,
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which is a good approximation when the right hand side is below 40%. If any detectives are added to the game, Braverman et al. proved that the number of Mafiosi must remain at a fixed proportion of the total number of players for their chance of winning to remain constant.
In 2008, Erlin Yao derived specific analytical bounds for the mafia's win probability when there are no detectives.
In a paper from 2010, the exact formula for the probability that the mafia wins was found. Moreover, it was shown that the parity of the initial number of players plays an important role. In particular, when the number of mafiosi is fixed and an odd player is added to the game (and ties are resolved by coin flips), the mafia-winning chance do not drop but rise by a factor of approx. <math> \sqrt{\pi/2}</math> (equality in the limit of the infinite number of players).
Results in live play
In live (or videoconference) real-time play, the innocents typically win more often than game theory suggests. Several reasons for this have been advanced:
- The physiological stress of sustained lying degrades the initial ability of mafioso to deceive the innocents, much more than a model of perfect play would predict, especially if the innocents can get the town emotionally involved in the game's outcome: