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In its simplest form, Mafia consists of two sides: the Mafia, and the Town. The Mafia's goal is to kill the Town members, while the Town's goal is to kill the Mafia.
At the beginning of the game, the Moderator will secretly send everyone their role. The Mafia will know each other, while (in the basic game) everyone else will only know their own role.
The game has two phases. Usually the first is Day. During the Day, the Town must choose someone to Lynch; they are trying to eliminate the Mafia, but the Mafia can lead them astray by casting suspicion elsewhere. Generally, players will vote for someone they think is Mafia (the Mafia will vote so that they seem to be doing this as well, and might even vote for each other to confuse the Town); when a player gets a majority of the votes, they are lynched. Their role is revealed by the Moderator, and it is once again Night.
During Night, the thread will be closed and the Town can not communicate, but the Mafia can discuss and choose someone to murder. After the Moderator has received the name of the victim, the Moderator reveals the dead player, and Day begins.
The game alternates between Night and Day until either the Mafia are eliminated, or the Mafia have control of the towns vote, by equalling the amount of live Town members in number.
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Hope that helps