[OLD] Puzzle: Claiming Strategy

[mith]

Consider the following hypothetical three-player setup:
1 Mafia
1 SuperCop
1 Townie
Pre-Game, the Cop is told who the Mafia is.

The Mafia would like to claim SuperCop with a result on the actual SuperCop, reducing the setup to a coinflip for the Townie. However, he doesn't know which is the SuperCop, so picking right is a coinflip as well if he is forced to claim without information. The town can in theory do this by having everyone point at the guilty party (with the SuperCop pointing at Mafia and the others pointing randomly) at exactly the same time. The EV in this case is 75% for the town (in order for scum to win, he has to guess right and the Townie has to guess wrong).

However, suppose the Mafia has incredible reflexes, and is able to see the others point and make his decision without being detected. (Like that RPS robot.) What is the EV now?

[F-16_Fighting_Falcon]

The Supercop will always point at the mafia so it reduces to whether the other townie will point at him or not:

a) If the townie points at him (50% chance), then he doesn't know who the supercop is and has to choose randomly. The EV in this case is 75% for the town (in order for scum to win, he has to guess right and the Townie has to guess wrong).
b) If the townie doesn't point at him, he knows who the supercop is and will point to them. Then the likelihood of the townie picking him and hence the EV is 50% for the town.

Overall, the EV for town would be 62.5%.

[chamber]

Best I can come up with at a glance is 7/12s town win.

[FakeGod]

Mafia sees either:

a) One player pointing at him or
B) Two players pointing at him

If a), then Mafia points at the guy pointing at him, and town EV is 1/2.
If b), then Mafia points at random, if mafia guesses wrong, he loses. If mafia guesses right, then town's EV is 1/2.

Final EV: 5/8 = 62.5%

[AngryPidgeon]

^ That sounds right. Barring cases where the actual cop lies for WIFOM in this stage, etc.

[F-16_Fighting_Falcon]

Suppose the actual cop lies (and everyone assumes he is telling the truth), the cop would be pointing at the townie:

a) If the townie points at the scum (50% chance), then the scum will think he knows who the supercop is (the townie) and fake-claim a guilty on that townie. The supercop will then choose correctly. That's a 100% EV for the town.
b) If the townie doesn't point at the scum and points at the supercop (50% chance), the EV in this case is 75% for the town (in order for scum to win, he has to guess right and the Townie has to guess wrong).

That gives town an 87.5% chance of winning. So optimal strategy for the supercop is to lie always.

[Scarab]

F-16, if it's optimal for the supercop to always lie, then scum should act as though the supercop will always lie, which reduces town win percentage. I'm working on generalizing that right now (i.e. if cop lies a certain percent of the time, what percent of the time should scum point at each person?).

[Scarab]

Wait, I just read your scenarios. If the supercop lies and points at the townie, why should the townie believe the supercop when he suddenly reveals he was lying? The supercop will just get himself lynched.

[Cheery Dog]

Assuming this is f2f (since the pointing) mafia wins due to the mod taking the pointing as votes. (unless both town players point at him)

[mith]

Obviously when I am modding F2F I require Vote Tags.



(The answer is higher than 62.5%.)

[Siveure DtTrikyp]

The mafia has incredible reflexes, so he's able to see the slight hesitation the townie has as to who to pick whereas the cop instantly picks the mafia, so he knows which is which, so he always picks the cop.

50% EV.

[mith]

EV assumes optimal play by all parties - including, for example, the Townie having already decided who he will randomly point at if he is the Townie before the game has started, or the Cop giving off slight hesitation tells, or both. The Mafia's only advantage in this puzzle is to decide who to point after knowing who the others are pointing at if they try to do the simultaneous pointing thing.

[FakeGod]

Do other players know about Mafia's extraordinary reflexes?

[Siveure DtTrikyp]

Yeah. I was joking.

So they don't do the simultaneous pointing, and there is

1/3 cop points at mafia - mafia points at him I think, 1/2 town win.
1/6 mafia points at cop - 1/2 town win.
1/6 mafia points at town - town wins
1/6 town points at mafia - mafia points at them, town wins.
1/6 town points at cop - mafia points elsewhere - 1/2

2/3 town win?

[Siveure DtTrikyp]

Uh, um, in my post they pick someone at random to go first, who then points, then they pick someone else.

[mith]

Do other players know about Mafia's extraordinary reflexes?

Yes.

[Siveure DtTrikyp]

Wait, i haven't thought that one through fully.

They pick someone first, then they pick the person the 1st one pointed at, and you get...

A) 1/3 cop first, always points at mafia, mafia points randomly, can't point back or other cases lose harder 3/4 EV
B) 1/6 town first who points at mafia, mafia points randomly, 3/4 EV
C) 1/6 town first who points at cop, mafia points back at cop, 1/2 EV.
D) 1/3 mafia first - 3/4 EV

So chances are 5/6 of 3/4 EV and 1/6 of 1/2 EV = 17/24 EV?

[Scarab]

I don't think they should have access to provable randomness once the game starts.

Also, I stopped working on figuring stuff out. I'll probably pick it back up tomorrow.

[FakeGod]

Do other players know about Mafia's extraordinary reflexes?

Yes.

Oh.

[FakeGod]

Then it's easy.

All three of them will write their name down and who they think the mafia is on a piece of paper, and reveal them at the same time. Reflexes negated.

EV is 75%.

[FakeGod]

Although I'm guessing that there's a more elegant solution mith is looking for.

[Cogito Ergo Sum]

We don't do anything fancy, just the pointing. So 50% the Townie guesses correctly initially and if he guesses right, Mafia has a 50% chance of guessing correctly.

25% the Townie points at the Mafia and the Mafia points at the Townie -> town win.
25% the Townie points at the Mafia and the Mafia points at the Supercop -> 2 people are pointing at the Mafia, 1 at the Supercop.
50% the Townie points at the Supercop and the Mafia points at the Supercop -> 2 people are pointing at the Supercop, 1 at the Mafia.

Thus, instead of having the Townie guess when faced with a successful counterclaim, you just lynch the guy the Townie didn't point at for a 75% EV.

[mith]

FakeGod, obviously they are locked in a room with no paper, or anything else but a carefully timed supply of oxygen (can't have our days running on too long). What kind of terrible mod do you think I am??

Anyway, CES has it. You can also do it without the "simultaneous" pointing, but it amounts to the same thing - the Townie has gained information in that the guy he's pointing at didn't point back, so there's a greater chance he's pointing at the SC. This negates the Mafia's advantage entirely (the only time the Mafia isn't forced to point randomly, it doesn't matter because he has already lost).

Next EV problem, which I haven't worked out the answer to yet:

1 Mafia
1 Cop <-- the regular old boring sane variety
2 Townies
Night Start.

Obviously, if the Mafia shoots the Cop, the EV is 33%

If the Mafia doesn't shoot the Cop, there's a 2/3 chance the Cop will have the identity of the Mafia (if he didn't investigate the victim). If this information were broadcast as part of the night results ("Townie X is dead; Cop did not investigate Townie X!"), EV would be 75% as above. There is a 1/3 chance the Cop did investigate the victim, in which case, if that information is also known, EV would be 50% (Mafia can claim whichever).

So the total EV is 1/3*1/3 + 2/3*(2/3*3/4 + 1/3*1/2) = 5/9.

What if the mod doesn't provide that extra Cop information, though? Can either side do better?

[kuribo]

The townie makes an offering to his dark master Satan in exchange for telepathy, he then joins the supercop in voting the mafia.

100% town win.

[mith]

That breaks the outside influences and outside communication rules. Modkill, town loses.