The mathematics of One Night Ultimate Werewolf

Suppose you like playing Werewolf, or "Mafia",  as this game is sometimes called. Is there a randomized optimal strategy you can follow if you are an innocent villager? When the total number of players is large, what is the size of the group of werewolves that guarantees comparable probabilities of winning for the werewolves and the villagers? How does the parity of the total number of players affect the game? We'll answer these questions based on the results of Braverman, Etesami, Mossel, Yao, and Midgal assuming the absence and then the presence of detectives.