Live Action Mafia

A game of sneakiness and paranoia
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 Post subject: Goodwill voting schemes
PostPosted: Sun Mar 27, 2016 3:55 am 
I feel we should have a list of them, and a discussion specifically about them. These are the ones so far:
-> means votes for. <-> means votes for each other

1: (ksedlar)

P1 <-> CT1
P2 <-> CT2
P3 <-> CT3
P4 <-> DA
everyone else: themself
Idea: For each of these 4 pair of people, the P either claims to have received goodwill from the CT or DA that voted for them, or creates a beef with whoever voted for them. This is essentially equivalent to each of the 4 Ps making alignment checks of their respective voters.
Additionally, the key people can claim a 5-way beef with all four of the CTs and DAs by claiming to have received goodwill, since if and only if any of the CTs or DAs are bad will a goodwill be given to someone other that a P, and KP is the next in the alphabetical tiebreak.
The main problem is that most of the Ps are probably bad, and having a mafia make an alignment check on a CT or DA doesn't give town much information.

2: (brunnerj)

CT -> CT -> CT -> DA -> K
everyone else: themself
Idea: same as ksedlar's, but has the advantage of creating a trust chain instead. It ends in a K since we think Troy is the most likely innocent person, since otherwise there cannot be more than 1 K in game. It could end in someone else if we found them more trusted.
The main problem is that it doesn't give any Ps goodwill, and we need to give goodwill to Ps to ask questions.

3: (aok)

everyone votes for a random P.
Idea: We get some information on various subsets of town since if any of the four people that randomly received the most goodwill votes doesn't claim to have received goodwill, then either they are mafia or someone who voted for them is.
aok can you explain more? I am not convinced that this scheme is nearly as good as ksedlar's right now.

I think schemes 1 and 2 should happen in some order on days 1 and 2 at this point.


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PostPosted: Sun Mar 27, 2016 1:00 pm 
Note that we can run scheme 1 multiple days in a row, permuting the psychiatrists. Given a world in which one CT is bad and half of the psychiatrists are bad, an innocent result 3 days in a row is incorrect with probability (1/3)*(4/8)*(3/7)*(2/6) = .0238. Additionally, note that a mafia psychiatrist who lies for their mafia CT buddy usually gets themself into a beef as well, which gives town even more info. There's also the n=2 psychiatrist question which checks whether the set of CTs as a whole is good or not. (I didn't stick aok in the math because I think he is much more likely to be innocent than a CT).

We need psychiatrists to be able to ask questions to find culprits, and we need to identify culprits to win the game. Getting a slightly quicker alignment check on the CTs/aok is not worth sacrificing psychiatrist questions, IMO.

Also, we can prevent shenanigans with mafia "randomizing" the voting matchups by having Troy pick them (plus whichever psychiatrist aok wants to banana bread). If Troy is not town, then it will be obvious very soon, especially due to the n=1 #killers psychiatrist question.


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PostPosted: Sun Mar 27, 2016 2:14 pm 
Scheme 1a: Modified version of scheme 1

P1 <-> CT1
P2 <-> CT2
P3 <-> CT3
P4 <- P6, P7, P8
P5 <-> DA
P4 -> P6
everyone else (KPs and K): themself

P1-4 get goodwill, unless there's a beef, in which case P5 gets a goodwill.

Instead of alignment checking aok, this scheme alignment checks if a group of 3 psychiatrists contains more than 2 mafia in it. If we estimate that 4/8 psychiatrists are mafia, then there is a 50% than any group of 3 psychiatrists contains at least 2 mafia. If we think aok is more than 50% likely to be town (I think he is way more likely than that due to the dumbness of 0 town DAs), then this alignment check is even more likely to produce a beef.

FURTHERMORE, it means that we are much less likely to lose out on a psychiatrist question. In scheme 1, any beef results in the loss of a psychiatrist question. Also, it is often not even obvious that a goodwill was lost during the process (we notice only if both KPs are town and one of them is the culprit for some day). Here, the 5th psychiatrist probably ends up with the goodwill, saving a psychiatrist question, and making it obvious that a beef existed.


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PostPosted: Sun Mar 27, 2016 2:49 pm 
Also, scheme 1a doesn't fall apart if aok gets shanked, which is likely. We can probably just sub in votes from psychiatrists 1-4.


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PostPosted: Sun Mar 27, 2016 7:09 pm 
I think I like scheme 1a most right now.
My analysis of it:
A goodwill is given to P1, P2, P3 if CT1, CT2, CT3 are good respectively.
A goodwill is given to P4 if (all 3 of {P6, P7, P8} are good) or (2 of {P6, P7, P8} are good and (at least one of {P4, CT1, CT2, CT3, DA} is bad))
A goodwill is given to P5 if DA is good and ((at least 1 of {P6, P7, P8} are bad and P4 is good) or (at least 2 of {P6, P7, P8} are bad) or 1 of {CT1, CT2, CT3} are bad))
A goodwill is given to KP if KP is good and ((at least 3 of the following: CT1 is bad; CT2 is bad; CT3 is bad; DA is bad; at least 2 of {P6, P7, P8} are bad) or ((at exactly 2 of the following: CT1 is bad; CT2 is bad; CT3 is bad; DA is bad; at least 2 of {P6, P7, P8} are bad) and (1 KP is bad or at least 1 KP is a culprit)))
A goodwill is given to K if K is good and (at least 4 of the following: CT1 is bad; CT2 is bad; CT3 is bad; DA is bad; at least 2 of {P6, P7, P8} are bad; KP1 is bad; KP2 is bad) or ((at exactly 2 of the following: CT1 is bad; CT2 is bad; CT3 is bad; DA is bad; at least 2 of {P6, P7, P8} are bad) and all KP are good and all KP are not culprits)
Also, letting Troy choose the assigning the labels P1-P8 to the psychiatrists and CT1-CT3 to the conspiracy theorists is a good idea.


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PostPosted: Sun Mar 27, 2016 8:00 pm 
Actually, I realized that in scheme 1 (and by extension 1a), the tiebreak goes towards giving goodwill to mafia before it goes towards people voted for by alphabetically early strings. So no goodwill can be received by town key people / killers.

I also realized that if one of the CTs or the DA is bad, then we have no idea whether P6-8 contain 1 mafia or 2+ mafia. This is because P4 would win a tiebreak against any single-voted person by having more mafia votes. So our check on P6-8 only happens if all of CTs and DA are good, which is unlikely. :( This is still strictly better than scheme 1 though, because it means that we don't lose out on a psychiatrist getting a goodwill if a beef happens.


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PostPosted: Sun Mar 27, 2016 8:31 pm 
Scheme 1b: Modified version of scheme 1a. Vote counts have been added for clarity, but they are the same as before. A vote of "n+" indicates that there are n votes, but that the vote wins tiebreaks.

P1 <- CT1 (3)
P2 <- CT2 (3)
P3 <- CT3 (3)
P4 <- P6, P7, P8 (3 if all good, 2+ if 1 bad, 1+ if 2 bad)
P5 <- P1, P2, P3 (3 if all good, 2+ if 1 bad, 1+ if 2 bad)
P6 <- DA (2)*
P4,P5 -> P6, P8
everyone else (KPs and K) ->: themself

This scheme is even better at not losing out on any goodwill for psychiatrists (it's only lost if at least 3 beefs happen, which seems very unlikely). It is also roughly as good at innocence testing the CTs as schemes 1 and 1a (the only thing that would make the tests not work would be if ALL of P1-3 AND P6-8 were town, which we think is highly unlikely).

I'm trying to think of a good way to explain probabilities and failing right now, but I think this scheme works better as an alignment check on groups of psychiatrists if we expect that at least one of the CTs is mafia, and scheme 1a works better if we think the CTs are all good.

* Can replace with P4, P5 if DA is dead.


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PostPosted: Mon Mar 28, 2016 9:42 pm 
Sorry, type: P4 should vote for P7.


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PostPosted: Tue Mar 29, 2016 1:05 am 
I think scheme 1b was a resounding success. For reference, we ended up using a modified version that looks like this:

P1 <- CT1 (3)
P2 <- CT2 (3)
P3 <- CT3 (3)
P4 <- P4, P5, P6 (3 if all good, 2+ if 1 bad, 1+ if 2 bad)
P5 <- P1, P2, P3 (3 if all good, 2+ if 1 bad, 1+ if 2 bad)
P6 <- DA (2)
everyone else (P7, P8, KPs and K) ->: themself

I suggest that on future votes, we keep doing the same thing, switching out specific P and CT identities, as well as permuting the location of P1-6 in the groups. I think letting two psychiatrists vote for themselves is much better, because it allows us to effectively deal with people being unavailable near day end. I had previously underestimated the issues that could be caused by inactivity.


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