Live Action Mafia

Jamb poked me to create a forum to post my thoughts on game during it, so people can see them later. I'll post a bit of what I remember from creating it.

There are essentially five ways for town to get information. If I've balanced it right, the town should need to use all the non-psychological information well to win. This is a massive post, mostly so I can refer into it in later posts.

In order from least to most obvious:

1. Alibis.
1A. If a mafia key person is killed, they won't report who did it, and alibis are relevant to narrow down possible killers.
1B. This is how the game is "live action" and not just a series of mechanical actions done in person.
1C. I expect town to ignore this source of information for a while, focusing on the more mechanical information. The sooner the town realizes it, the better for them.
1D. At some point, someone "pointed out" that alibis don't matter because the dead people can just say who killed them in the next loop. I gloated internally and made some lame reply like "maybe catching a killer would save you a loop" or "oh, right" to avoid hinting at this source of information.
1E. If there are two mafia killers alive, there's very little information to be gained here, because even an alibi doesn't prove someone not a mafia killer. I'll call this a "weak use for alibis".
1F. If there's only one mafia killer alive, then anyone who's alibi'd correctly is proven not the other mafia killer. I'll call this a "strong use for alibis". This is likely to come up; see section 5B3b.
1G. If the mafia key person is suspected enough, the town can follow them around watching for people killing them and reporting immediately if they die. I'll call this a "likely use for alibis".

2. Psychological information.
2A. I don't know how to measure this. I'm assuming it'll help very little.

3. Goodwill voting.
3A. There's a tradeoff between this and psychiatrist information, in that the best schemes to out mafia by goodwill voting give slightly less goodwill (in expectation) to the psychiatrists.
3B. Goodwill voting easily reveals the alignment of truly claimed conspiracy theorists and truly claimed double agents, because the swing between the vote of a mafia conspiracy theorist and a town conspiracy theorist is 2, enough to swing from losing a goodwill vote without any tiebreakers to winning one without any tiebreakers.
3C. Goodwill attempts to reveal the alignment of anyone else are necessarily vulnerable to mafia meddling: the swing between the vote of a mafia and town is only 1, so such a scheme needs to arrange to either have a tie if the suspect is mafia or have a tie if the suspect is town. An outed mafia who knows about the whole plan can cast their vote tiebreaking vote in such a way that the result is the same whether the suspect is mafia or town.
3C1. If there are as many outed mafia as people whose alignment is still uncertain, the outed mafia can keep any more information from escaping any deterministic goodwill voting system they know about.
3C1a. A soft upper bound on the amount of information that comes out of goodwill voting is 4 outed mafia and a 4-way beef for the other two.
3C1b. When there are two or three outed mafia (whether from goodwill voting or anything else), goodwill voting should prove about a townie a day, by my rough attempts to come up with a scheme.
3C2. Randomness or limited communication in goodwill voting makes it hard for mafia to meddle, at the cost of reducing the chance of finding real information.
3C3. The town can infer something from what parts of the scheme outed mafia try to meddle with.
3D. I have less idea how much information the town can get from goodwill voting than from the rest of these things.

4. Psychiatrist questions.
4A. Psychiatrist questions are the most likely way to find out information about incidents, because mafia are likely to win by killers in the first loops.
4B. There's a tradeoff between information from psychiatrist questions and information from goodwill voting, because the best goodwill voting schemes aren't guaranteed to give goodwill to psychiatrists.
4C. The more mafia are claimed psychiatrists, the less the town can trust its psychiatrist information.
4D. The town has just enough goodwill to let every true psychiatrist ask a question, so if the town gets all the psychiatrists correct and isn't attempting to win, the psychiatrists can all have something to do.
4E. Town conspiracy theorists can make psychiatrists slightly more useful, at the cost of possibly losing loops.
4E1. To use a current town proposal as an example: if the town triggers the day 1 incident right away without even having any revealed killers, their psychiatrist questions become pretty useful for finding the days 3 and 4 culprits. This decreases by quite a bit the amount of mechanical information they'd get from the mafia that loop; see section 5.
4F. As the amount of goodwill increases, the amount of information a psychiatrist question can reveal about roles and alignments grows quickly, and the amount of information they can reveal about culprits grows only linearly---catch a culprit by asking about the right person.
4F1. Goodwill 1 questions can't possibly prove anything about non-culprits' alignments or roles except that there's at least two of each.
4F2. Goodwill 2 questions can check whether someone is a key person or double agent (the roles of which there are only two): for instance, "double agent and (playername)" is too big unlessss (playername) is a double agent. I don't think they can prove anything else about non-culprits' alignments or roles.
4F3. Two goodwill 3 questions can test an alignment: "((playername) and mafia) OR killer" is too big iff (playername) is a non-killer mafia; "((playername) and town) OR killer" is too big iff (playername) is a non-killer town. (You can't test easily test the alignment of someone who's a claimed killer with goodwill 3 questions, but that's unlikely to matter.) That is, a goodwill 3 question gives about half a bit of information.
4F4. Any question of goodwill 4 or more can give close to the one bit of role information about non-culprits that's the theoretical maximum.
4G. The earlier the mafia end a loop, the less information gets out from psychiatrists.
4G1. If the mafia end a loop on day 2, the town only gets up to one bit's worth of psychiatrist information (from two goodwill 3 questions or a goodwill 4 question, from two banana breads), assuming they pick the right psychiatrists.
4G2. If the mafia end a loop on day 3, the town gets up to 5 bits' worth of psychiatrist information (from three goodwill 4 questions and four goodwill 3 questions), assuming they pick the right psychiatrists.
4G3. The rapid rise in useful psychiatrist information makes it really important for the mafia to end at least the early loops early.

5. Mechanical actions.
5A. The town can get only a little information from mechanical actions.
5A1. All role claims except psychiatrist can be checked.
5A2. Conspiracy theorists and double agents can get a bit of information about culprits by putting paranoia on people, at the cost of probably losing a loop whenever they do.
5A3. Conspiracy theorists and double agents can maybe work with psychiatrists to get a bit more information from psychiatrist questions, but this is risky.
5B. The mafia need to win, and this requires some mechanical actions.
5B1. The longer the loop, the less information they have to reveal (but the more time the town has to get information from other sources).
5B2. No mafia win on day one is possible without town help.
5B2a. If there were a mafia win on day one without town help, the mafia could do it (possibly racing), and the town couldn't have a provable win.
5B2b. The only at all reasonable way for a one-day loop to happen is if the town killer kills a townie and a mafia killer kills the town double agent. This could even reasonably come up: see 5B3b2v.
5B3. The mafia can win a two-day loop at a cost comparable to outing a killer.
5B3a. A hidden killer can win a loop at the cost of putting the killer in the worse end of a beef (henceforth "outing"), by killing the town key person after the mafia double agent unprotects them.
5B3b. There's a whole family of scenarios by which a mafia key person and a hidden killer can win a loop after deactivating the town double agent. These all put the mafia key person in the better end of a beef and either give the town either a strong use for alibis (see the alibis section) or require that the town not have figured out the day 1 culprit.
5B3b1. A hidden conspiracy theorist adds paranoia to the day 1 culprit and the town double agent, outing themself and giving some information about the day 1 culprit. The hidden killer kills the mafia key person and the mafia blames someone else. This outs a conspiracy theorist and puts the mafia key person on the better end of a beef.
5B3b2. If there are two outed conspiracy theorists, two hidden killers, and the day 1 culprit isn't outed, the conspiracy theorist who isn't killed by the town killer adds paranoia to the day 1 culprit and the town double agent. The hidden killer kills the mafia key person and the mafia blames someone else.
5B3b2i. The town gets no more information than a weak use for alibis and a key person on the better end of a beef.
5B3b2ii. The town deserves a low-information loss because both conspiracy theorists are already outed but the town still hasn't figured out the day 1 culprit.
5B3b2iii. If the mafia trigger the day 1 incident before the town killer kills the day 1 culprit, then they get the same low-information win. Jamb discovered this and I consider it game-breaking for the mafia to be able to win a loop at the cost of only a weak use for alibis. This is why the rules changed to give consensual actions a 9-hour mechanical advantage overnight.
5B3b2iv. If both conspiracy theorists add paranoia to the town double agent and the town double agent can't be banana breaded, then they get the same low-information win. Jamb discovered this and I consider it game-breaking for the mafia to be able to win a loop at the cost of only a weak use for alibis. This is why the rules changed to let double agents banana bread themselves.
5B3b2v. If the day 1 culprit is outed, the town killer can kill them before the incident triggers, and then even two conspiracy theorists and possibly a double agent putting paranoia on the town double agent aren't enough to deactivate them the next day. The mafia can either out a killer on the town double agent to win the day 1 execution vote or ignore this sequence of possibilities entirely.
5B3b3. An outed killer who wins a race to kill the town double agent before the town killer kills them and a hidden killer who kills the mafia key person on day 2 can win at the cost of a strong use for alibis.
5B4. The mafia should have no trouble winning in three or four days with relatively little mechanical information loss, as long as the town hasn't figured everything out.

To balance the game, I tried to calculate from two directions: by counting bits of information the town needs and can get, and by simulating possible loops and strategies. Here's the former:

To win, the town needs to figure out the following by the start of loop 6 (or possibly during it, if the game really comes down to the wire):
13 bits of alignment information (a little less than 15 because the number of mafia is almost certainly 5 or 6, and the role claims will make one or two people almost certainly town)
15 bits of culprit information (a little less than 4 bits per culprit because the culprits are distinct. Also, they technically don't need to know the day 1 culprit to win, but they practically do because knowing the day 1 culprit keeps the mafia from doing their best win, 5B3b2.)
3 bits of role information (which mafia the killers are).

I expect the 3 bits of role information to be trivial: probably the killers will out themselves as killers at the same time as they out themselves as mafia to win a loop, and even if they manage not to, psychiatrist questions aiming at alignments will probably out them.

The culprit information should be relatively easy too: the town will have more goodwill 1 and 2 questions than it knows what to do with. The goodwill 1 questions find the day 1 culprit; the goodwill 2 questions find the culprits for day 2. The mafia are likely to reveal the day 3 culprit for a win at some point, but if they don't, goodwill 2 questions of the form "(day 1 culprit) OR (person to check)" check people for day 3 culpritness and day 4 culpritness at the same time. And that's without any incidental information from goodwill 3+ questions (which should mostly go to the alignment hunt).

The 13 bits of alignment information are the tricky part, and the part I think I have the best handle on.
-The town gets up to 1 bit of information from one use of strong alibis (finding a mafia by that they're a killer). My estimate for a good town performance is 1 bit from two uses of strong alibis. (They could also get the role information this way: after the first killer is outed, narrowing down the 7 remaining psychiatrists to one killer.) (This is a particularly bottlenecky bit: if the town doesn't have it, the mafia can win in two days; if the town has it, the mafia probably needs 3.)
-Assuming the lengths of the loops are 2, 2, 2, 2, 3 (my best guess), the town gets up to 9 bits of psychiatrist information from those loops, as calculated in 4F. My estimate for a reasonably good town performance (after the town loses some bits to fake psychiatrists, double-checking, inefficiency, and spending questions on culprit information instead of alignment information) is 3 bits.
-The town gets up to 13 bits from psychological evidence, but my estimate is 0.
-Each loop that the mafia end on day 2 either reveals a mafia or gives a strong use for alibis. I expect two of the latter (counted above) and two of the former.
That's two revealed mafia, for a total of about 2 new bits.
-The town gets up to about 10 bits from goodwill voting schemes (see 3C1a). 4 of those bits are basically guaranteed (since there are 9 bits of entropy left after the claimed double agents and conspiracy theorist are checked). My estimate for a good town performance is about 7 bits.

The previous posts were really abstract. Another way of thinking about it is to count mafia wins:
-Each hidden killer is 1 win, by killing the town double agent and then a key person.
-The key person is worth 1-2 more wins: one win from getting killed and not naming the killer, maybe another if the town doesn't get suspicious enough or its act together enough to keep a close eye for who kills them the next time.
-The day 3 incident is worth about 0-1 wins: it'll probably either trigger or make the town waste an execution vote and lose on day 4, but either of those gives the town more psychiatrist questions than the mafia would like.
-The conspiracy theorists are worth about 1 more win between them, either by disabling the town double agent so the key person can die or triggering an incident.

Here's a sample set of loops:

Loop 1:
One conspiracy theorist outs themself on the day 1 culprit and and the town double agent to disable them. Hidden killer kills the mafia key person. One outed conspiracy theorist, one beef favoring the key person, but the first psychiatrist questions are about enough to resolve that against them.
Loop 2:
The other conspiracy theorist outs themself the same way, and a hidden killer kills the mafia key person. The mafia key person is thoroughly outed, but the mafia manage to make the kill cleanly enough that the killer isn't outed.
Loop 3: the killer who's close to being outed kills the town key person on day 2.
Loop 4: the other killer kills the town key person on day 2.
Loop 5: Both killers die on day 1 (to the town killer and the execution vote), but meanwhile the conspiracy theorists put enough paranoia on everyone to trigger every incident.
Loop 6: The town checks whether it's figured everything out.

Jamb: "How long did it take you to make the game?" -- About 20 hours?

The mafia have been planning about as well as I'd've expected. I think claiming conspiracy theorist was a mistake, since their alignment is checkable.

I think they're overly excited about their last-minute switch to killing the mafia key person, because they haven't thought at all about how much evidence alibis give. Also, even being on the better side of a beef with someone is a big deal; that someone will be quite insistent, and there'll be at least two psychiatrist-3 questions, each of which lets a new claimed psychiatrist join the beef.

The mafia are really excited about how little mechanical information they have to reveal to win loops, so they think they're going to win, but they don't realize the alibi information and aren't really thinking about how much they'll lose to goodwill voting and psychiatrists.

The town's really excited about how much information they can get from goodwill voting and psychiatrists, so they think they're going to win, but they don't realize just how constrained their correct game is, so they think they're going to win.

I'm happy that everyone thinks before the game starts that they're going to win.

I really want someone to point out how Troy can still be mafia. Lotta already made most of the points except that she hasn't realized Katie being a mafia kp who got a gun is possible. This would require aok and Jackie both being mafia, oh and I guess this is no longer possible because then the loop wouldn't end.

Ok yeah I no longer think this would stand up past the end of the night but this might be a valid argument up until the loop doesn't end. I haven't looked at claims made about goodwill last night to figure out who needs to be mafia for this to be consistent.

5B3b2iii. If the mafia trigger the day 1 incident before the town killer kills the day 1 culprit, then they get the same low-information win. Jamb discovered this and I consider it game-breaking for the mafia to be able to win a loop at the cost of only a weak use for alibis. This is why the rules changed to give consensual actions a 9-hour mechanical advantage overnight.
5B3b2iv. If both conspiracy theorists add paranoia to the town double agent and the town double agent can't be banana breaded, then they get the same low-information win.

I don't understand why this is game-breaking at all. As long as the double agent is alive, town gets at two checks on either alignment or killers per loop (from n=3 questions). Also, town gets a lot of goodwill information. Depending on the loop, town can get alignment info about either the mafia double agent and/or the mafia key person.

Also, Pesto, out of curiosity, did you consider the following balance things:
1. Mafia double agent can't hide at all. If they claim DA, then they get goodwill tested, and if they don't, their lynch vote can be traced back to some mafia on the day 2 lynch where the loop is ending. The only way to prevent this is by having a living, outed mafia, which isn't going to happen in early loops.
2. "Goodwill attempts to reveal the alignment of anyone else are necessarily vulnerable to mafia meddling" - Not necessarily relevant meddling. If you do the standard 4 groups getting a max of 3 votes and one person being voted for by the town double agent, all you need is for the double agent to vote for a non-public person, and mafia can't mess with things by adding themselves in as a tiebreaker. Also, you can easily replace the town double agent with two of any trusted non-conspiracy theorists you want to get the same result.
3. Having two people with beef is extremely exploitable. Consider the standard 4 groups with max 3, double agent as backup goodwill voting scheme. If they both vote for person X, and person Y votes for person X, then person X gets a full alignment check on person Y. You can continue doing this for goodwill voting as long as both people in the beef cooperate.
Note: See the L2D1 voting scheme for an example of 2 and 3 in practice, though it didn't end up being relevant.

If the information flow had been better balanced so that alibis were a more significant part of the town's information, then that would have been too much. As it was, it'd've been fine.

1. I knew that mafia double agents couldn't hide at all; that's "4 of those bits are basically guaranteed (since there are 9 bits of entropy left after the claimed double agents and conspiracy theorist are checked)."

2. Yes, I didn't think town could both get enough people participating in a goodwill voting scheme to make it worth it and not let the mafia know; I didn't realize that telling people their orders right before day end had the same effect. This is most of the difference between my pregame estimate of a loop 6 town win and my estimate for if I ran it again of a loop 5 town win.

3. I think this is a subcase of point 2. If the mafia knew who the town double agent was going to vote for and there were an outed mafia, they could vote with the town double agent and keep the town from getting any information.