How much more valuable than a townie is a confirmed cop?

[Aelyn]

Simple question, really: How much more valuable, to the town, is a confirmed cop compared to an unknown townie? For simplicity's sake, let's say in a newbie game

I ask simply because it's occured to me that game theory states that, even with a confirmed cop, it's sometimes right for doctors not to protect him. And it's sometimes right for the scum to shoot at him.

After all, consider the situation from the doc's eyes. He knows the scum will believe he'll protect the cop overnight, and so won't try to kill him. Therefore, it would seem that it's an error of sorts for the doc to protect the cop.

However, the scum can use similar logic, and figure that sometimes it's right for the doctor to not protect the cop. Therefore, it's perhaps worth the scum gunning for the cop.

The only question needed before the ratio of protecting cop to not, and gunning for cop to not, can be figured out, is how much more is a confirmed cop worth to the town than a regular townie? And please excuse the mind-aching grammar in that sentence.

[Locus Cosecant]

Depends on the situation, unfortunately. Cops are more valuable in mid-game than anywhere else.

[Save The Dragons]

Townies are worthless. Unknown townies are worth a little more.

The following is strictly for newbie games.

Scum should never think the doctor will not protect the cop. And the doctor should protect the cop. Like I said, the townies are worthless. It's a stalemate for the mafia, but if the only chance the scum has is if the cop dies.

In other games, the scum should look for the doctor and then get the cop. Often times doctors get lynched.

And I'd say a confirmed cop at the end of the game is really valueable. That's untainted insight, a confirmed innocent, and possible results at endgame.

[Narninian]

[quote="Save The Dragons"]

Scum should never think the doctor will not protect the cop. And the doctor should protect the cop. Like I said, the townies are worthless. It's a stalemate for the mafia, but if the only chance the scum has is if the cop dies.
quote]

Thats exactly why, its *almost* safe to protect a townie rather than the cop. It gives another confirmed innocent, and one less dead the next -- its tricky situation - but a townie being saved gives town an advantage it could use to win the game (Cop has an extra investigation, Doc also has a confirmed innocent, and nobody dies).

[Locus Cosecant]

It's clear that unless cop protecting is the dominant strategy, strategy mixing is appropriate. However, what mix to use is up in the air.

[Save The Dragons]

What I meant is that scum should always try to kill the cop, because having 4 people at the end hurts the town more than it hurts the scum. Because if there's 3, then of the 3 people (scum, doc, dead townie), both a cop result on the doc or scum will end in a town win, where with 4, only the scum will be a sure bet for the town. Also, the scum will be able to pick between claiming doc or townie.

[Locus Cosecant]

Yes, in that single situation, perhaps. More generally, your statement does not apply.

[jeep]

I generally think of townie as the baseline: 1
Then I think of sane cops worth about 3.

I was going to model this and figure out the numbers (assuming perfect play), but I haven't been able to get to it recently.

The numbers I ran came up to something like: protect the cop 99% of the time and a random other person 1% and the scum should target the cop 15% of the time and someone else the rest. (That's 100% from memory, so if you find conflicting information, go with the other.) It should be posted in the archive somewhere.

-JEEP

[Save The Dragons]

Yes, in that single situation, perhaps. More generally, your statement does not apply.

Yeah, I know, but without knowledge of which roles are in play, it makes it a lot harder to figure out.

[halo freak]

He knows the scum will believe he'll protect the cop overnight, and so won't try to kill him

not always once when i was mafia i knew that i was screwed because we had a confirmed doc confirmed cop and me + a townie i knew that the only way i could save myself was to kill the cop so i tried to kill them even though the doc protected the. It was a last resort if the doc was playing tricks on me then i might win but there was nothing else i could do. So you can't always make that assumation depends on the situation.

[halo freak]

gold rush newbie 57 i think for anyone interested

[inHimshallibe]

Yeah, that was the one, halo freak.

[halo freak]

you were in it, i forgot. i think the only way out of that situation is a strong count cop claim, annoyingpest's wasn't up to much. But i knew when you lynched annoyingpest that the end was near.

[Mr. Flay]

I'd consider a sane, verified cop (or a known insane one) almost infinitely more valuable than a confirmed townie. Which is not to say the townie is unvalued, you need them...but the cop has additional skills in addition to everything a townie has (

i.e.

their vote).

Because of this, I don't believe the doctor should *ever* gamble on not protecting the cop, unless they know something bizarre about the mafia's night strategy (and if you know that much, you should know who they are already). I like jeep's numbers...1% chance of the doc not protecting the cop making sense, slightly greater chance for the mafia to attack a known cop anyway. Of course with two killing groups, you can sometimes override the doc's protection, which is a whole other kettle of fish...

[Locus Cosecant]

Treatise on Targeting

Let us set up a scoring system such that the mafia achieves an expected value of 1 from killing a townsperson. Let us further define X to be the expected value of killing a confirmed cop, Y to be the value of killing the doctor, Z to be the number of non-cop players remaining in the game, and W to be the number of remaining scum. Assume that, other than scum, one cop, and one doc, all roles are vanilla townies. It is trivial to prove that, if the doc does not protect the cop, sans any other information, the doc should protect a random element of the set which Z enumerates, and, similarly, if the scum do not attack the cop, they should attack a random non-scum non-cop player.

Since mafia is a zero-sum game between scum and town, we can set up a table of expected values for scum, and the town's values will be the same, but with inverse sign.

If mafia targets cop, and doc targets cop: EV=0.
If mafia targets cop, and doc targets elsewhere: EV=X.

Those are the easy ones. More difficult is what comes when mafia does not target the cop, because there's a doc out there to be killed as well.

If mafia targets elsewhere, and doc targets cop: The kill always goes through, so the expected value is the average value of all remaining targets. There are Z-W remaining targets(scum won't target scum), so the EV is (Y+Z-W-1)/(Z-W).

If both target elsewhere: The kill might be blocked. If the mafia targets the doc, the kill can't be blocked, so they're guaranteed an EV of at least Y/(Z-W). Assume they target a non-doc, which they do with a probability of (Z-W-1)/(Z-W). The odds that the doc will block that targeting at random are 1/(Z-1), since the doc does not know who mafia are. Thus, the odds it will go unblocked are (Z-2)/(Z-1), so the total EV is Y/(Z-W)+((Z-W-1)(Z-2))/((Z-W)(Z-1)) which is equivalent to Y/(Z-W)+(Z^2-WZ-3Z+2W+2)/(Z^2-WZ+W)

To summarize, then:
Both target cop: 0.
Neither target cop: Y/(Z-W)+(Z^2-WZ-3Z+2W+2)/(Z^2-WZ+W) (For convenience, let us call this value Q)
Mafia targets cop: X
Doc targets cop: (Y+Z-W-1)/(Z-W) (For convenience, let us call this value K)

The doctor needs to make it so that the mafia realizes no advantage by either targeting

or

failing to target the cop. We will define the probability with which the doc protects the cop to be A. Thus, (1-A)X=AK+(1-A)Q, so X-AX=AK+Q-AQ, so X-Q=AX+AK-AQ, so A=(X-Q)/(X+K-Q).

Similarly, the mafia needs to ensure that the doctor realizes no advantage from either of his choices. We will define the probability with which the mafia targets the cop to be B. Thus, (1-B)K=BX+(1-B)Q, so, similarly, K-BK=BX+BQ+Q, and K-Q=BX+BK+BQ, and thus B=(K-Q)/(X+K+Q).

We are left with the following formulae:
Chance doc targets cop: (X-Q)/(X+K-Q)
Chance mafia attacks cop: (K-Q)/(X+K+Q)
Calculating Q:Y/(Z-W)+(Z^2-WZ-3Z+2W+2)/(Z^2-WZ+W)
Calculating K:(Y+Z-W-1)/(Z-W)

To put this into an example, let's take jeep's value of 3 townies for a cop as X, and take the standard newbie setup after a day 1 cop claim and no-lynch. Let us further assume that the two power roles are of equal power, and thus Y is also 3.

X=3
Y=3
Z=6
W=2
Thus, we can calculate Q and K.
Q=3/(6-2)+(36-12-18+4=2)/(36-12+2)= 59/52
K=(3+6-2-1)/(6-2)=3/2
From these values we can calculate the chance of each group targeting the cop.
Doc protects cop:(3-59/52)/(3+3/2-59/52) 97/175, which works out to be about a 55.4% chance.
Mafia attacks cop:(3/2-59/52)/(3+3/2+59/52) 19/286 which works out to be about a 6.64% chance.

Based on this evidence, I would disagree with jeep's assertion that the cop is worth 3 townies.

[Phoebus]

Half tongue in cheek, half serious:

How many people read ALL of that ^^^^?

At the title, I jumped to see if there was a new post.

What is this? A game and fun?
OR math, on which winning and losing, and in turn, the future of the world, depends on?

[Internet Stranger]

I was going to add my opinion, but then you dorks had to pull out the fancy math.

[Mr. Flay]

Hey, I'm usually the dork with the fancy math, but he lost *me*...

In the long run, mafia is about social interaction, not mathematical certainty (as much as it pains me to say that). We can make all of the logic trees and probability calculations we want, but in the end it comes down to what we believe our opponent will do.

Which is probably a good thing, since I'm a math hack by inclination, but a sociologist/criminologist by trade...

[mole]

Locus: I don't think you can make that kind of calculation and completely disregard what the cop does about all of this. The role is only worth anything because it has abilities that can hurt the mafia--your cop doesn't do anything at all, but still the mafia want to kill him.

What does "value" mean in a game? There hasn't yet been a scum group who's been able to make the choice between killing 3 townies or killing the cop, and while we do lynch sometimes based on the fear of forcing a more powerful role to claim, again this is between one townie (or one-shot vigilante, etc) and the cop, not three.

Let's look at things from the mod's perspective, since he's the only one in the position to act on such a judgment, and he doesn't have to worry about the social interaction--he wants the two sides to have an equal chance of winning in theory, so the more skilful or luckier side will win out. So we have a choice between adding a cop to his game, or adding three townies.

This will probably be way too simplistic (I doubt we'll get a constant value this way), but we could figure out the odds of the mafia winning a Newbie Game, then remove the cop and add townies until we get the same odds.

[Locus Cosecant]

My calculations should be correct if you grant my assumptions, but I will admit that a lack of a decent way to calculate the value of cops and doctors is apparent in evaluating the above post. Your plan for calculating the value is interesting, but it seems as though it would quickly become unworkable at larger gamesizes. I mean, we could cobble together some kind of combinatoric or Monte Carlo computerized odds calculator, but I would think even that would have some issues as the size of the game grew. Off the top of my head, it doesn't seem like it would run effectively in polynomial time. (You can always make a Monte Carlo fit polynomial time, but the results would just keep getting worse.)

[Narninian]

Players may change the 'ideal setup' drastically...

for example: I recently played on efnet - IRC mafiagame devoted to playing mafia - they often play with a 7 player setup

2 mafia
4 townies
1 cop

the town/mafia win rate is just about even. Granted, their method of play is a little different (votes are secret until the voting records are revealed at the end of the day) - but mafia wins at the same rate town does - so it seems quite balanced. If out newbie game is balanced as well(which adds a doc) - I dont think we can calculate a perfectly balanced game - but it depends on players..

[mith]

Huh. I don't check MD very much these days, but I happened to check today to start a thread somewhat along these lines.

I suspect, of course, that these sort of values are not linear, but anyway.

I have to go now, but I have a few ideas, and may take a stab at some simulation programming this weekend.