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.
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