.

Sure, but mafia doesn't work that way. A more compelling scenario would be to have the mod start modkilling one person at a time until you either have a guaranteed town win or a guaranteed scum win; in that situation the odds of a mafia win are much higher then you suggest.

And it's not even that good for the town, as in a more realistic simulation, every other mod-kill could not even hit a scum (since they control the night-kills).

I note that in the Theory Post on the Wiki, figuring random Lynchings but non-random Night-killings and with 9 "humans" and 3 "wolves": humans (Town) win 0.164.

Although I suppose the point of this argument is that truly "random" Night-killings via a vigilante favor the town. And yes, they favor the town over the non-random Night-killings of mafia. But whether they are always mathamatically the correct decision I don't know.

[Glork]

I'd make an aggressive vig. Then again, if you couldn't guess that by now, you've probably never played with me ever.