Miss du Chat resigned herself to her fate, sitting alone on the lifeboat with the cat now willingly on her lap for warmth. She considered the others, wondering which of the three would regret their vote when they realised they'd lost and which were the two Mafiosi, who would be rejoicing. Her thoughts on interfering parrots that had not been around when needed, but added unnecessary comments from beyond the grave were not publishable.
As everyone has voted, I shall declare the result: Miss du Chat is duly lynched. She was the doctor.
Fletcher Christian and Rover prepare to go to sleep. As they look round them suspiciously they notice that Phereniki and Jonathon Fffffortyscue-Fyfffye are looking at the pair of them with an unfriendly sort of grin on their faces while sharpening that butterknife.
As there are equal numbers of innocents and mafia left and no doctor or other role remaining, I declare the Mafia the winners. Congratulations to not entirely me and Wet Kipper.
I was pretty certain that Phereniki/notentirelyme was Mafia, this round, once I knew Gwai was innocent, and seriously wondered about nominating her instead of Boogie. But if I was right and Boogie/Rover was innocent, she was voting to lynch me anyway, so we'd lost whatever I did.
I was very pleasantly surprised when you and Boogie nominated each other, which meant we didn't need to rush to defending either one of us, nor did we need to nominate an innocent in case we had been nominated.
we could then choose at our leisure who to vote for.
Well as soon as Wet Kipper followed my last posts with his nomination I was pretty sure he was guilty (though still thought it might be him and the cat)... Sorry Ck!
(Between root canal funs and other health things I couldn’t play as much as I’d have liked)
Yeah, I was pretty sure that Boogie was innocent from her posting - she'd also said so after one of the deaths. But not sure enough to say so outright. I was hoping she'd pick up from my kamikaze posting to vote with me. Plus the way Wet Kipper and notentirelyme reacted to me putting all the cards on the table labelled them as Mafia.
There's a bit of me wondering why I didn't support lynching Rover in the first round.
Now that the game is over, I guess I may chip in as a mathematician. This "we must lynch someone in the first round to improve the innocents' chances" really is BS.
Now that the game is over, I guess I may chip in as a mathematician. This "we must lynch someone in the first round to improve the innocents' chances" really is BS.
I don’t agree. The innocents can only win by lynching.
Yeah, but by lynching accurately - which is why I agreed to lynch both Lambie and Amy. I thought Lambie was acting very suspiciously and I'd wrongly guessed Fletcher was the detective and knew something, had checked Amy overnight to be certain, plus Amy's insistence on lynching, innocent or not, really does worsen the odds for the innocents. And it does the Mafia's job for them, Why should we help the Mafia?
These are the calculations I made (ignoring the doctor and the detective):
There are three scenarios from an innocent's point of view: Scen. 1: No lynching
Scen. 2: Lynching (turns out to be innocent)
Scen. 3: Lynching (turns out to be mafia)
The game starts with 8 innocents, 2 mafia. After the mafia lynches another innocent at night, we are left with:
If people do choose to lynch, there is 80% probability you'll end up in scen. 2, 20% that you'l end up in scen. 3. So, statistically you'll end up with 6.2 innocents and 1.8 mafia.
That's worse for the innocents than 7 innocents and 2 maffia, both in terms of difference (4.4 vs 5) as in terms of proportion (3.44 vs 3.5).
The argument for no lynching only works if you think there'll be more information tomorrow. So I guess you'd have to be arguing that the votes from day 1, or the detective's nightly check, would provide enough extra info to be worth giving up one of our chances at catching a mafia.
When I play this in real life, my usual group tends to have pretty much everyone have some kind of power role, and we play pretty logically — we don't do so much looking for tells in people's tone of voice, etc; rather it's going through the permutations of what could have possibly happened on each night. The tradeoff I'm used to is, is the extra information worth one dead innocent? Usually the answer is no — you still wouldn't be able to know anything for sure, so it's worth diving in with a guess.
Yes, I get the argument "let's lynch someone to get more information". And then of course, whether the high first-round chance of lynching an innocent (80%) is worth this information is a valid discussion.
But "we must lynch because it will statistically work out better for the innocents" is not.
If people do choose to lynch, there is 80% probability you'll end up in scen. 2, 20% that you'l end up in scen. 3. So, statistically you'll end up with 6.2 innocents and 1.8 mafia.
That's worse for the innocents than 7 innocents and 2 maffia, both in terms of difference (4.4 vs 5) as in terms of proportion (3.44 vs 3.5).
OK, so we don't lynch on day 1, and we're on 7 innocents 2 mafia.
This maths is faulty because, taken to its logical conclusion, the villagers do not win.
The following is faulty: "We must lynch someone because statistically after the first round, it is better for the innocents."
A valid argument could be: "We must lynch someone. Statistically, it will be worse for the innocents after the first round, but it will give us some information and I think that's worth it."
The argument made in the game felt very much like the former (and I suspect some players interpreted it this way).
This maths is faulty because, taken to its logical conclusion, the villagers do not win.
The following is faulty: "We must lynch someone because statistically after the first round, it is better for the innocents."
A valid argument could be: "We must lynch someone. Statistically, it will be worse for the innocents after the first round, but it will give us some information and I think that's worth it."
The argument made in the game felt very much like the former (and I suspect some players interpreted it this way).
I think the problem is the use of "better" and "worse".
I still think it is better for the innocents to lynch in the first round. That's not the same as saying statistically there will be more innocents alive after the first round, which (you're right) is not true…
I still think it is better for the innocents to lynch in the first round. That's not the same as saying statistically there will be more innocents alive after the first round, which (you're right) is not true…
Yeah, and I think it was interpreted in this way. Not to criticise you in the Circus, but you started with "The Honourable Miss Alton has been calculating the probabilities." I think that's what got other players on this train of thought.
PS After you made that post I was sure you were mafia. I was very surprised when you turned out to be innocent. It was a fun game to watch, thanks to you all.
I was going for "if we lynch, the first kill has a small chance of being a werewolf. If we don't lynch, the first kill has a zero chance of being a werewolf."
This particular method of playing the game online is also odd because of the amount of roleplaying we do. It was a bit hard to know how much characterisation I should do and how much that should affect logical argument
Having said that, I still have a gut reaction that voting for no lynching is a sign of werewolf-y tendencies. I've seen this in many real life games and I'll find it hard to let that one go.
But as the doctor, I was finding things out overnight. Having worked out that the odds were better if innocents stayed alive so was reluctant to lynch anyone I wasn't certain of, because doing the Mafia's job for them. I was also hoping that keeping as many people alive as long as possible would mean that the detective would also start identifying Mafia.
Amy's right. The town should always lynch in the early game. It may be true that the first lynching will, on average, make things worse for the innocents, but the more important truth is (as one of my IC personae put it) "This is a game we win by fucking killing people".
With 11 players the town can count on getting 4 opportunities to kill mafia, and each time the odds will be bad. This is a reason to take every one of those opportunities and not waste them. Neglecting to lynch on day one is basically offering to start the actual game one player down.
On LeRoc's model of 10 players, lynching on day 1 means:
Lynch 1, kill 1, 8 left;
Lynch 1, kill 1, 6 left;
Lynch 1, kill 1, 4 left;
Lynch 1, 3 left - mafia win unless at least one of the 4 lynchings was a hit.
The chance of all four lynchings missing (assuming lynchings are random, which of course they aren't) is 8/10 x 6/8 x 4/6 x 2/4 = 1 in 5. The town has a (notional) 80% chance* of not getting bowled for a duck.
Kill 1, 9 left;
Lynch 1, kill 1, 7 left;
Lynch 1, kill 1, 5 left;
Lynch 1, kill 1, 3 left - mafia win unless at least one of the 3 lynchings was a hit.
The chance of a mafia whitewash is now 7/9 x 5/7 x 3/5 = 1 in 3. That's a 66.6% chance for the town to take it to extra time, which is worse.
(*notional because reason and detective work shift the odds in favour of the town, and obfuscation, error, and the fact that the mafia are voting with full knowledge, not randomly, shift the odds against them.)
It's in the late game that the town should consider unorthodox strategies, because then the special roles (if they've lasted that far) are simultaneously more valuable and more vulnerable, and buying an extra night action might be worth forgoing a day action for. But it usually isn't.
Well played to the Mafia on this one. From an outsider's perspective, I was suspicious of CK at the outset, and when she hinted at being the doctor I was almost certain she was guilty. But after Gwai was eliminated, I thought FC had to be assumed innocent because the most likely reason for Gwai (the detective) to have nominated Chelley (who was innocent) after being so suspicious of Fletcher, was if she'd investigated Fletcher and cleared him. She'd obviously just cleared someone, because if she'd investigated a mafioso, she would have nominated one.
So when CK made her declaration also definitively clearing Fletcher I thought it very likely to be genuine - but I think the town has to trust uncontradicted role claims by default - the real doctor would have known if she was lying and should have said so. Boogie's nomination of CK was a mistake, but it would have been even more of a mistake for the mafia, and therefore one that they would be unlikely to make because they would have been certain that CK was telling the truth and that nominating her (and making the known innocents suspect them) was worse for them than making the innocent players guess which of the unknowns were guilty. Whereas Boogie didn't know for sure that CK was being truthful, but did know that she was herself, and the only way for Boogie to be sure that an innocent player had nominated anyone was for her to do it. So I thought Boogie the least likely of the unknowns to be guilty, but up until that point I couldn't have called it.
First night I missed playing because I suck. Second night I found Rover innocent. Third night I found out someone innocent who was also that night killed by the mafia. Then I confirmed that not always me was mafia. And died. By the way, the line about that explains a lot was supposed to be a hit because Dafyd said I could drop one.
Curious though about theories of who to analyze as the detective since we are meta-gaming.
I had fun but, being a complete Mafia novice, I had no idea what I was doing. So I bounced around much like Rover. I like to think I added a good measure of confusion to proceedings.
I was relieved to have been chosen as an innocent as I’d have had no idea what the job of Doctor etc would have been.
May I thank everyone for playing.
One has an entirely different sense of how the game is going when one knows everything that is happening (aside from any intra-mafia deliberations). After Curiosity saved Fletcher Christian and Gwai rejoined the game it seemed to me that the mafia victory was much more precarious than it must have looked to everyone in the game.
Missed the call notification. In the absence of information (either detective or gossip), with an independent method of chosing the lynchee. Then I think there is a strong call to no-lynch if the population is even and lynch if the population is odd.
Lynching wrongly on 2 Mafia-4 Villagers is just as fatal as 2-3 but has worse odds, but a no lynching with a murder gets you to the 2-3 game and a no lynch no murder is harmless.
While a 2/5 game does have worse odds of getting someone that turn but even a wrong lynching and murder still gets you to the 2/3 game. etc...
(and similarly for 1/2 1/3)
This of course is not the case (detective and gossip), however I'm yet to be convinced that there is an actually better strategy that drawing lots. Until the detective makes a strong claim (or dies having dropped some hints).
I had Fletcher pegged as the detective until Amy was lynched, and he was proved wrong in his assumptions. That's what made me think Amy was Mafia I was also working on Gwai as Mafia until she was murdered and revealed as the detective. Gwai obviously checked out Kookoosint - No_Prophet, as he was the person to die around then, and I was wondering about him, too.
When Gwai was killed I then realised not entirely me was Mafia, and was fairly sure that Wet Kipper was too, but wasn't 100% sure as Rover/Boogie wasn't being predictable.
I reckoned the Mafia thought I was being too useful to them and one round I really thought I'd be the next to be lynched - not entirely me did try for it, but no-one else picked up on it.
On LeRoc's model of 10 players, lynching on day 1 means:
Lynch 1, kill 1, 8 left;
Lynch 1, kill 1, 6 left;
Lynch 1, kill 1, 4 left; [Don't the mafia win here??? as in this game]
Lynch 1, 3 left - mafia win unless at least one of the 4 lynchings was a hit.
The chance of all four lynchings missing (assuming lynchings are random, which of course they aren't) is 8/10 x 6/8 x 4/6 x 2/4 = 1 in 5. The town has a (notional) 80% chance* of not getting bowled for a duck.
More thoughts with considering the detective and naive strategies. If the detective identifies all mafia with 2 turns to go the village ought to win. If the detective identifies the mafia with 1 turn to go then it's a contest of believability. If the mafia get the detective then it resolves to the earlier game.
After a number of turns, the initial Population is divided into Murdered, Lynched and Survivors.
The detective has inspected D of them (in addition a number have been Cleared by being murdered/lynched before inspection, and the detective can also win by identifying all the mafia or all the non mafia).
The chances of the mafia having got the detective are (M+L)/P.
While the odds of the detective having got a given mafia member are at least (D+C)/P, this is clearly between (M)/P and (2M+L)/P
With 30 players, after 5 murders the odds of a mafia victory* changes linearly from 17% to 33% as the lynchings changes from 0-5. While the odds of a detective against a single mafia victory* change from 17%~33% to 17%~50%. The odds of the detective winning before the mafia seems highest when the lynchings is zero**. But this ignores the endgame, and the finer details of how C behaves.
*this includes posthumous ones (which is why I've gone for a high number of players, so I can try to pretend the chances are low)
**Argument on the possibly dubious basis that the odds of a mafia win doubles as lynchings increases, while the detective one doesn't.
The problem is that it assumes that the mafia neither give themselves away or successfully manipulate things. Which could be forced for situations (e.g. in a live game by rolling dice and committing to it) but would be boring and potentially miss better strategies. It's still fun.
Talking of manipuating things, one of our tactics was to try and find scapegoats for what was going on, and make them look suspicious. The hounding of Miss Amy was of particular benefit to us but we also tried not to look for the most obvious person.
I think there was a lot of luck involved that we did so well, especially the way that all the innocents seemed so wary of each other, without too much nudging from us.
Lynching wrongly on 2 Mafia-4 Villagers is just as fatal as 2-3 but has worse odds,
True, but the power of the Mafia block vote increases as the population shrinks. On a 2-3 vote, assuming the Mafia don't help them (and why would they?) ALL of the innocents have to guess right, and agree with one another.
Under the rules in this game (with the tie-breaker provision) a 2-4 vote might be more likely to go the town's way - the agreement of any three of the four innocents could be enough.
Lynching wrongly on 2 Mafia-4 Villagers is just as fatal as 2-3 but has worse odds,
True, but the power of the Mafia block vote increases as the population shrinks. On a 2-3 vote, assuming the Mafia don't help them (and why would they?) ALL of the innocents have to guess right, and agree with one another.
Under the rules in this game (with the tie-breaker provision) a 2-4 vote might be more likely to go the town's way - the agreement of any three of the four innocents could be enough.
I know.
If you're officially (and cynically) voting at random anyway you can avoid that block vote by going for an agreed external source of randomness (e.g. lottery numbers) and then all innocents (and probably the mafia) vote in sync. So I think it's a useful null strategy.
Attempting to naively account for a block vote in the critical stage. In a mafia v innocent candidature (which is of course already dubious) and with the obvious votes. 2 bystanders have a 25% chance of both being lucky.
3 bystanders+casting have a 6.3% chance of all being lucky, and a 25% chance of only one being wrong, giving overall a 31% chance of survival (although the odds of getting two innocents on the scales are presumably higher).
Our null strategy had a 40% chance of success, so hopefully the bystanders shouldn't be purely guessing (they only need to be 64% certain, though). Also the final round in either thoughtful case better than a 50% chance (assuming the mafia didn't vote for his colleague) whereas the random method is 25% or 33%. So a controlled lynch in this case could show information. Although presumably then the mafia would presumably sometimes risk sacrificing their colleague.
Comments
I vote miss du chat
Fletcher Christian and Rover prepare to go to sleep. As they look round them suspiciously they notice that Phereniki and Jonathon Fffffortyscue-Fyfffye are looking at the pair of them with an unfriendly sort of grin on their faces while sharpening that butterknife.
As there are equal numbers of innocents and mafia left and no doctor or other role remaining, I declare the Mafia the winners. Congratulations to not entirely me and Wet Kipper.
we could then choose at our leisure who to vote for.
(Between root canal funs and other health things I couldn’t play as much as I’d have liked)
There's a bit of me wondering why I didn't support lynching Rover in the first round.
I don’t agree. The innocents can only win by lynching.
There are three scenarios from an innocent's point of view:
Scen. 1: No lynching
Scen. 2: Lynching (turns out to be innocent)
Scen. 3: Lynching (turns out to be mafia)
The game starts with 8 innocents, 2 mafia. After the mafia lynches another innocent at night, we are left with:
Scen. 1: 7 innocents, 2 mafia
Scen. 2: 6 innocents, 2 maffia
Scen. 3: 7 innocents, 1 maffia
If people do choose to lynch, there is 80% probability you'll end up in scen. 2, 20% that you'l end up in scen. 3. So, statistically you'll end up with 6.2 innocents and 1.8 mafia.
That's worse for the innocents than 7 innocents and 2 maffia, both in terms of difference (4.4 vs 5) as in terms of proportion (3.44 vs 3.5).
The argument for no lynching only works if you think there'll be more information tomorrow. So I guess you'd have to be arguing that the votes from day 1, or the detective's nightly check, would provide enough extra info to be worth giving up one of our chances at catching a mafia.
When I play this in real life, my usual group tends to have pretty much everyone have some kind of power role, and we play pretty logically — we don't do so much looking for tells in people's tone of voice, etc; rather it's going through the permutations of what could have possibly happened on each night. The tradeoff I'm used to is, is the extra information worth one dead innocent? Usually the answer is no — you still wouldn't be able to know anything for sure, so it's worth diving in with a guess.
But "we must lynch because it will statistically work out better for the innocents" is not.
OK, so we don't lynch on day 1, and we're on 7 innocents 2 mafia.
after lynching, we could end up with:
6 innocents, 2 mafia (78%)
7 innocents 1 mafia (22%).
then the mafia lynch another at night, so we're at
5 innocents 2 mafia (78%)
6 innocents 1 mafia (22%)
Giving us an expected value of 5.22. Which is worse than 6 innocents 2 mafia. So we don't lynch on day 2.
And thus the cycle continues. This maths is faulty because, taken to its logical conclusion, the villagers do not win.
A valid argument could be: "We must lynch someone. Statistically, it will be worse for the innocents after the first round, but it will give us some information and I think that's worth it."
The argument made in the game felt very much like the former (and I suspect some players interpreted it this way).
I think the problem is the use of "better" and "worse".
I still think it is better for the innocents to lynch in the first round. That's not the same as saying statistically there will be more innocents alive after the first round, which (you're right) is not true…
This particular method of playing the game online is also odd because of the amount of roleplaying we do. It was a bit hard to know how much characterisation I should do and how much that should affect logical argument
Having said that, I still have a gut reaction that voting for no lynching is a sign of werewolf-y tendencies. I've seen this in many real life games and I'll find it hard to let that one go.
I suppose it implies I'm quite good at it?
With 11 players the town can count on getting 4 opportunities to kill mafia, and each time the odds will be bad. This is a reason to take every one of those opportunities and not waste them. Neglecting to lynch on day one is basically offering to start the actual game one player down.
On LeRoc's model of 10 players, lynching on day 1 means:
Lynch 1, kill 1, 8 left;
Lynch 1, kill 1, 6 left;
Lynch 1, kill 1, 4 left;
Lynch 1, 3 left - mafia win unless at least one of the 4 lynchings was a hit.
The chance of all four lynchings missing (assuming lynchings are random, which of course they aren't) is 8/10 x 6/8 x 4/6 x 2/4 = 1 in 5. The town has a (notional) 80% chance* of not getting bowled for a duck.
Kill 1, 9 left;
Lynch 1, kill 1, 7 left;
Lynch 1, kill 1, 5 left;
Lynch 1, kill 1, 3 left - mafia win unless at least one of the 3 lynchings was a hit.
The chance of a mafia whitewash is now 7/9 x 5/7 x 3/5 = 1 in 3. That's a 66.6% chance for the town to take it to extra time, which is worse.
(*notional because reason and detective work shift the odds in favour of the town, and obfuscation, error, and the fact that the mafia are voting with full knowledge, not randomly, shift the odds against them.)
It's in the late game that the town should consider unorthodox strategies, because then the special roles (if they've lasted that far) are simultaneously more valuable and more vulnerable, and buying an extra night action might be worth forgoing a day action for. But it usually isn't.
Well played to the Mafia on this one. From an outsider's perspective, I was suspicious of CK at the outset, and when she hinted at being the doctor I was almost certain she was guilty. But after Gwai was eliminated, I thought FC had to be assumed innocent because the most likely reason for Gwai (the detective) to have nominated Chelley (who was innocent) after being so suspicious of Fletcher, was if she'd investigated Fletcher and cleared him. She'd obviously just cleared someone, because if she'd investigated a mafioso, she would have nominated one.
So when CK made her declaration also definitively clearing Fletcher I thought it very likely to be genuine - but I think the town has to trust uncontradicted role claims by default - the real doctor would have known if she was lying and should have said so. Boogie's nomination of CK was a mistake, but it would have been even more of a mistake for the mafia, and therefore one that they would be unlikely to make because they would have been certain that CK was telling the truth and that nominating her (and making the known innocents suspect them) was worse for them than making the innocent players guess which of the unknowns were guilty. Whereas Boogie didn't know for sure that CK was being truthful, but did know that she was herself, and the only way for Boogie to be sure that an innocent player had nominated anyone was for her to do it. So I thought Boogie the least likely of the unknowns to be guilty, but up until that point I couldn't have called it.
Good game folks.
Curious though about theories of who to analyze as the detective since we are meta-gaming.
I was relieved to have been chosen as an innocent as I’d have had no idea what the job of Doctor etc would have been.
One has an entirely different sense of how the game is going when one knows everything that is happening (aside from any intra-mafia deliberations). After Curiosity saved Fletcher Christian and Gwai rejoined the game it seemed to me that the mafia victory was much more precarious than it must have looked to everyone in the game.
I am interested in whether the mafia also had CK pegged as the doctor. I was pretty sure after she said that about knowing Fletcher was innocent.
Lynching wrongly on 2 Mafia-4 Villagers is just as fatal as 2-3 but has worse odds, but a no lynching with a murder gets you to the 2-3 game and a no lynch no murder is harmless.
While a 2/5 game does have worse odds of getting someone that turn but even a wrong lynching and murder still gets you to the 2/3 game. etc...
(and similarly for 1/2 1/3)
This of course is not the case (detective and gossip), however I'm yet to be convinced that there is an actually better strategy that drawing lots. Until the detective makes a strong claim (or dies having dropped some hints).
I had Fletcher pegged as the detective until Amy was lynched, and he was proved wrong in his assumptions. That's what made me think Amy was Mafia I was also working on Gwai as Mafia until she was murdered and revealed as the detective. Gwai obviously checked out Kookoosint - No_Prophet, as he was the person to die around then, and I was wondering about him, too.
When Gwai was killed I then realised not entirely me was Mafia, and was fairly sure that Wet Kipper was too, but wasn't 100% sure as Rover/Boogie wasn't being predictable.
I reckoned the Mafia thought I was being too useful to them and one round I really thought I'd be the next to be lynched - not entirely me did try for it, but no-one else picked up on it.
More thoughts with considering the detective and naive strategies. If the detective identifies all mafia with 2 turns to go the village ought to win. If the detective identifies the mafia with 1 turn to go then it's a contest of believability. If the mafia get the detective then it resolves to the earlier game.
After a number of turns, the initial Population is divided into Murdered, Lynched and Survivors.
The detective has inspected D of them (in addition a number have been Cleared by being murdered/lynched before inspection, and the detective can also win by identifying all the mafia or all the non mafia).
The chances of the mafia having got the detective are (M+L)/P.
While the odds of the detective having got a given mafia member are at least (D+C)/P, this is clearly between (M)/P and (2M+L)/P
With 30 players, after 5 murders the odds of a mafia victory* changes linearly from 17% to 33% as the lynchings changes from 0-5. While the odds of a detective against a single mafia victory* change from 17%~33% to 17%~50%. The odds of the detective winning before the mafia seems highest when the lynchings is zero**. But this ignores the endgame, and the finer details of how C behaves.
*this includes posthumous ones (which is why I've gone for a high number of players, so I can try to pretend the chances are low)
**Argument on the possibly dubious basis that the odds of a mafia win doubles as lynchings increases, while the detective one doesn't.
I think there was a lot of luck involved that we did so well, especially the way that all the innocents seemed so wary of each other, without too much nudging from us.
True, but the power of the Mafia block vote increases as the population shrinks. On a 2-3 vote, assuming the Mafia don't help them (and why would they?) ALL of the innocents have to guess right, and agree with one another.
Under the rules in this game (with the tie-breaker provision) a 2-4 vote might be more likely to go the town's way - the agreement of any three of the four innocents could be enough.
If you're officially (and cynically) voting at random anyway you can avoid that block vote by going for an agreed external source of randomness (e.g. lottery numbers) and then all innocents (and probably the mafia) vote in sync. So I think it's a useful null strategy.
Attempting to naively account for a block vote in the critical stage. In a mafia v innocent candidature (which is of course already dubious) and with the obvious votes. 2 bystanders have a 25% chance of both being lucky.
3 bystanders+casting have a 6.3% chance of all being lucky, and a 25% chance of only one being wrong, giving overall a 31% chance of survival (although the odds of getting two innocents on the scales are presumably higher).
Our null strategy had a 40% chance of success, so hopefully the bystanders shouldn't be purely guessing (they only need to be 64% certain, though). Also the final round in either thoughtful case better than a 50% chance (assuming the mafia didn't vote for his colleague) whereas the random method is 25% or 33%. So a controlled lynch in this case could show information. Although presumably then the mafia would presumably sometimes risk sacrificing their colleague.
[ETA no idea about singing]