Mafia Strategy when Cop Claims...

[LoudmouthLee]

Scenario:

You are a member of a 15 person Mafia game. Night one, you kill a random townie. However when day breaks... one of your mafia buddies is fingered by a claim of a cop.

After a quick lynch, it is proven, the cop nailed a mafia scum.

Problem:

What is the play next night? Should the mafia target the cop? or someone who they get a "doctor tell" from?

Further Wrinkles:

If it is a sound idea that the mafia should target someone else... what stops the doctor from NOT defending the cop?

This sounds an awful like the Prisoner's Dilemma.

Discussion here would be helpful. By the way, if this was discussed elsewhere in a previous thread, I apologize.

[Yoko Kurama]

If you think you know who the doc is in my opinion you should go after them hands down or anyone that might be rather than the cop.

Though the doctor may not protect the cop because he figures mafia will not target the cop since the doc will protect him, he may think thats what the mafia is thinking and protect the cop anyway. However it goes you should not go after the cop in my opinion.

[Narninian]

I could swear I've seen this topic come up twice before, but I can't seem to find it.
Anyway, *someone* determined optimal strategy was for doc to protect the cop X% of the time (higher than 50) and for the mafia to try to kill the cop Y% of the time (low number).

[LoudmouthLee]

As much as that's great.. what's the math behind it?

[Narninian]

wish I could find the thread... wondering myself if is sound or not..

Well it does *make sense*, but you have to still determine how ballsy you think the doc is, and it really is like the Princess Bride Wine Scene.

The doc is *obviously* going to protect the cop, so its a bad idea to try to kill the cop, but because of this the doc may think s/he is safe to try and protect somebody else, but the mafia will *know* the doc may be thinking this so may and try to kill the cop anyway. The doc may *know* the mafia will try this so may end up protecting the cop, making the mafia waste the kill.

If these scenarios are going through both the mafia's and Doc's head as possiblities, then the risk/rewards are different for each side.

The Risk of not protecting the cop for the Doc is: Losing the Doc, a very important figure in the game.

The Reward is: Potentially protecting somebody else which, for the doc, clears that person of being mafia, and wastes a mafia kill - given the town an advantage,


For Mafia- Risk is losing a mafia Kill
Reward is killing the cop.

Now for the mafia the Reward is a lot better, so if say, the doc would protect the cop 50% of the time then Ballpark figure then it would be smart of them to target the cop most of the time.

For the Doc, the Risk vs. Reward seems much less fruitful, and if the mafia would target the cop 50% of them time, it would be smart of them to almost always protect the cop.

I'm sure whoever posted numbers in the thread before started with that, and figured out what % of the time each party should target the cop, if both sides used 'optimal strategy'.

[Axelrod]

If it's night two, as per your example, I can't see the Doc protecting anyone other than a confirmed cop. What's the alternative? Protecting a random person who probably won't be targeted anyway and might even be mafia? The benefit to be gained from this is greatly out weighed by the benefit of protecting a cop and guaranteeing them another inspection. And as long as the Mafia don't identify you as the Doc, you will be able to keep protecting him.

The only time this changes is when another role is revealed that might also merit Doc protection, then you have to ue your best judgment about who to protect (and then still probably protect the cop.)

[Seol]

If it's night two, as per your example, I can't see the Doc protecting anyone other than a confirmed cop. What's the alternative? Protecting a random person who probably won't be targeted anyway and might even be mafia? The benefit to be gained from this is greatly out weighed by the benefit of protecting a cop and guaranteeing them another inspection. And as long as the Mafia don't identify you as the Doc, you will be able to keep protecting him.

If the Mafia know the doc is protecting the cop, then they won't target the cop. If the cop isn't being targetted, there's no point protecting him - you might as well protect a random townie and hope for no deaths. There's a delicate balance, though.

[Axelrod]

If it's night two, as per your example, I can't see the Doc protecting anyone other than a confirmed cop. What's the alternative? Protecting a random person who probably won't be targeted anyway and might even be mafia? The benefit to be gained from this is greatly out weighed by the benefit of protecting a cop and guaranteeing them another inspection. And as long as the Mafia don't identify you as the Doc, you will be able to keep protecting him.

If the Mafia know the doc is protecting the cop, then they won't target the cop. If the cop isn't being targetted, there's no point protecting him - you might as well protect a random townie and hope for no deaths. There's a delicate balance, though.

Well, yes, if you knew the Mafia wasn't going to target the Cop it would be pointless to protect him. But exactly when do you ever know this? What I'm saying is that the benefit to be gained from protecting a random person is outweighed by the benefit of protecting your cop. Just in case the Mafia is thinking like you are thinking. Bluffs and double bluffs are fine, but you should have a better reason for risking it than protecting someone at random. At least the way I see it.

[Narninian]

if you take a newbie setup (for example) - protecting a non cop can in, some cases, be a game breaker.

It gives the doc a confirmed innocent. If the Cop finds a confirmed innocent - thats 4 confirmed innocents barring counter claims.

In the 15 person mafia example, thats less the case, but it still denies mafia a kill, and gives the doc a confirmed innocent(well at least a confirmed non mafia). In 15 person games there are often other power roles in play other than doc/cop that are useful for town.

[Seol]

Well, yes, if you knew the Mafia wasn't going to target the Cop it would be pointless to protect him. But exactly when do you ever know this? What I'm saying is that the benefit to be gained from protecting a random person is outweighed by the benefit of protecting your cop. Just in case the Mafia is thinking like you are thinking. Bluffs and double bluffs are fine, but you should have a better reason for risking it than protecting someone at random. At least the way I see it.

This is where the game theory comes in. Assuming the cop isn't

infinitely

more valuable than the other townies (and he's not), it's

occasionally

worth the risk of targetting someone else, and it can be calculated how frequently that should be. If you pick the right frequency, the occasional times you lose the cop will be outweighed by the number of times you end up with no deaths. I'm not going to try and produce the numbers, though.

[Axelrod]

Well, yes, if you knew the Mafia wasn't going to target the Cop it would be pointless to protect him. But exactly when do you ever know this? What I'm saying is that the benefit to be gained from protecting a random person is outweighed by the benefit of protecting your cop. Just in case the Mafia is thinking like you are thinking. Bluffs and double bluffs are fine, but you should have a better reason for risking it than protecting someone at random. At least the way I see it.

This is where the game theory comes in. Assuming the cop isn't

infinitely

more valuable than the other townies (and he's not), it's

occasionally

worth the risk of targetting someone else, and it can be calculated how frequently that should be. If you pick the right frequency, the occasional times you lose the cop will be outweighed by the number of times you end up with no deaths. I'm not going to try and produce the numbers, though.

That's an interesting point, but it still seems to me that if you concede that protecting the cop is usually the correct play, then for all intents and purposes it should always be the play.

I'll use a blackjack analogy. Basic strategy dictates you hit when you have a 16 and the dealer is showing a 7, 8, 9, or 10. You know, when you take the hit, that you will probably bust, but you also know that, over the long haul, by taking the hit, you will win more hands than you would by standing. Basic Strategy doesn't say "you will win 25% of the time when you hit, and 10% of the time when you stand, therefore you should hit more than you stand (but still stand sometimes)", instead it says you should always hit.

The only exception to that is when you're counting cards and know the odds are actually tilted the other way. It doesn't seem to make sense to say that, if protecting the cop is right 85% of the time and wrong 15%, you should protect the cop in 85% of the cases and a random person 15% of the time. Chances are, you will do worse than if you just protected the cop every time.

[Mgm]

Of course, you should make sure there is a doc in the first place.
I was in a game once where the cop and the doc where masons and he couldn't protect him. In such cases going round trying to kill the doc is suicide.

[Narninian]

I see your point, but here's the thing. We're talking about 'optimal' play. the percentages we get would be how often they should do something, not how often their will be a succesful result.

If its 'optimal' play for doc to protect the cop 100% of the time,
then 'optimal' play for mafia would be to kill the cop 0% of the time.

Since the mafia is attacking the cop 0% of the time, it cant be ideal strategy to defend the cop 100% of the time, becuase its a wasted protect. There is where the numbers meet in the middle somewhere.

Lets say the 'optimal' strategy for doc would be to protect the cop 85% of the time. That doesnt mean 85% of the time it would have a better result - it mean the best play it to protect it that amount of time. The 15% of the time they are not protecting changes mafia 'optimal strategy' too. It might be 35% of the time or so that they should attack the cop. That doesnt mean that 35% of the time they will have a better result, but if they attack the cop 35% of the time - they'll have the best results over time.

Im making these numbers up - cuz I dont know where to begin calculating actual numbers, but Im sure they are out there.

[Dav]

Additional wrinkle: What if we have the above scenario, except 2 random townies end up dead, suggesting an SK? Does this really change anything?

My first instinct is that as doc I'd be that much more likely to protect the cop. Of course now we have the additional factor that the doc protection is not guaranteed to save someone, as they may be double-targeted. But of course in that case it doesn't matter WHAT I do, so I don't really need to worry about it.

As mafia or SK though, I have to consider whether teaming up to kill the cop is a good idea...hm. Probably too much of a risk to guess whether we'd think alike. Though of course if the doc doesn't protect the cop we score anyhow. And thinking as the SK, if I attack and I'm thwarted, assuming this isn't a game where people are told they were attacked, that would bring us down to one kill which at least has the benefit of misdirection--maybe it was a foolish night 1 vig and there's no SK...

[Cubsfan4ever]

Hmm... in Newbie 62 we had a scenario something like that

a townie was lynched Day 1
The Doc protected the would-to-be victim Night 1 (who was a townie)
The Cop found an innocent Night 1 (the OTHER townie)

[Mr. Flay]

Found it! (well, the most recent version, with some numbers): How much more valuable than a townie is a confirmed cop?

As for the idea of a double-kill on a known cop, I actually tried to arrange this in Greek Mafia over on GL recently, as LoudmouthLee probably saw. The cop was confirmed and there was a good chance there was a doc, but only one doc. Mafia and SK had accidentally double-killed the other cop the previous night. Unfortunately the lynchee for the day turned out to *be* the Serial Killer, which sort of borked my strategy and got me lynched the next day. So I don't actually know how successful it would be...but I was willing to trade my life in the effort.

[Axelrod]

Now observe as I embarass myself with my poor math skills.

If we assume that there is some kind of "optimal" strategy, by which I mean there is an ideal % that the Doc should protect the Cop. How does this play out? If you take a % at random, say the 85% figure I threw out before. If you play so as to protect the Cop 85% of the time and a random person 15% of the time, don't you come out worse than just protecting the Cop everytime?

Take the 85% of the times that you do protect the Cop. If this is the correct play 85% of the time, then your result is 85% x 85% or 72.25%. You must add to this the fact that 15% of the 15% of times you protect a random person it will be the correct play. This = .0225%.

72.25% + .0225% = 75%

So by "gaming the system," if you will, you end up with the correct result 75% of the time. Whereas if you just protected the Cop everytime you would be right 85% of the time.

I think this works with most sets of numbers as long as you assume that protecting the Cop is usually the correct play. If you throw out that assumption, however, then I guess anything goes. This is maybe the point I was trying to make with my original post.

[Locus Cosecant]

The point of gaming the system is that you make it so that

whatever the other side does

you come out with your maximal result. If you always protect the cop, then the scum have the easy strategy of killing non-cop every time, and you're left looking pretty stupid. At the equilibrium point, however, it doesn't matter what the scum do, your EV remains constant.

[StevieT92]

i think the scum should always go after someone else, becuase then after 2 nights, the town will start to get suspiciuos if there are no blocked kills and the *cop* is still alive.
also, if the mafia are good actors, then the cop will have no reason to target them.

[Yoko Kurama]

72.25% + .0225% = 75%

72.2725 actually. Which under most cases (if by chance you were rounding) would go down to 70 to the nearest five or 72 to the nearest one.

I wish we could post as guests in here still... I couldn't let the error slip it bugged me too much, but I didn't want to be the one shown correcting it.

[Aelyn]

72.25% + .0225% = 75%

72.2725 actually. Which under most cases (if by chance you were rounding) would go down to 70 to the nearest five or 72 to the nearest one.

I wish we could post as guests in here still... I couldn't let the error slip it bugged me too much, but I didn't want to be the one shown correcting it.

It was obviously just a typo. It should have been 2.25%; he forgot to convert from fractions to percentages.

[Ythill]

What is the play next night? Should the mafia target the cop? or someone who they get a "doctor tell" from?

These questions offer a limited strategy set, and any scum worth their salt will have created other options.

The most obvious strategy -the fingered scum should claim a protective role before the lynch- is too common to effectively flush out the doc. However, there are other options...

Sullying the cop's reputation can dissuade protection, especially if you can paint a realistic fake-claim-bus scenario without seeming like you believe it yourself. Presenting other protection options is also a viable strategy. Vig fake-claims come to mind.

If the scum have a gf, he should get himself investigated and claim a protective role within the next night/day cycle, probably better if the claim comes before the investigation. Non gfs need to keep themselves clean to keep the cop away from them.

I've had a lot of luck buddying to the cop. The thing about keeping a cop alive is that you know the town will follow him, so it makes the game predictable. If you can keep his investigations on those who will come up innocent and then NK his confirmed innocents, you can keep the playing field even until the doc is flushed or until it is determined that there is no protective role.

All you need is a day or two to push for mass-claim, and the game is in the bag.

The cop has to be killed before endgame, but targeting him that night is a risky move that is likely to cripple your team no matter who he investigates.

[hasdgfas]

[The Fonz]

The point of gaming the system is that you make it so that

whatever the other side does

you come out with your maximal result. If you always protect the cop, then the scum have the easy strategy of killing non-cop every time, and you're left looking pretty stupid.

How do you look stupid if whilst the scum are killing off townies, the cop is either reducing the pool of potential lynchables by confirming other innocents, or actually nailing scum?

The loss of a townie is nothing compared with simultaneously losing an investigation AND the credibility of your doc claim.

In that scenario, it's a no-brainer to protect the cop the night he claims, because the scum DON'T KNOW there is one, so the chances of them targetting the cop are significantly higher. Basically, I wouldn't EVER not protect the cop, until it has worked once, unless it's an open game. Until the scum know there is a doc, they are more likely to gamble that there isn't one.

Once you have protected successfully once, THEN it might be a good idea to protect the confirmed innocent like, 10% of the time.

[PokerFace]

If you

know

there is a doc. Try to kill the doc. If you don't know who the doc is in this scenario, then you should kill someone the cop has already confirmed since they won't be lynched now and they could be the doc. If you have a scum blocker, they should block the cop.

If you think there is no doc or the doc is already dead. Kill the Cop. If you have no proof a doc exists, shoot the cop in the head.