[EV] Vanilla Multiball

[LicketyQuickety]

Just posting here so I have it in my "view my posts" section.

[LicketyQuickety]

Town EV 1/2(2+m+w); Mafia EV w/(2+m+w) + 3/4(2+m+w) = (3+4w)/4(2+m+w); Werewolf EV (3+4m)/4(2+m+w)

That means if we were always going to lynch, m->0 and w->0, as expected.

I understand everything up until this point.. No idea where 1/2 and 3/4 come in...

[LicketyQuickety]

I think the confusion here is from me being lazy with brackets and order of operations.

In the 1:1:2 setup:

If scum is lynched first (which has probability of w/(2+m+w) or m/(2+m+w), where m and w are the likelihood that Mafia and Werewolf would offer themselves as the lynch), the other scum wins (EV 0 for town, EV 1 for that scum group).
If town is lynched first (which has a probability of 2/(2+m+w), where m and w are the likelihood that Mafia and Werewolf would offer themselves as the lynch), town's EV is 1/4 (it's a 1:1:1 Night with unknown alignments, and town only wins if Mafia and Werewolf shoot each other). Mafia and Werewolf each win 3/8 of the time (counting the both-shoot-town as a draw and EV 1/2). So town EV is 1/4 * 2/(2+m+w) = 1/(2(2+m+w)), since this is the only way they can win.

Mafia EV is 1 * w/(2+m+w) + 3/8 * 2/(2+m+w), with the first part coming from Werewolf lynch and the second from town lynch, this reduces to (3+4w)/(4(2+m+w)). Similarly for the Werewolf EV.

Thanks for explaining that.