Here are all the percentages:
1) If the BG reveals himself, the Gs have
14% chance to whack the D (if Osman is a C)
2) if the BG reveals himself, the Gs have
12.5% to whack the D (if Osman is a G)
3) If the BG doesn't reveal himself, the Gs have
12.5% to whack the D (if Osman is C)
4) If the BG doesn't reveal himself, the Gs have a
11% chance to whack the D (if Osman is a G)
So,
the Gs chance to whack the D increases by 1.5% if the BG reveals himself
5) If the BG doesn't reveal himself, the Gs have
12.5 chance to waste a whack if Osman is a C, and
11% to waste a whack if Osman is a G.
6) If the BG reveals himself, the D has a
30% chance to investigate a G.
7) if the BG doesn't reveal himself, the D has a
27% chance to investigate a G.
What should prevail? The 3% better chance for the D to get it right, or the 1.5% risk to lose the D?
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One more thing...
What if the D investigates a G tonight?
What does he do? Go public and tell us?
Remember that the Gs will find out via the crooked cop who was investigated and the investigated G can claim to be the D and confuse us all. We might even lynch a C tomorrow.
I did this already as a G once or twice.
That's why we need the BG to go public. The D will tell him via PM who he'll investigate and if the investigated person turns out to be a G, the BG will tell us this information, protecting the D's identity during the process.
@piotrr Percentage wise it makes sense to investigate Osman, but I would prefer if the D simply follows his instinct. He should investigate who he finds most suspicious and not who we tell him to.