Mafia can be described as an experiment in human psychology and mass hysteria, or as a game between informed minority and uninformed majority. Focus on a very restricted setting, Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2] showed that in the mafia game without detectives, if the civilians and mafias both adopt the optimal randomized strategy, then the two groups have comparable probabilities of winning exactly when the total player size is R and the mafia size is of order Sqrt(R). They also proposed a conjecture which stated that this phenomenon should be valid in a more extensive framework. In this paper, we first indicate that the main theorem given by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2] can not guarantee their conclusion, i.e., the two groups have comparable winning probabilities when the mafia size is of order Sqrt(R). Then we give a theorem which validates the correctness of their conclusion. In the last, by proving the conjecture proposed by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2], we generalize the phenomenon to a more extensive framework, of which the mafia game without detectives is only a special case.

https://arxiv.org/abs/0804.0071

tl;dr in a blue/nilla only game, with random kills and lynches, town and mafia will have equal chances of winning if mafia has square root of total population, and the original formula in Mossel doesn't state equal but rather proportional to the square root, it should be fairly obvious that sqrt(R) isn't a 50% mafia victory with the rather trivial example of sqrt(4) where victory is 100% for mafia.