For the purpose of pair investigations, one can assign each player a sketchiness rating such that the probability each player is guilty is proportional to their rating (assuming that exactly one player is guilty). For example, if X, Y, Z has rating 1, 2, 100, then the probabilities are 1/103, 2/103, 100/103 respectively.
Then, if a pair investigation on {A, B} returns A as innocent, then the effect is to set A’s rating to 0 and double B’s rating. e.g. if {Y,Z} returns Z as innocent in the above example, then the ratings become 1, 4, 0 and probabilities are 1/5, 4/5, 0. For actual application, since investigations can be faked, it might make more sense to change the multipliers from x0 and x2 to something like x0.25 and x1.75.
One important corollary: if there are N players and M pair investigations agains a certain player X on a kill, then assuming all players being equally sketchy a priori and there is exactly one person guilty of the kill, the probability that X is actually guilty is (2^M)/(2^M+(N1M)).
Useful pointers: when M=log_2(N), the probability is about 1/2, so before log_2(N) investigations each nonnegative investigation roughly doubles the sketchiness of that player and after that it’s diminishing return.
