In 2006, the computer scientists Braverman, Etesami and Mossel proved that without detectives and with perfect players the randomized strategy is optimal for both citizens and mafia. They showed that when there are a large number of players to give both groups similar probability of winning, the initial number of mafiosi

*m*need to be proportional to the square root of the total number of players

*P*, that is

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.[Note 3]