Here’s a first cut at it:
Premise: The probability of someone claiming a particular “unusual” role is much greater than the probability of a random player having that role. This is because sanctions are taken against false claims. For the actual numbers we can examine this thread and the previous Werewolf thread, but I have not done so. I’ll suggest some numbers later; if anyone cares to, they can dig up the right numbers.
Let c be the probability that a role-claim is true. Then the probability that it is false is 1-c.
On investigation, the Beat Cop will “correctly” confirm a true role-claim 1/2 of the time. The other 1/2 of the time he will still “incorrectly” confirm it 1/2r of the time, where r is the number of roles. (He gets a “wrong” result but then picks the right role randomly out of* r* roles.) Multiply both of these by the probability of a true role-claim, c.
This gives us a total of c*/2 + c/2r * “true” confirmations. This simplifies to c(r+1)/2r.
The Beat Cop will “confirm” a false role-claim if these conditions are met: the claim is false (*1 - c), * the Cop misses his ID (1/2), and randomly picks the claimed role (1/r). This multiplies out to (1-c)/2r.
We can cancel the common factor *1/2r * from both these results, giving us a true positive: false positive ratio of c(r+1):(1-c).
Suppose there are 7 roles in the game. Then we have the following illustrative numbers:
If the stats show that one out of two role-claims tend to be true, the Cop’s check-out ratio will be 1/2(8):1/2, or 8:1.
If the stats show that only one out of four role-claims tend to be true, the Cop’s check-out ratio will be 1/4(8):3/4, or 8:3.
Conclusion: If we have confidence that a claim of Cop is believable and that, say, true role-claims represent at least a quarter of all role-claims, a Cop-backed role-claim is much more likely to be true than either a random Cop investigation or an unverified role-claim, and also much more likely to be true than not.
I suggest we therefore consider carefully the possibility that Aguecheek’s role-claim is correct, for his claimed results are reasonably likely, although his claim that Queuing is the SK must still be viewed as probably incorrect (for the reasons previously stated).
Btw I think the earlier formula needs a little tweaking, but it is still broadly correct. (It neglected the possibility that a Cop might correctly but accidentally pick a player’s actual role.)
Open to peer-review.
