((All of this is ooc, in too much hurry to post this as note paper))
Ash, you either did not understand or you are deliberately trying to mislead the population here. Perhaps you have a hidden agenda?
The point is not about the absolute hit probability. If it were, it would be easy to attain 100% accuracy: just pass until all are mafia! The point is that there is one "extra" townie who brings absolutely no advantage whatsoever to the townie side, but helps the mafia side by decreasing the odds of hitting a mafia member with guesswork alone. Thus, while you managed to post "hey, this way we are getting higher numbers" in your carefully crafted examples, the end result is worse.
For those of you, who are still in doubt, I'll explain it in watertight way.
1) The game is about townies maintaining their majority.
Once it is gone (even a tie at start of a day round), the game is lost. Mafia's ability to win majority / block with a tie (or cause both sides to lose one player, depending on Winston's mystery rule) during day leaves just the mafia to do the killings.
2) Percentages are irrelevant. The number of people making up the difference is what counts.
If you have two-princess advantage at 101v99 (50.5% majority) you are equally close to losing as you'd be at 3v1 (75% majority), as a wrong guess in both cases would give the mafia the winning position.
3) There is no way for the townies to increase their majority during a full day-night cycle.
Even if townies manage to kill a mafia member during the day, the following night mafia gets one townie.
4) If townies pass the day, their lose 1 person from their majority during the day-night cycle.
Obvious death during night is obvious.
5) If townies kill one of themselves, they lose 2 persons from their majority during the day-night cycle.
This is obvious, as well.
6) If townies kill one of the mafia, they simply maintained their majority during that cycle.
However, this is the only way for the townies to win the game. They need to get this outcome n times, where n is the number of mafia members before they lose their majority.
7) The current difference between the two sides is even (multiple of 2).
8 left. Either 5v3, 6v2, 7v1. Differences are 2, 4 and 6. All even.
Now for the important part:
8) One townie is useless.
5v3 loses its majority with one bad voting round. So does 4v3 - not any sooner!
6v2 loses its majority with two bad voting rounds. So does 5v2 - not any sooner!
7v1 loses its majority with three bad voting rounds. So does 6v1 - not any sooner!
In each and every example above, the latter alternatives are better off. They have exactly the same amount of time remaining to make the correct guesses, but they all have better chances of making those guesses.
The game in nutshell is about making enough correct guesses before making too many wrong ones. It's not about making enough correct guesses as soon as possible.
Unfortunately I have to rush off now, so I can't bash the vile faulty "math" in ash's post any, but essentially what he is doing there to deceive (!) the good townsfolk is to compare apples and oranges. And the trump card argument is nullified by the good approach of voting whoever voted first - though this time I'd simply vote ash if I were alive.
Ash, you either did not understand or you are deliberately trying to mislead the population here. Perhaps you have a hidden agenda?
The point is not about the absolute hit probability. If it were, it would be easy to attain 100% accuracy: just pass until all are mafia! The point is that there is one "extra" townie who brings absolutely no advantage whatsoever to the townie side, but helps the mafia side by decreasing the odds of hitting a mafia member with guesswork alone. Thus, while you managed to post "hey, this way we are getting higher numbers" in your carefully crafted examples, the end result is worse.
For those of you, who are still in doubt, I'll explain it in watertight way.
1) The game is about townies maintaining their majority.
Once it is gone (even a tie at start of a day round), the game is lost. Mafia's ability to win majority / block with a tie (or cause both sides to lose one player, depending on Winston's mystery rule) during day leaves just the mafia to do the killings.
2) Percentages are irrelevant. The number of people making up the difference is what counts.
If you have two-princess advantage at 101v99 (50.5% majority) you are equally close to losing as you'd be at 3v1 (75% majority), as a wrong guess in both cases would give the mafia the winning position.
3) There is no way for the townies to increase their majority during a full day-night cycle.
Even if townies manage to kill a mafia member during the day, the following night mafia gets one townie.
4) If townies pass the day, their lose 1 person from their majority during the day-night cycle.
Obvious death during night is obvious.
5) If townies kill one of themselves, they lose 2 persons from their majority during the day-night cycle.
This is obvious, as well.
6) If townies kill one of the mafia, they simply maintained their majority during that cycle.
However, this is the only way for the townies to win the game. They need to get this outcome n times, where n is the number of mafia members before they lose their majority.
7) The current difference between the two sides is even (multiple of 2).
8 left. Either 5v3, 6v2, 7v1. Differences are 2, 4 and 6. All even.
Now for the important part:
8) One townie is useless.
5v3 loses its majority with one bad voting round. So does 4v3 - not any sooner!
6v2 loses its majority with two bad voting rounds. So does 5v2 - not any sooner!
7v1 loses its majority with three bad voting rounds. So does 6v1 - not any sooner!
In each and every example above, the latter alternatives are better off. They have exactly the same amount of time remaining to make the correct guesses, but they all have better chances of making those guesses.
The game in nutshell is about making enough correct guesses before making too many wrong ones. It's not about making enough correct guesses as soon as possible.
Unfortunately I have to rush off now, so I can't bash the vile faulty "math" in ash's post any, but essentially what he is doing there to deceive (!) the good townsfolk is to compare apples and oranges. And the trump card argument is nullified by the good approach of voting whoever voted first - though this time I'd simply vote ash if I were alive.