gritch said:
So basically the most specific a question the Spy asks the more likely they are to get information but the less useful that information normally is.
Yes, but going by the metric of number of suspects reduced, asking balanced question reduces the number of suspects the most, under the assumption that all suspects are equally suspicious and likely to be the killer. Continuing the previous example of 10 suspects:
If you ask about a trait only 10% of the users have the number of suspects is reduced by:
0.01 * 9 + 0.81 * 1 + 0.18 * 0 = 0.9 [footnote]The decimals are percentages of likelihood, so 0.01 is a 1% likelihood, and the whole numbers are measures of suspects reduced, so there is a 1% chance the number of suspects is reduced by 9, a 81% chance that the number of suspects is only reduced by 1 and a 18% chance that the number of suspects isn't reduced at all, leading to an average of 0.9 suspects less per question.[/footnote]
If you ask about a trait only 25% of the users have the number of suspects is reduced by:
0.0625 * 7.5 + 0.5625 * 2.5 + 0.375 * 0 = 1.875
If you ask about a trait 50% of the users have the number of suspects is reduced by:
0.5 * 5 + 0.5 * 0 = 2.5
So if we calculate this, we see that the most efficient manner of reducing the number of suspects is to ask questions with a 50/50 split.
Interestingly, the doppelgänger doesn't affect the decision-making process at all. So if we just look at the probabilities without taking the doppelgänger-effect into account.
If you ask about a trait only 10% of the users have the number of suspects is reduced by:
0.1 * 9 + 0.9 * 1 = 1.8
If you ask about a trait only 25% of the users have the number of suspects is reduced by:
0.1 * 9 + 0.9 * 1 = 3.75
If you ask about a trait 50% of the users have the number of suspects is reduced by:
0.5 * 5 + 0.5 * 5 = 5
As you can see, the Doppelgänger reduces the number of suspects the spy gets rid off by precisely 50%.[footnote]0.9*2=1.8, 1.875*2=3.75 and 2.5*2=5[/footnote] But this is quite intuitive, as he simply makes 50% of all questions ineffective.
What was rather unintuitive to me was the fact that this 50/50 split reduces the number of suspects more efficiently. My first theory was that the specificity of the spy's question wouldn't even have an impact, but after crunching the numbers, it doesn't look that way. And I'm pretty sure my math is correct. So my recommendation to spies would be to ask balanced questions, unless somebody finds a mistake here.