[OLD] Puzzle: Claiming Strategy

[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

[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)

[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).

[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).

[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%).