What if everyone played one game

[Bicephalous Bob]

Right, the faction of the new slot might change. It's balanced in the long run but not immediately in the short run. (Presumably the way it would work is that each player would be replaced once, but not necessarily immediately; you'd start out with a pool of fake players waiting to be replaced, when someone is lynched their faction is added to the pool and a random faction from the pool is taken to determine the next entrant's alignment.)

If the game is large enough you can mathematically guarantee that this never produces a scum majority, while making it impossible to predict the alignment of a new entrant unless there have been

n

lynches of the same alignment in a row (where

n

can be arbitrarily large).

how large does the game have to be for that?

[Bicephalous Bob]

and what would be the starting alignment distribution?