[OLD] Puzzle: Claiming Strategy

[kuribo]

well played, mith


you've won this round




...for now

[Glass]

Without the extra cop information I believe that the EV with optimal play is still 5/9.

The strategy is to assume that if the cop lived it is essentially a supercop, and go from there.

1/3 chance of the cop dying, which leads to a 1/3 chance of town winning.
2/3 chance of the cop surviving, and 2/3 of those times he has a result on a non-dead person (and is thus essentially a super-cop) and 1/3 of the time he does not.

Probability of winning when cop actually has results: 3/4, from the previous problem.

We need to look at the odds of town winning when the ignorant cop is pointing randomly.

-If the not-super-cop points at the mafia, we again have a 3/4 chance of winning (because it is the same result as if the cop were a supercop).
-If the not-super-cop points at the VT, we need to look at what the VT randomly points at.
-If the VT points at the cop, there is a 1/2 chance that the mafia will point at the cop (0% chance of winning for town) and a 1/2 chance that the mafia will point at the VT (50% chance of winning for town)
-if the VT points at the mafia, the mafia has 2 possible strategies:
1. Vote for the person who points at him
2. Vote randomly

It is actually better for the mafia to vote randomly in this case (leads to 56.9% vs 5/9), so we have a 50% chance that mafia points at the VT -> (50% chance of town winning) and 50% that mafia points at the cop (0% chance of town winning).

Total:
1/3*1/3 + 2/3*(2/3*3/4 + 1/3*(1/2*3/4 + 1/2*(1/2*(1/2*1/2 + 1/2*0) + 1/2*(1/2*0 + 1/2*1/2))) = 5/9

[Cogito Ergo Sum]

Probability of winning when cop actually has results: 3/4, from the previous problem.

This isn't necessarily true any more, since the scum can claim ignorant cop now instead of having to counterclaim the supercop fully.

There's also the added complication that in a certain scenario, getting the scum to claim first is probably beneficial for the town.

[Glass]

The strategy I proposed requires town to lynch any ignorant cop claim if there is a supercop claim pointing at an ignorant cop.

But you are correct in that it actually may be better for mafia to claim VT in certain situations (I assume that is what you meant), which I neglected to look into. Further analysis needs to be done on that front.

Edit: Ya, there are definitely situations where it is better for the mafia to claim VT (for 1 example, when cop/VT pointed at mafia and mafia pointed at VT. Claiming cop is suicide but claiming VT allows for the chance that the cop is ignorant)

[Cogito Ergo Sum]

The strategy I proposed requires town to lynch any ignorant cop claim if there is a supercop claim pointing at an ignorant cop.

I don't see that anywhere?

And if that's true, how does the following hold:

-If the not-super-cop points at the mafia, we again have a 3/4 chance of winning (because it is the same result as if the cop were a supercop).

Surely, if supercop claims get believed over ignorant cop claims, the mafia has a 50% chance of winning in this scenario by claiming supercop. From a larger perspective, it's also not clear this situation is distinguishable for the town from the other scenarios (if ignorant cop points at mafia and mafia claims ignorant cop pointing at townie, the town has no way of knowing the made up result was correct).

[Glass]

Surely, if supercop claims get believed over ignorant cop claims, the mafia has a 50% chance of winning in this scenario by claiming supercop. From a larger perspective, it's also not clear this situation is distinguishable for the town from the other scenarios (if ignorant cop points at mafia and mafia claims ignorant cop pointing at townie, the town has no way of knowing the made up result was correct).

I guess I was unclear in my proposed strategy, so I will elaborate a bit:

1. Everyone points at somebody (supercop points at mafia, ignorant cop points at random)
2. Everyone claims cop/VT (mafia gets to see other claims before claiming)
3. All cop claims will claim super/ignorant
4. If a supercop is pointing at an ignorant cop, ignorant cop claim is lynched (this only will happen if supercop points at mafia and mafia points at VT, or if ignorant cop points at VT and mafia points at ignorant cop. The latter is far less likely)

Obviously if an ignorant cop is pointing at mafia and the mafia claims cop the real cop will just lie and say he is super. Again, I am aware that I have mistakes in my analysis because I haven't accounted for mafia claiming VT (which is optimal when the mafia is pointing at the VT).

[Cogito Ergo Sum]

Yeah, that is a lot of detail to leave out. I'd reconsider point 2 since town could benefit from arranging it so the scum claims first in certain scenarios (specifically, if there are two people pointing at him).

[Fishythefish]

That's a fun little problem. We're looking for 2/3*3/4 + 1/3*1/2 = 2/3. Town can achieve this, even if scum get the secret investigation info from the mod and town don't.

1) Everyone point, as before. Townies without info point randomly.

2a) Suppose the pointing is in a circle A -> B -> C -> A. All claim supercop or not, in any order. If one supercop (say A), we lynch B unless C is the cop, in which case we know A is lying and lynch him. If two (A, B), we'll know B is lying and lynch him. If none, we all claim and lynch randomly out of counterclaims.
b) Next suppose that A and B are pointing to C, and C to B; A -> B = C. Then B claims. If B claims supercop, we lynch C, unless C is a townie (because then we know B is lying again). If B claims ignorant cop or townie, everyone claims, and we lynch randomly out of counterclaims.

Now, as scum in your nanosecond of reflex time with your extra information, you see one of four things:

1) A->B->you, there's a supercop. You're dead. If you point at A, B's supercop information is enough to lynch you. If you point at B, we'll be in scenario 2b) above, and B's supercop claim will again lynch you.
2) A->B->you, no supercop. If you point at A, and claim supercop, you win with 50% - if and only if A is the cop. If you point at A, and don't claim supercop, it's 50% between you and your counterclaim. If you point at B, B isn't going to claim supercop, so we are in scenario 2b) and you have a 50% shot.
3) A->you<-B, supercop. You can point at A/B and claim supercop; in this case you have to guess the real cop. Or you can claim ignorant cop/townie; either way, you have a 50% shot.
4) A->you<-B, no supercop. Identical. 50%

So, you have a 50% shot if there's no supercop, and if there is you have to hope that the townie is right and you win a coinflip, which is 75%. That's exactly the situation in the "mod reveals the investigation result" scenario.

From the other side, the scum can always achieve 30% (less than the 33.3% he can get in the above) without using his reflexes. Before the vote, pick the dead townie with probability 20%, and each of living ones with probability 40%. In whatever crazy scheme the town choose, simulate a cop with the corresponding result (either ignorant or a guilty on the townie you picked). In the 40% of cases you pick the wrong living townie; you lose immediately. In the other 60%, you are indistinguishable from a real cop; 1/3 of the time you are ignorant, 2/3 of the time you are supercop with a guilty on the other cop. Hence the townie has no way of telling you apart from the cop, and you win exactly half the time, so 30% overall.

I can't see how to close that gap atm.

[kuribo]

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

WAIT


it's not outside influence or outside communication because i'd be INSIDE the mind of the other players which are part of the game assuming the players are sentient beings!

therefore, 100% win rate



check. and. mate, mr mith.

[Glass]

That's a fun little problem. We're looking for 2/3*3/4 + 1/3*1/2 = 2/3. Town can achieve this, even if scum get the secret investigation info from the mod and town don't.

1) Everyone point, as before. Townies without info point randomly.

2a) Suppose the pointing is in a circle A -> B -> C -> A. All claim supercop or not, in any order. If one supercop (say A), we lynch B unless C is the cop, in which case we know A is lying and lynch him. If two (A, B), we'll know B is lying and lynch him. If none, we all claim and lynch randomly out of counterclaims.
b) Next suppose that A and B are pointing to C, and C to B; A -> B = C. Then B claims. If B claims supercop, we lynch C, unless C is a townie (because then we know B is lying again). If B claims ignorant cop or townie, everyone claims, and we lynch randomly out of counterclaims.

Now, as scum in your nanosecond of reflex time with your extra information, you see one of four things:

1) A->B->you, there's a supercop. You're dead. If you point at A, B's supercop information is enough to lynch you. If you point at B, we'll be in scenario 2b) above, and B's supercop claim will again lynch you.
2) A->B->you, no supercop. If you point at A, and claim supercop, you win with 50% - if and only if A is the cop. If you point at A, and don't claim supercop, it's 50% between you and your counterclaim. If you point at B, B isn't going to claim supercop, so we are in scenario 2b) and you have a 50% shot.
3) A->you<-B, supercop. You can point at A/B and claim supercop; in this case you have to guess the real cop. Or you can claim ignorant cop/townie; either way, you have a 50% shot.
4) A->you<-B, no supercop. Identical. 50%

So, you have a 50% shot if there's no supercop, and if there is you have to hope that the townie is right and you win a coinflip, which is 75%. That's exactly the situation in the "mod reveals the investigation result" scenario.

From the other side, the scum can always achieve 30% (less than the 33.3% he can get in the above) without using his reflexes. Before the vote, pick the dead townie with probability 20%, and each of living ones with probability 40%. In whatever crazy scheme the town choose, simulate a cop with the corresponding result (either ignorant or a guilty on the townie you picked). In the 40% of cases you pick the wrong living townie; you lose immediately. In the other 60%, you are indistinguishable from a real cop; 1/3 of the time you are ignorant, 2/3 of the time you are supercop with a guilty on the other cop. Hence the townie has no way of telling you apart from the cop, and you win exactly half the time, so 30% overall.

I can't see how to close that gap atm.

You didnt account for the situation where the mafia sees:
5) A = B you

Which would lower the town win %. (also you forgot the part where there is a 1/3 of the cop dying at night, but that is a fairly minor issue).

[Fishythefish]

Oops - I had accounted for 5, but forgot to write it down. Thanks.

5) A=B you - no supercop. Doesn't matter who you point to, say B, so the diagram is as in townie strategy 2b). B claims ignorant cop or townie, and whatever you claim it comes down to a coinflip.

So it's still true that you get 50% if there's no supercop, 75% if there is. And yeah, I left out the cop dying at night deliberately - that's unrelated to the rest of the situation, and really just confounds the numbers.

[Glass]

Looks good, I do not see any other issues with it (in #3 mafia wouldn't be able to claim townie, but supercop/ignorant cop would both give 50%).

[mith]

Nice work, Fishy! I'd be really surprised if that gap couldn't be closed and town can do better somehow. I haven't had time to think about it though.

[Fishythefish]

I can't even decide which of the two EVs I think is right. Hopefully it's 70%, and there's a better town strategy - because I have no idea what else to do for a scum strategy.

[mith]

The most interesting result would be that they are both right, and the scum can only get it down to 2/3 by reflex cheating.

[AngryPidgeon]

Huh, I think this is actually comparable logically to the Monty Hall problem.

[mith]

Yup.

[Fishythefish]

The most interesting result would be that they are both right, and the scum can only get it down to 2/3 by reflex cheating.

Yes, that would be nice. And not at all implausible. I feel like the town should be able to rig up a simultaneous claim scheme where all the scum can do is take that 30%. But with reflex cheating, scum should surely be able to use something about the town's claims to do something realistic.

[mith]

Well, I've at least managed to show that the Mafia doesn't need extra information or cheaty reflexes to get to 2/3 in the simultaneous pointing plan outlined for the town by Fishy. After pointing, and seeing A->B<->C, Mafia claims SuperCop 80% of the time and NoResultCop 20% of the time (unless the Townie claims first from the B position, in which case he claims NoResultCop or Townie, knowing the Cop didn't get a result); other than the cases where he claims SC with a result on the Townie, the Townie is presented with a bunch of 50-50 paired cases.

The gap is closed because of cases h and i; since the Townie claims first in these cases, the Mafia can adjust his claim so that he's not immediately caught. I don't see a way for the Town to avoid having this happen some of the time, so I think I am satisfied the answer is 2/3.

Spoiler: tree

Code: Select all

I. A->B<->C
=== 1/6 a. T, SC, M
======= 4/5 i. SM->C ========== A
======= 1/5 ii. NM ============ B
=== 1/6 b. T, M, SC
======= 4/5 i. SM->C ========== A
======= 1/5 ii. NM ============ C
=== 1/6 c. SC, M, T
======= 4/5 i. SM->T ========== WIN
======= 1/5 ii. NM ============ D
=== 1/24 d. T, NC, M
======= 4/5 i. SM->C ========== C
======= 1/5 ii. NM ============ E
=== 1/24 e. T, M, NC
======= 4/5 i. SM->C ========== B
======= 1/5 ii. NM ============ E
=== 1/24 f. NC, M, T
======= 4/5 i. SM->T ========== WIN
======= 1/5 ii. NM ============ F
=== 1/24 g. M, NC, T
======= 4/5 i. SM->C ========== D
======= 1/5 ii. NM ============ F
=== 1/24 h. M, T, NC ========== G
=== 1/24 i. NC, T, M ========== G

II. A->B->C->A
=== 1/6 SC, M, T (EV 1)
=== 1/24 NC, M, T (EV 1/2)
=== 1/24 NC, T, M (EV 1/2)

[mith]

Five players:

A. 1 Mafia, 3 Townies, 1 Cop, Daystart.

D1:
1/5 Mafia (EV 1)
1/5 Cop (EV 1/2)
3/5 Townie (EV 5/9)

19/30


B. 1 Mafia, 3 Townies, 1 Cop known to have 1 Innocent Result, Daystart.

Cop Claims Immediately, EV 2/3

Cop Waits:

D1:
1/5 Mafia (EV 1)
2/5 Cop or Confirmed Townie (EV 2/3)
2/5 Townie
N1:
1/3 NK Cop (EV 1/3)
2/3 NK Townie (EV 3/4) <- guaranteed SuperCop this time
11/18
32/45 > 2/3

Six Players:

1 Mafia, 4 Townies, 1 Cop, Nightstart

1/5 Mafia kills Cop (EV 7/15)
4/25 Mafia kills Townie, Cop investigates Mafia (EV 1)
4/25 Mafia kills Townie, Cop investigates same Townie (A)
16/25 Mafia kills Townie, Cop investigates different Townie (B)

If it's known which case happened EV is 914/1125... If not, then from the Mafia's perspective, it's slightly more likely that if it gets down the three player scenario there's a SuperCop, but after going through the previous exercise I'm confident that the Mafia can just adjust their claiming strategy slightly to account for this and keep the EV the same regardless of whether it's known or not.

[Fishythefish]

I like the mafia reflexes pairing proof. So, that tells us that no town strategy starting "everyone point, and supercop points honestly" does better than 2/3 from town. I agree, 2/3 now seems likely.

I'd be fairly surprised if that worked for all larger numbers, though. It seems fairly coincidental that the mafia and town ignorance cancel out in this case.