[OLD] Puzzle: Claiming Strategy

[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.

[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.

[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.

[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.

[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.